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The author discusses the theoretical background and demonstrates the \ practical application of the fractional calculus to the tautochrone problem \ formulated by Abel and a fractional relaxation problem in Weyl calculus." }], "Subtitle"], Cell[TextData[StyleBox["by Gerd Baumann", FontWeight->"Plain"]], "Author", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "The term ", StyleBox["fractional calculus", FontSlant->"Italic"], " is by no means new. It is a generalization of the ordinary \ differentiation by non-integer derivatives. The subject is as old as calculus \ itself and goes back to the time when Leibniz, Gauss, and Newton invented \ differentiation as a new tool. 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Born: 1 July 1646 in Leipzig, Died: 14 November 1716 in Hannover. \ Leibniz developed the present day notation for the differential and integral \ calculus. He never thought of the derivative as a limit. " }], "Caption", TextJustification->1], Cell[TextData[{ "Can the meaning of derivatives with integer order ", Cell[BoxData[ \(TraditionalForm\`\(\(d\^n\) \(y(x)\)\)\/dx\^n\)]], " be generalized to derivatives with non-integer order; with, in general, \ ", Cell[BoxData[ \(TraditionalForm\`n\)]], " being a complex number? The story goes that L`Hopital was somewhat \ curious about that question and replied with another question to Leibniz. \ What if ", Cell[BoxData[ \(TraditionalForm\`n\ = \ 1\/2\)]], "? In a letter dated September 30, 1695 Leibniz replied: ", StyleBox["Il y a de l'apparence qu'on tirera un jour des consequences bien \ utiles de ces paradoxes, car il n'y a guere de paradoxes sans \ utilit\[EAcute]. ", FontSlant->"Italic"], "The translation reads: \"It seems that useful consequences shall be one \ day drawn from these paradoxes, as there are no paradoxes that do not prove \ useful.\" " }], "Text"], Cell["\<\ The question raised by Leibniz for a fractional derivative has been \ an ongoing topic during the last 300 years. Many mathematicians (e.g. \ Liouville, Riemann, and Weyl) made major contributions to the theory of \ fractional calculus. \ \>", "Text"], Cell[TextData[{ "Let us consider Leibniz's question in connection with a special set of \ functions, say powers. In this case a fractional derivative is useful and can \ be expressed again by powers. For example let us consider the ", StyleBox["n", FontSlant->"Italic"], "th derivative of ", Cell[BoxData[ \(TraditionalForm\`x\^m\)]], ". We know that the general expression for the ", StyleBox["n", FontSlant->"Italic"], "th derivative is given by" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\(d\^n\) x\^m\)\/dx\^n = \ \(\(m!\)\/\(\((m - n)\)!\)\) \(\(x\^\(m - n\)\)\(.\)\)\)]]], "NumberedEquation"], Cell[TextData[{ "We also know that the factorial is connected with Euler's \ \[CapitalGamma]-function by ", Cell[BoxData[ \(TraditionalForm\`n != \[CapitalGamma](n + 1)\)]], ". Replacing the factorials in (1) with \[CapitalGamma]-functions, we are \ able to write" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\(d\^n\) x\^m\)\/dx\^n = \ \(\(\[CapitalGamma](m + 1)\)\/\(\[CapitalGamma](m - n + 1)\)\) x\^\(m - n\)\)]], "." }], "NumberedEquation"], Cell[TextData[{ "This representation is equivalent to equation (1). However, equation (2) \ contains the potential of a generalization. We know that the \ \[CapitalGamma]-function is defined for continuous arguments over the complex \ domain of numbers. If we change the integer value of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " to a number ", Cell[BoxData[ \(TraditionalForm\`q \[Element] \ \[DoubleStruckCapitalC]\)]], ", we are able to generalize an integer differentiation to a noninteger \ one. We are even able to define a complex differentiation. Replacing ", Cell[BoxData[ \(TraditionalForm\`n\)]], " by ", Cell[BoxData[ \(TraditionalForm\`q\)]], " in (2) results into the general expression" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\(d\^q\) x\^m\)\/dx\^q = \ \(\(\[CapitalGamma](m + 1)\)\/\(\[CapitalGamma](m - q + 1)\)\) x\^\(m - q\)\)]], "." }], "NumberedEquation"], Cell[TextData[{ "From the mathematical point of view, relation (3) has a well defined \ meaning. However, this meaning is restricted to the special class of powers ", Cell[BoxData[ \(TraditionalForm\`x\^m\)]], ". If we try to fractionally differentiate such simple functions in ", StyleBox["Mathematica", FontSlant->"Italic"], ", we end up with the following result " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_{x, 1/2}\ x\^2\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(D::"dvar" \( : \ \) "Multiple derivative in \!\({x, 1\/2}\) does not have the form {x, n}."\ \)], "Message"], Cell[BoxData[ \(\[PartialD]\_{x, 1\/2}x\^2\)], "Output", CellLabel->"Out[1]="] }, Open ]], Cell[TextData[{ "This means that ", StyleBox["Mathematica", FontSlant->"Italic"], " is not capable of dealing with fractional differentiation. However, the \ developers of ", StyleBox["Mathematica", FontSlant->"Italic"], " designed the system in a way that the user can extend the definition of \ derivatives. The extension of the derivative D[] will be our next task. \ Telling ", StyleBox["Mathematica", FontSlant->"Italic"], " that fractional derivatives of powers are useful mathematical constructs \ is realized by the following lines" }], "Text"], Cell[BoxData[ \(\(Unprotect[D]; \)\)], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \(D[x_\^m_. , {x_, q_}] := \(Gamma[m + 1]\/Gamma[m - q + 1]\) x\^\(m - q\) /; Head[q] == Real\ || \ Head[q] == Rational || Head[q] == Complex\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \(\(Protect[D]; \)\)], "Input", CellLabel->"In[59]:="], Cell["\<\ The definition of the fractional derivative of powers is based on \ equation (3) and restricts the order of differentiation to a rational, a \ real, or a complex number. A derivative for a negative rational number can \ now be calculated by\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_{x, \(-\(1\/2\)\)}\ x\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ \(\(4\ x\^\(3/2\)\)\/\(3\ \@\[Pi]\)\)], "Output", CellLabel->"Out[61]="] }, Open ]], Cell[TextData[{ "If we set the order of differentiation ", Cell[BoxData[ \(TraditionalForm\`q\)]], " to a real number, we find" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_{x, 2.1}\ x\^2\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(1.87155744182574519`\/x\^0.100000000000000088`\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ If we use complex numbers as the differentiation order, we get the \ result\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_{x, 11.5 + I}\ x\^4\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(\((\(\(57152.0744388545853`\)\(\[InvisibleSpace]\)\) - 143371.109440624255`\ I)\)\ x\^\(\(-7.5`\) - I\)\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[TextData[{ "This kind of formulas were first discussed by ", ButtonBox["Lacroix ", ButtonData:>"Lacroix-1819", ButtonStyle->"Hyperlink"], "in 1819 [", ButtonBox["1", ButtonData:>"Lacroix-1819", ButtonStyle->"Hyperlink"], "]. In retrospect, formula (3) is the first analytical answer on Leibniz's \ question on fractional derivatives. The story on the fractional calculus was \ continued with contributions from Fourier, Abel, Liouville, Riemann and Weyl. \ For a historical survey, the reader may consult the books of ", ButtonBox["Oldham and Spanier", ButtonData:>"Oldham-1974", ButtonStyle->"Hyperlink"], " [", ButtonBox["2", ButtonData:>"Oldham-1974", ButtonStyle->"Hyperlink"], "] or ", ButtonBox["Miller and Ross", ButtonData:>"Miller-1993", ButtonStyle->"Hyperlink"], " [", ButtonBox["3", ButtonData:>"Miller-1993", ButtonStyle->"Hyperlink"], "]. The historical developments culminated in two calculi which are based \ on the work of ", ButtonBox["Riemann", ButtonData:>"Riemann-1892", ButtonStyle->"Hyperlink"], " [", ButtonBox["4", ButtonData:>"Riemann-1892", ButtonStyle->"Hyperlink"], "] and ", ButtonBox["Liouville", ButtonData:>"Liouville-1832a", ButtonStyle->"Hyperlink"], " [", ButtonBox["5", ButtonData:>"Liouville-1832a", ButtonStyle->"Hyperlink"], "] on the one hand and on the work of ", ButtonBox["Weyl", ButtonData:>"Weyl-1917", ButtonStyle->"Hyperlink"], " [", ButtonBox["6", ButtonData:>"Weyl-1917", ButtonStyle->"Hyperlink"], "] on the other. The formulations are connected and Weyl's calculus (W) is \ a subset of the Riemann-Liouville (RL) calculus. In the following section, we \ will introduce the RL calculus. We will also outline the connection between \ the W and RL calculi. The application of the two calculi to two physical \ problems will demonstrate the usefulness of the methods." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". The Riemann-Liouville and Weyl calculi " }], "Section", CellTags->"The Riemann-Liouville and Weyl calculus"], Cell[TextData[ "In the above discussion, we motivated the fractional derivative by \ heuristics using properties of Euler's \[CapitalGamma]-function. Such a \ discussion is restricted to a special class of functions; i.e. powers. In \ this section we will motivate and introduce a calculus allowing the \ calculation of fractional derivatives for a more general class of functions. \ The results will be two calculi mainly conceived by Riemann and Liouville, \ and Weyl. Paradoxically the basis of the two calculi is not a derivative but \ an integral. However, we can understand integration as an inverse \ differentiation if we define the symbols "], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\(d\^\(-1\)\/dx\^\(-1\)\) \(f(x)\)\ := \ \[Integral]\_0\%x\( f(t)\) \[DifferentialD]t\)]]], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(d\^\(-2\)\/dx\^\(-2\)\) \(f(x)\)\ := \ \[Integral]\_0\%x \(\[Integral]\_0\%t\( f(s)\) \[DifferentialD]s \[DifferentialD]t\)\)]], " \[Ellipsis]" }], "NumberedEquation"], Cell[TextData[{ "The negative order of differentiation means nothing more than an \ integration. Higher orders of negative differentiation are calculated by \ nesting the integrals on the right hand side. We will abbreviate this kind of \ recursion by the symbol ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_\(0, x\)\%\(-n\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is a positive integer, and 0, ", Cell[BoxData[ \(TraditionalForm\`x\)]], " denote lower and upper boundaries of the integral. Thus equation (4) is \ reduced to" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(0, x\)\%\(-1\)\) \(f(x)\)\ = \ \ \[Integral]\_0\%x\( f(t)\) \[DifferentialD]t\)]], "." }], "NumberedEquation"], Cell[TextData[{ "The symbol ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_\(0, x\)\%\(-n\)\)]], " contains the complete information to carry out the calculation of the \ negative differentiation. The lower two indices denote the lower and upper \ boundaries on the integral. The superscript indicates the order of \ \"differentiation\". A generalization of the notation is an arbitrary \ starting point ", Cell[BoxData[ \(TraditionalForm\`a\)]], " at the lower boundary in the integral; meaning" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(a, x\)\%\(-1\)\) \(f(x)\)\ = \ \ \[Integral]\_a\%x\( f(t)\) \[DifferentialD]t\)]], "." }], "NumberedEquation"], Cell[TextData[{ "If we consider the ", StyleBox["n", FontSlant->"Italic"], "th negative derivative ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_\(a, x\)\%\(-n\)\)]], " of an arbitrary function ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ", we write" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(a, x\)\%\(-n\)\) \(f(x)\)\ = \ \[Integral]\_a\%x \(\[Integral]\_a\%x\_\(n - 1\)\(f(x\_0)\) \[DifferentialD]x\_\(n - 1\) \[Ellipsis] \[DifferentialD]x\_0\)\)]], "." }], "NumberedEquation"], Cell["Now remember Cauchy's integral formula ", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(d\^n\/dx\^n\) \(f( x)\) = \ \(\(n!\)\/\(2 \[Pi]i\)\) \(\[Integral]\_C\(\((\[Zeta] - z)\)\^\(\(-n\) - 1\)\) \(f(\[Zeta])\)\ \[DifferentialD]\[Zeta]\)\)]], "," }], "NumberedEquation"], Cell["and applying (9) to (8), we can reduce equation (8) to", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(a, x\)\%\(-n\)\) \(f(x)\)\ = \ \(1\/\(\((n - 1)\)!\)\) \(\[Integral]\_a\%x\(\((x - x\_0)\)\^\(n - 1\)\) \(f(x\_0)\) \[DifferentialD]x\_0\)\)]], "." }], "NumberedEquation"], Cell[TextData[{ "Introducing again the \[CapitalGamma]-functions for the factorials, we can \ generalize the result to an arbitrary order of fractional differentiation by \ ", Cell[BoxData[ \(TraditionalForm\`\(n!\) \[Rule] \[CapitalGamma](n + 1)\)]], ". The general formula for arbitrary negative derivatives reads" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalD]\_\(a, x\)\%\(-q\)\) \(f( x)\)\ = \ \(1\/\(\[CapitalGamma]( q)\)\) \(\[Integral]\_a\%x\(\((x - x\_0)\)\^\(q - 1\)\) \(f( x\_0)\) \[DifferentialD]x\_0\ \ \ \ \ \ \ \ \ \ \ \ \ q\) > 0\)]], "," }], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`q\)]], " is a fractional, real, or complex number. This kind of operator is known \ as the Riemann (R) version of the fractional integral [", ButtonBox["3]", ButtonData:>"Miller-1993", ButtonStyle->"Hyperlink"], ". The Liouville (L) version of this operator follows if we replace the \ lower boundary ", Cell[BoxData[ \(TraditionalForm\`a\)]], " in (11) by -\[Infinity]; i.e. ", Cell[BoxData[ \(TraditionalForm\`\(\(\[ScriptCapitalD]\_\(\(-\[Infinity]\), x\)\%\(-q\)\) \(f(x)\)\(\ \)\)\)]], " is the Liouville fractional integral. The case with ", Cell[BoxData[ \(TraditionalForm\`a = 0\)]] }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(0, x\)\%\(-q\)\) \(f(x)\)\ = \ \(1\/\(\[CapitalGamma](q)\)\) \(\[Integral]\_0\%x\(\((x - x\_0)\)\^\(q - 1\)\) \(f(x\_0)\) \[DifferentialD]x\_0\ \ \ \ \ \ \ \ \ \ \ \ \ q\) > 0\)]]], "NumberedEquation"], Cell["\<\ is called the Riemann-Liouville (RL) fractional integral. We observe that the \ various definitions of fractional integrals differ only in the lower boundary \ of the integral. You may think that this small difference is of minor \ importance. The following section will demonstrate that this assumption is \ not correct. The change of the lower boundary has far reaching consequences \ in the calculation of fractional derivatives. \ \>", "Text"], Cell[TextData[{ "So far we defined the notation of the fractional integral. A fractional \ derivative is connected with a fractional integral by introducing a positive \ order of differentiation in the operator ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_\(a, x\)\%\(-q\)\)]], ". This shift in the order of differentiation can be achieved by \ introducing an ordinary differentiation followed by a fractional integration. \ We thus define a fractional differentiation by" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\[ScriptCapitalD]\_\(a, x\)\%s\) \(f(x)\) := \ \((d\^n\/dx\^n)\) \(\[ScriptCapitalD]\_\(a, x\)\%\(-\((n - s)\)\)\) \(f(x)\)\)]], " with ", Cell[BoxData[ \(TraditionalForm\`n\ \[Element] \[DoubleStruckCapitalN], \ s > 0, \ n - s\ > 0\)]], "." }], "NumberedEquation"], Cell[TextData[{ "In this Riemann notation the fractional derivative depends on a lower \ boundary ", Cell[BoxData[ \(TraditionalForm\`a\)]], " of the integral. This dependence disappears if we consider only the RL \ operator with ", Cell[BoxData[ \(TraditionalForm\`a = 0\)]], ". Now we know in principle how general functions are treated in a \ fractional calculus. Before we apply the given notation to fractional \ differential equations, we state some general properties of the fractional \ derivative necessary to realize this operator in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ The main properties needed in an implementation are linearity and the \ composition rule. These are two basic properties besides the Leibniz rule of \ differentiation and the chain rule. From a practical point of view only the \ property of linearity and the composition of derivatives are needed. The \ first property of linearity reads\ \>", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\[ScriptCapitalD]\_\(a, x\)\%s\ \((\[Alpha]\ \(f(x)\) + \ \[Beta]\ \(g(x)\))\)\ = \ \[VeryThinSpace]\[Alpha]\ \(\[ScriptCapitalD]\_\(a, x\)\%s\) \(f(x)\) + \ \[Beta]\ \(\[ScriptCapitalD]\_\(a, x\)\%s\) \(g(x)\)\)]]], "NumberedEquation"], Cell[TextData[{ "with ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\)]], " real constants. The composition rule of fractional derivatives combines \ two derivatives with different orders ", Cell[BoxData[ \(TraditionalForm\`s\)]], " and ", Cell[BoxData[ \(TraditionalForm\`p\)]], " in the form" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\[ScriptCapitalD]\_\(a, x\)\%s\ \(\[ScriptCapitalD]\_\(a, x\)\%p\) \(f(x)\)\ = \ \(\[ScriptCapitalD]\_\(a, x\)\%\(s + p\)\) \(f(x)\)\)]]], "NumberedEquation"], Cell[TextData[{ "with ", Cell[BoxData[ \(TraditionalForm\`p < 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " finite at ", Cell[BoxData[ \(TraditionalForm\`x = a\)]], ". These properties are sufficient to define a function ", StyleBox["RiemannLiouville[]", "Input", FontWeight->"Plain"], " in ", StyleBox["Mathematica", FontSlant->"Italic"], ". The following lines contain the main steps of the definitions necessary \ to implement the RL operator." }], "Text"], Cell[BoxData[{ \(\n\n (*\ \(\(--\(-\ treatment\)\)\ of\ \(constants\ --\)\)\(-\)\ *) \n\ \(RiemannLiouville[f_, {x_, order_, a_: 0}] := f\ RiemannLiouville[1, {x, order, a}] /; FreeQ[f, x] && \ Not[SameQ[f, 1]];\)\n\t\n (*\ \(--\(-\ remove\)\)\ constants\ from\ the\ \(argument\ --\)\ *) \), "\n", \(\(RiemannLiouville[c_\ f_, {x_, order_, a_: 0}] := c\ RiemannLiouville[f, {x, order, a}]\ /; \ FreeQ[c, x];\)\n\n (*\ \(--\(-\ \(linerity\ --\)\)\)\(-\)\ *) \), "\n", \(RiemannLiouville[f_\ + \ g_, {x_, order_, a_: 0}] := RiemannLiouville[f, {x, order, a}]\ + \ RiemannLiouville[ g, {x, order, a}]\n\t\n (*\ \(--\(-\ \(composition\ --\)\)\)\(-\)\ *) \), "\n", \ \(RiemannLiouville[\ RiemannLiouville[f_, {x_, order1_, a_: 0}], {x_, order2_, a_: 0}] := RiemannLiouville[f, {x, order1 + order2, a}]\ /; order1 < 0\n\n (*\ \(--\(-\ \(Identity\ --\)\)\)\(-\)\ *) \), "\n", \(RiemannLiouville[f_, {x_, 0, a_: 0}] := f\n\n\n (*\(\(--\(-\ main\)\)\ \(function\ --\)\)\(-\)\ *) \), "\n", \(\(\(RiemannLiouville[ f_, {x_, order_, a_: 0}]\)\(:=\)\(Block[{n, int, $y1, q}, \n\t\tIf[ NumericQ[order] && Simplify[order > 0], n = Floor[order]; \ q = \ order - n, q = order; n = 0]; \n\t\tint\ = \ Integrate[\(\((x - $y1)\)\^\(\(-q\) - 1\)\) \((f /. x \[RuleDelayed] $y1)\), {$y1, a, x}, GenerateConditions \[Rule] False]; \n\t\tD[ int\ /\ Gamma[\(-q\)], {x, n}]\ /; \ FreeQ[int, $y1]\n\t\t]\)\(\n\)\(\t\t\t\)\)\)}], "Input", CellLabel->"In[1]:="], Cell[TextData[{ "The above program lines show that it is sufficient to use the properties \ and the definition of the RL operator given in equation s(11)-(15). The \ symbolic notation introduced in the discussion above can be generated in ", StyleBox["Mathematica", FontSlant->"Italic"], " by the utilities package ", StyleBox["Notation", "Input", FontWeight->"Plain"], "." }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[6]:="], Cell["\<\ We define the symbolic pattern on the left hand side and the function on the \ right hand side.\ \>", "Text"], Cell[BoxData[ RowBox[{"Notation", "[", RowBox[{ TagBox[\(\(\[ScriptCapitalD]\_\(a_, x_\)\%order_\) f_\), NotationBoxTag, Editable->True], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[\(RiemannLiouville[f_, {x_, order_, a_}]\), NotationBoxTag, Editable->True]}], "]"}]], "Input", CellLabel->"In[7]:="], Cell["A small palette allows us to generate the symbolic notion:", "Text"], Cell[BoxData[ ButtonBox[ \(\(\[ScriptCapitalD]\_\(\[Placeholder], \[Placeholder]\)\%\[Placeholder]\) \[Placeholder]\), ButtonStyle->None]], "Input"], Cell[TextData[{ "where the placeholders have the meaning given in the definition of the \ notation. Application of the symbol ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_\(0, x\)\%\(\(-1\)/2\)\)]], " to the power function ", Cell[BoxData[ \(TraditionalForm\`x\)]], " gives the same result as calculated with the Lacroix definition (3)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalD]\_\(0, x\)\%\(\(-1\)/2\)\) x\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(\(4\ x\^\(3/2\)\)\/\(3\ \@\[Pi]\)\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ "An example frequently discussed in the literature is the differentiation \ of a constant [", ButtonBox["4", ButtonData:>"Oldham-1974", ButtonStyle->"Hyperlink"], "]. From ordinary calculus we know that the differentiation of a constant \ vanishes. Surprisingly this is not the case in fractional calculus. Using the \ palette notation of the RL operator, we can write for a semi-differentiation" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalD]\_\(0, x\)\%\(1/2\)\) 1\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(1\/\(\@\[Pi]\ \@x\)\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ "This result contradicts the general knowledge that the differentiation of \ a constant vanishes. This behavior can be understood if we go back to the \ definition of a fractional derivative based on an integral. Another \ interesting example of a fractional derivative occurs if we replace the lower \ limit of integration by a constant, say ", Cell[BoxData[ \(TraditionalForm\`a\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalD]\_\(a, x\)\%\(1/2\)\) 1\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(1\/\(\@\[Pi]\ \@\(\(-a\) + x\)\)\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[TextData[{ "The result shows that the lower boundary in the integral has some \ influence on the result. We also see that the results of the RL operator and \ the Riemann representation are different. The result for the Liouville \ definition is only accessible if we consider the limit of ", Cell[BoxData[ \(TraditionalForm\`a \[Rule] \(-\[Infinity]\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\(\[ScriptCapitalD]\_\(a, x\)\%\(1/2\)\) 1, a \[Rule] \(-\[Infinity]\)]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["\<\ We observe that the result of the fractional differentiation mainly depends \ on the definition of the fractional operator. Another example demonstrating \ the exceptional behavior of the fractional derivative is given by\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalD]\_\(0, x\)\%q\) \((a + x)\)\^p\)], "Input", CellLabel->"In[65]:="], Cell[BoxData[ \(\(-\(\(a\^p\ x\^\(-q\)\ Hypergeometric2F1[1, \(-p\), 1 - q, \(-\(x\/a\)\)]\)\/\(q\ Gamma[\(-q\)]\)\)\)\)], "Output", CellLabel->"Out[65]="] }, Open ]], Cell[TextData[{ "The fractional differentiation of a general polynomial of order ", Cell[BoxData[ \(TraditionalForm\`p\)]], " results in a hypergeometric function ", Cell[BoxData[ \(TraditionalForm\`F\_\(2, 1\)\)]], ". So far, we have demonstrated some properties of the Riemann-Liouville \ calculus. Another fractional operator with a broad physical application is \ the fractional Weyl operator. Weyl defined the fractional derivative in a way \ similar to Riemann and Liouville [", ButtonBox["6", ButtonData:>"Weyl-1917", ButtonStyle->"Hyperlink"], "]. In 1917 ", ButtonBox["Weyl", ButtonData:>"Weyl-1917", ButtonStyle->"Hyperlink"], " stated his operator as " }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(\(\[ScriptCapitalW]\&+\)\_t\%\(-q\)\) \(f(t)\)\), "TraditionalForm"], FormBox[\(\ \( = \ \(1\/\(\[CapitalGamma](q)\)\) \[Integral]\_t\%\[Infinity]\(\((\[Zeta] - t)\)\^\(q - 1\)\) \(f(\[Zeta])\) \[DifferentialD]\[Zeta]\)\), "TraditionalForm"]}], TraditionalForm]]], " with ", Cell[BoxData[ \(TraditionalForm\`\(\ \ \ Re(q) > 0, \ t > 0\)\)]], "." }], "NumberedEquation"], Cell["\<\ This definition is related to the Liouville definition by the relation\ \>", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`\(\(\[ScriptCapitalW]\&+\)\_t\%\(-q\)\) \(f(t)\)\ = \ \ \(\((\(-1\))\)\^q\) \(\[ScriptCapitalD]\_\(\[Infinity], \ t\)\%\(-q\)\) \(f(t)\)\)]], "." }], "NumberedEquation"], Cell[TextData[{ "The Weyl and Liouville definitions differ in the change of limits in the \ integral and the factor ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^q\)]], ". Knowing this behavior, we can define the Weyl operator by" }], "Text"], Cell[BoxData[ \(WeylPlus[f_, {t_, q_}] := RiemannLiouville[\(\((\(-1\))\)\^q\) f, {t, q, \[Infinity]}]\)], "Input",\ CellLabel->"In[9]:="], Cell["The related template is defined by", "Text"], Cell[BoxData[ RowBox[{"Notation", "[", RowBox[{ TagBox[\(\(\[ScriptCapitalW]\&+\)\_t_\%q_[f_]\), NotationBoxTag, Editable->True], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[\(WeylPlus[f_, {t_, q_}]\), NotationBoxTag, Editable->True]}], "]"}]], "Input", CellLabel->"In[10]:="], Cell[BoxData[ ButtonBox[ \(\(\[ScriptCapitalW]\&+\)\_\[Placeholder]\%\[Placeholder][ \[Placeholder]]\), ButtonStyle->None]], "Input"], Cell[TextData[{ "Examples for some Weyl differentials and integrals follow. The first \ example is concerned with the ", StyleBox["q", FontSlant->"Italic"], "th differentiation of a negative power:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%q[x\^\(-2\)]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\(-\(\(\[Pi]\ \((1 + q)\)\ x\^\(\(-2\) - q\)\ Csc[\[Pi]\ q]\)\/Gamma[ \(-q\)]\)\)\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[TextData[{ "The second example deals with a general polynomial of ", StyleBox["p", FontSlant->"Italic"], "th order:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%q[\((x + a)\)\^p]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(\(\((a + x)\)\^\(p - q\)\ Gamma[\(-p\) + q]\)\/Gamma[\(-p\)]\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[TextData[{ "The Weyl fractional derivative is again a polynomial of order ", Cell[BoxData[ \(TraditionalForm\`p - q\)]], ". A semi-Weyl integration of the power ", Cell[BoxData[ \(TraditionalForm\`\(-1\)/2\)]], " of ", Cell[BoxData[ \(TraditionalForm\`1/x\)]], " results in" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\(\(-1\)/2\)[1\/x]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(\@\[Pi]\/\@x\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[TextData[{ "The general Weyl derivative of \[Nu]th order of ", Cell[BoxData[ \(TraditionalForm\`x\^\(-\[Mu]\)\)]], " gives us again a power relation of order -\[Mu]-\[Nu]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\[Nu][x\^\(-\[Mu]\)]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(\(x\^\(\(-\[Mu]\) - \[Nu]\)\ Gamma[\[Mu] + \[Nu]]\)\/Gamma[\[Mu]]\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[TextData[ "The fractional integral of Weyl for the same function results in a power \ behavior with -\[Mu]+\[Nu]"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\(-\[Nu]\)[x\^\(-\[Mu]\)]\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(\(x\^\(\(-\[Mu]\) + \[Nu]\)\ Gamma[\[Mu] - \[Nu]]\)\/Gamma[\[Mu]]\)], "Output", CellLabel->"Out[15]="] }, Open ]], Cell["\<\ Other examples that may be obtained by fractional Weyl integration are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\(-\[Nu]\)[ \(x\^\(\(-1\)/2\)\) Exp[\(-\[Alpha]\)\ x\^\(1/2\)]]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(\(2\^\(1\/2 + \[Nu]\)\ x\^\(1\/4\ \((\(-1\) + 2\ \[Nu])\)\)\ \[Alpha]\^\(1\/2 - \[Nu]\)\ BesselK[\(-\(1\/2\)\) + \[Nu], \@x\ \[Alpha]]\)\/\@\[Pi]\)], "Output",\ CellLabel->"Out[16]="] }, Open ]], Cell["\<\ containing the modified Bessel function of the second kind. More \ complicated integrals may be obtained by other functions in the argument, \ such as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\(-\[Nu]\)[ \(x\^\(\[Nu] - 1\)\) Sin[\[Alpha]\ x]]\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(\(1\/Gamma[\[Nu]]\(( 2\^\(\(-1\) - 2\ \[Nu]\)\ x\^\(\(-\(1\/2\)\) + \[Nu]\)\ \((\[Alpha]\^2)\)\^\(1\/4 - \[Nu]\/2\)\ \((2\^\(2\ \[Nu]\)\ \@\[Pi]\ BesselJ[1\/2\ \((1 - 2\ \[Nu])\), 1\/2\ x\ \[Alpha]\ Sign[\[Alpha]]]\ Cos[\(x\ \[Alpha]\)\/2]\ Gamma[\[Nu]]\ Sign[\[Alpha]] + 2\ BesselJ[1\/2\ \((\(-1\) + 2\ \[Nu])\), 1\/2\ x\ \[Alpha]\ Sign[\[Alpha]]]\ Gamma[1\/2 - \[Nu]]\ Gamma[2\ \[Nu]]\ Sin[\(x\ \[Alpha]\)\/2] + 4\ BesselJ[1\/2\ \((1 - 2\ \[Nu])\), 1\/2\ x\ \[Alpha]\ Sign[\[Alpha]]]\ Gamma[3\/2 - \[Nu]]\ Gamma[\(-1\) + 2\ \[Nu]]\ Sin[\(x\ \[Alpha]\)\/2]\ Sin[\[Pi]\ \[Nu]])\))\)\)\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell["\<\ using the Bessel J function for representation of the result. \ Another non trivial example is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\&+\)\_x\%\(-\[Nu]\)[\(x\^\(-\[Mu]\)\) Log[x]]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(\(x\^\(\(-\[Mu]\) + \[Nu]\)\ Gamma[\[Mu] - \[Nu]]\ \((Log[x] + PolyGamma[0, \[Mu]] - PolyGamma[0, \[Mu] - \[Nu]]) \)\)\/Gamma[\[Mu]]\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[TextData[{ StyleBox["where PolyGamma[", "MR"], StyleBox["n", "TI"], StyleBox[",", "MR"], " ", StyleBox["z", "TI"], StyleBox["]", "MR"], " gives ", Cell[BoxData[ \(TraditionalForm\`n\)], "InlineFormula"], Cell[BoxData[ \(TraditionalForm\`th\[Null]\^\*"\<\"\"\>"\)], "InlineFormula"], " derivative of the digamma function ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\^\((n)\)\)(z)\)], "InlineFormula"], ". ", StyleBox["PolyGamma[", "MR"], StyleBox["n", "TI"], StyleBox[",", "MR"], StyleBox["z", "TI"], StyleBox["]", "MR"], " is given by ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\^\((n)\)\)( z) = \(d\^n\) \(\[Psi](z)\)/\(d z\^n\)\)], "InlineFormula", CellTags->"S3.2.10"], ". Note that the digamma function corresponds to ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\^\((0)\)\)(z)\)], "InlineFormula", CellTags->"S3.2.10"], ". The general form ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\^\((n)\)\)(z)\)], "InlineFormula", CellTags->"S3.2.10"], " is the ", Cell[BoxData[ \(TraditionalForm\`\((n + 1)\) th\)], "InlineFormula", CellTags->"S3.2.10"], Cell[BoxData[ \(TraditionalForm\`\[Null]\^\*"\<\"\"\>"\)], "InlineFormula", CellTags->"S3.2.10"], ", not the ", Cell[BoxData[ \(TraditionalForm\`n\)], "InlineFormula", CellTags->"S3.2.10"], Cell[BoxData[ \(TraditionalForm\`\[Null]th\^\*"\<\"\"\>"\)], "InlineFormula", CellTags->"S3.2.10"], " logarithmic derivative of the gamma function. The polygamma functions \ satisfy the relation ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\^\((n)\)\)( z) = \(\((\(-1\))\)\^\(n + 1\)\) \(n!\) \(\[Sum]\_\(k = 0\)\%\[Infinity] 1/\((z + k)\)\^\(n + 1\)\)\)], "InlineFormula", CellTags->"S3.2.10"], "." }], "Text"], Cell["\<\ So far, we implemented the basic properties of the \ Riemann\[Dash]Liouville and Weyl fractional calculi. We already observed that \ application of the RL and W operators to well-known functions delivers \ extraordinary results. In the following section, we will discuss how these \ results are useful in connection with physical applications.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Applications of the RL and W calculi", "Section", CellTags->"Applications of the RL and W calculus"], Cell["\<\ In this section we discuss some examples demonstrating the physical \ relevance of the fractional calculus. We consider approaches and arguments \ used by early researchers in their attempts to apply the fractional calculus \ to physical problems. We discuss Abel's problem of the tautochrone as an \ example of the Riemann-Liouville calculus and fractional relaxation in \ connection with the Weyl calculus. Both examples demonstrate the usefulness \ of the fractional calculus.\ \>", "Text"], Cell[CellGroupData[{ Cell["The Tautochrone problem", "Subsection", CellDingbat->None, CellTags->"The Tautochrone problem"], Cell[TextData[{ "Let us examine a particle of mass ", Cell[BoxData[ \(TraditionalForm\`M\)]], " moving in a vertical gravitational field along a path \[GothicCapitalC]. \ The initial velocity of the particle is zero. Now let us find the shape of \ the path \[GothicCapitalC] for which the time of descent \[Tau] from the \ starting point ", StyleBox["P", FontSlant->"Italic"], " to the origin is independent of the starting point. A curve \ \[GothicCapitalC] describing this kind of movement is called a ", StyleBox["tautochrone", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "The tautochrone problem should not be confused with the brachistochrone \ problem. That problem was to find the shape of the path \[GothicCapitalC] \ such that the time of descent from ", StyleBox["P", FontSlant->"Italic"], " to the origin would be a minimum. However, as we will see, both problems \ are solved by cycloids." }], "Text"], Cell[TextData[{ "We now proceed to formulate the physical problem. Let ", Cell[BoxData[ \(TraditionalForm\`s\)]], " be the arc length measured along \[GothicCapitalC] from the origin to an \ arbitrary point ", StyleBox["Q", FontSlant->"Italic"], " with coordinates (\[Xi],\[Eta]) on \[GothicCapitalC]. At this point \ (\[Xi],\[Eta]), the particle of mass ", Cell[BoxData[ \(TraditionalForm\`M\)]], " increases the force ", Cell[BoxData[ \(TraditionalForm\`\(-M\)\ g\ \(cos(\[Beta])\)\)]], " with \[Beta] an angle of inclination and ", Cell[BoxData[ \(TraditionalForm\`g\)]], " the gravitational constant. Newton's second law tells us " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm \`M \(\( d\^2\) \(s(t)\)\)\/dt\^2 = \ \(-M\)\ g\ \(cos(\[Beta])\)\)]], "." }], "NumberedEquation"], Cell[TextData[{ "The change of the vertical coordinate \[Eta] along the arc is given by ", Cell[BoxData[ \(TraditionalForm\`cos(\[Beta])\ = \ d\[Eta]/ds\)]], ". Thus the equation of motion can be expressed by" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\(d\^2\) s\)\/dt\^2\ = \ \(-g\)\ d\[Eta]\/ds\)]]], "NumberedEquation"], Cell[TextData[{ "Multiplying equation (19) by ", Cell[BoxData[ \(TraditionalForm\`ds/dt\)]], " and integrating with respect to ", Cell[BoxData[ \(TraditionalForm\`t\)]], " gives" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\((ds\/dt)\)\^2\ = \ \(-2\)\ g\ \[Eta]\ + \ c\_1\)]]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], " is a first constant of integration. Since the particle started from rest, \ ", Cell[BoxData[ \(TraditionalForm\`ds/dt\)]], " is zero when ", Cell[BoxData[ \(TraditionalForm\`\[Eta] = y\)]], ", and thus ", Cell[BoxData[ \(TraditionalForm\`c\_1 = 2 g\ y\)]], ". The velocity of the particle becomes ", Cell[BoxData[ \(TraditionalForm\`ds/dt = \(-\@\(2 \( g(y - \[Eta])\)\)\)\)]], ". Thus the time of descent \[Tau] from ", StyleBox["P", FontSlant->"Italic"], " to the origin is" }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"\[Tau]", " ", "=", " ", RowBox[{"-", FormBox[\(\(1\/\@\(2 g\)\) \(\[Integral]\_P\%O\( 1\/\@\(y - \[Eta]\)\) \[DifferentialD]s\)\), "TraditionalForm"]}]}], TraditionalForm]]], "." }], "NumberedEquation"], Cell[TextData[{ "It is obvious that the arc length ", Cell[BoxData[ \(TraditionalForm\`s\)]], " is a function of \[Eta], say ", Cell[BoxData[ FormBox[ RowBox[{"s", " ", "=", " ", FormBox[\(h(\[Eta])\), "TraditionalForm"]}], TraditionalForm]]], ". ", Cell[BoxData[ \(TraditionalForm\`h\)]], " defines the shape of path \[GothicCapitalC]. The line element ", Cell[BoxData[ \(TraditionalForm\`ds\)]], " in (21) is thus replaced by ", Cell[BoxData[ \(TraditionalForm\`ds = \ h' \((\[Eta])\)\ d\[Eta]\)]], " which results in" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm \`\@\(2\ g\)\ \[Tau]\ = \ \[Integral]\_0\%y\(\((y - \[Eta])\)\^\(\(-1\)/2\)\) h' \((\[Eta])\)\ \[DifferentialD]\[Eta]\)]]], "NumberedEquation"], Cell[TextData[{ "Introducing the abbreviation ", Cell[BoxData[ \(TraditionalForm\`f(\[Eta])\ = h' \((\[Eta])\)\)]], " allows us to introduce the RL operator on the right hand side of (22)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\@\(2\ g\)\) \[Tau]\)\/Gamma[1\/2] == \ \(\[ScriptCapitalD]\_\(0, y\)\%\(\(-1\)/2\)\) f[y]\)], "Input", CellLabel->"In[137]:="], Cell[BoxData[ \(\@g\ \@\(2\/\[Pi]\)\ \[Tau] == \(\[ScriptCapitalD]\_\(0, y\)\%\(-\(1\/2\)\)\) f[y]\)], "Output", CellLabel->"Out[137]="] }, Open ]], Cell[TextData[{ "This relation was first used by Abel to solve the tautochrone problem. He \ attacked the equation by applying the RL operator ", Cell[BoxData[ \(TraditionalForm\`\(\ \[ScriptCapitalD]\_\(0, \ y\)\%\(1/2\)\)\)]], " on both sides of the above equation" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalD]\_\(0, y\)\%\(1/2\)\) \((\(\(\@\(2\ g\)\) \[Tau]\)\/Gamma[1\/2])\) == \(\[ScriptCapitalD]\_\(0, y\)\%\(1/2\)\) \((\(\[ScriptCapitalD]\_\(0, y\)\%\(\(-1\)/2\)\) \((f[y])\))\)\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(\(\@2\ \@g\ \[Tau]\)\/\(\[Pi]\ \@y\) == f[y]\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[TextData[{ "The use of the composition relation allows to express the unknown function \ ", Cell[BoxData[ \(TraditionalForm\`f(y)\)]], " in an explicit way. The knowledge of ", Cell[BoxData[ \(TraditionalForm\`f\)]], " delivers the relation for ", Cell[BoxData[ \(TraditionalForm\`h'\)]], ". Since we know that ", Cell[BoxData[ \(TraditionalForm\`\(\ f = \(h'\ = \ \(ds/dy\ = \ \@\(1 + \((dx/dy)\)\^2\)\)\)\)\)]], ", we can write" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(equation\ = \@\(1 + \((\[PartialD]\_y\ x[y])\)\^2\) == \ \ \(\[ScriptCapitalD]\_\(0, y\)\%\(1/2\)\) \(\(\@\(2\ g\)\) \[Tau]\)\/Gamma[1\/2]\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ RowBox[{ SqrtBox[ RowBox[{"1", "+", SuperscriptBox[ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "y", "]"}], "2"]}]], "==", \(\(\@2\ \@g\ \[Tau]\)\/\(\[Pi]\ \@y\)\)}]], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[TextData[{ "The resulting first-order ordinary differential equation can be solved by \ ", StyleBox["DSolve[]", "Input", FontWeight->"Plain"], " to be" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol\ = DSolve[equation, x, y]\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \({{x \[Rule] \((\(\@#1\ \@\(\(-\[Pi]\^2\)\ #1 + 2\ g\ \[Tau]\^2\)\)\/\[Pi] - \(2\ g\ \[Tau]\^2\ ArcTan[\(\[Pi]\ \@#1\ \@\(\(-\[Pi]\^2\)\ #1 + 2\ g\ \[Tau]\^2\)\)\/\(\[Pi]\^2\ #1 - 2\ g\ \[Tau]\^2\)]\)\/\[Pi]\^2 + C[1]&)\)}, { x \[Rule] \((\(-\(\(\@#1\ \@\(\(-\[Pi]\^2\)\ #1 + 2\ g\ \[Tau]\^2\)\)\/\[Pi]\)\) + \(2\ g\ \[Tau]\^2\ ArcTan[\(\[Pi]\ \@#1\ \@\(\(-\[Pi]\^2\)\ #1 + 2\ g\ \[Tau]\^2\)\)\/\(\[Pi]\^2\ #1 - 2\ g\ \[Tau]\^2\)]\)\/\[Pi]\^2 + C[1]&)\)}}\)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell[TextData[{ "The solutions represented above can be plotted for specific values of the \ parameters \[Tau], ", Cell[BoxData[ \(TraditionalForm\`g\)]], ", and C[1]. The constant of integration C[1] can be chosen to be C[1]=0 \ since path \[GothicCapitalC] must intersect the origin. The gravitational \ constant takes the value of ", Cell[BoxData[ \(TraditionalForm\`9.81 m/s\^2\)]], " on earth. The time of descent \[Tau] is independent of the starting point \ and thus can be set to an arbitrary value ", Cell[BoxData[ \(TraditionalForm\`\((\[Tau] = 1)\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1\ = \ \(x[y]\ /. \ sol\)\ /. \ {g \[Rule] 9.81, \[Tau] \[Rule] 1, C[1] \[Rule] 0}\)], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \({\(\@y\ \@\(\(19.6200000000000009`\[InvisibleSpace]\) - \[Pi]\^2\ y\)\)\/\[Pi] - 1.98792162306266728`\ ArcTan[\(\[Pi]\ \@y\ \@\(\(19.6200000000000009`\[InvisibleSpace]\) - \[Pi]\^2\ y\)\)\/\(\(-19.6200000000000009`\) + \[Pi]\^2\ y\)], \(-\(\(\@y\ \@\(\(19.6200000000000009`\[InvisibleSpace]\) - \[Pi]\^2\ y\)\)\/\[Pi]\)\) + 1.98792162306266728`\ ArcTan[\(\[Pi]\ \@y\ \@\(\(19.6200000000000009`\[InvisibleSpace]\) - \[Pi]\^2\ y\)\)\/\(\(-19.6200000000000009`\) + \[Pi]\^2\ y\)]}\)], "Output", CellLabel->"Out[31]="] }, Open ]], Cell["\<\ The graphical representation of the two solution branches follows by\ \>", "Text"], Cell[BoxData[ 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The upper branch is the solution realized in nature. The lower branch is \ a another mathematical possibility." }], "Caption", TextJustification->1], Cell[TextData[{ "To find a more readable representation of the solution, let us replace \ coordinate ", Cell[BoxData[ \(TraditionalForm\`y\)]], " by the expression ", Cell[BoxData[ \(\(2\ g\ \(\[Tau]\^\(2\ \)\) Sin[\[Alpha]]\^2\)\/\[Pi]\^2\)]], ", corresponding to a parametric representation of the path by means of \ angle \[Alpha]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(asol\ = \ \(\(\((x[y]\ /. sol\ )\)\ /. \ {y \[Rule] \(2\ g\ \(\[Tau]\^\(2\ \)\) Sin[\[Alpha]]\^2\)\/\[Pi]\^2, C[1] \[Rule] 0} // Simplify\) // PowerExpand\) // Expand\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \({\(2\ g\ \[Alpha]\ \[Tau]\^2\)\/\[Pi]\^2 + \(2\ g\ \[Tau]\^2\ Cos[\[Alpha]]\ Sin[\[Alpha]]\)\/\[Pi]\^2, \(-\(\(2\ g\ \[Alpha]\ \[Tau]\^2\)\/\[Pi]\^2\)\) - \(2\ g\ \[Tau]\^2\ Cos[\[Alpha]]\ Sin[\[Alpha]]\)\/\[Pi]\^2}\)], "Output", CellLabel->"Out[33]="] }, Open ]], Cell["\<\ The result is a parametric representation of a cycloid with coordinates\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cycloid\ = \ Transpose[{ asol, {\(2\ g\ \(\[Tau]\^\(2\ \)\) Sin[\[Alpha]]\^2\)\/\[Pi]\^2, \(2\ g\ \(\[Tau]\^\(2\ \)\) Sin[\[Alpha]]\^2\)\/\[Pi]\^2}}]\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ \({{\(2\ g\ \[Alpha]\ \[Tau]\^2\)\/\[Pi]\^2 + \(2\ g\ \[Tau]\^2\ Cos[\[Alpha]]\ Sin[\[Alpha]]\)\/\[Pi]\^2, \(2\ g\ \[Tau]\^2\ Sin[\[Alpha]]\^2\)\/\[Pi]\^2}, { \(-\(\(2\ g\ \[Alpha]\ \[Tau]\^2\)\/\[Pi]\^2\)\) - \(2\ g\ \[Tau]\^2\ Cos[\[Alpha]]\ Sin[\[Alpha]]\)\/\[Pi]\^2, \(2\ g\ \[Tau]\^2\ Sin[\[Alpha]]\^2\)\/\[Pi]\^2}}\)], "Output", CellLabel->"Out[34]="] }, Open ]], Cell[TextData[ "The two solutions are plotted for different times of descent \[Tau] by"], "Text"], Cell[BoxData[ \(ParametricPlot[ Evaluate[Flatten[ Table[cycloid /. {g \[Rule] 9.81, \[Tau] \[Rule] i}, {i, .5, 1, .1}], 1]], {\[Alpha], 0, 2 \[Pi]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {RGBColor[0.996109, \ 0, 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.45714 .22643 L .45714 .22632 L .45714 .22603 L .45714 .2256 L .45713 .22503 L .45712 .22438 L .4571 .22293 L .45706 .22099 L .45691 .21509 L .45666 .20801 L Mistroke .45578 .19038 L .45409 .16623 L .45164 .14009 L .44804 .11088 L .44328 .08181 L .43789 .05724 L .43498 .04683 L .43174 .0372 L .42856 .02957 L .42557 .02391 L .42226 .01927 L .42041 .01739 L .41955 .01669 L .41864 .01607 L .41786 .01563 L .417 .01524 L .41663 .01511 L .41622 .01498 L .41584 .01489 L .41548 .01482 L .41507 .01476 L .41461 .01472 L .41416 .01472 L .41369 .01474 L .41325 .01479 L .41283 .01487 L .41245 .01495 L .41205 .01507 L .4112 .01539 L .41029 .01585 L .40865 .01698 L .40672 .01882 L .40497 .02095 L .40125 .02706 L .39783 .0346 L .39165 .05357 L .38657 .07527 L .38216 .09994 L .37823 .12859 L .37545 .15498 L .37338 .18092 L .3722 .20158 L .37184 .20992 L .3716 .21725 L .37153 .22005 L .37148 .22225 L .37144 .22517 L .37143 .2257 L .37143 .22593 L .37143 .22613 L Mistroke .37143 .22628 L .37143 .22641 L .37143 .2265 L .37143 .22656 L .37143 .2266 L .37143 .22661 L .37143 .22661 L .37143 .22659 L .37143 .22654 L .37143 .2265 L .37143 .22645 L .37143 .22631 L .37143 .22614 L .37143 .22596 L .37142 .22549 L .37142 .22494 L .3714 .22361 L .37136 .22175 L .37131 .21925 L .37115 .21357 L .3709 .20699 L .37001 .1894 L .3683 .16522 L .36586 .13948 L .36226 .11043 L .35783 .08321 L .35222 .05741 L .34938 .04722 L .34619 .03764 L .3433 .03056 L .34007 .02427 L .33861 .02197 L .33705 .01987 L .33559 .01824 L .33422 .01699 L .33293 .01607 L .33226 .01569 L .33154 .01534 L .33112 .01518 L .33074 .01505 L .33035 .01494 L .32999 .01486 L .32963 .0148 L .3293 .01475 L .32895 .01473 L .32857 .01472 L Mfstroke .996 0 0 r .5 .01472 m .50058 .01473 L .5011 .01478 L .5017 .01487 L .50228 .01499 L .50329 .01528 L .50439 .01573 L .5056 .01636 L .50688 .0172 L .50917 .01913 L .51151 .0217 L .51368 .02461 L .51853 .03305 L .52329 .04409 L .52747 .05616 L .53604 .08915 L .54241 .12261 L .54808 .16165 L .55219 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.62697 .0203 L .62935 .02321 L .63442 .03151 L .63906 .04178 L .64747 .0676 L .65439 .09714 L .66039 .13072 L .66574 .16971 L .66953 .20563 L .67234 .24094 L .67396 .26906 L .67444 .28042 L .67477 .29039 L .67486 .2942 L .67493 .29719 L .67499 .30117 L .67499 .30188 L .675 .30221 L .675 .30247 L Mistroke .675 .30268 L .675 .30286 L .675 .30298 L .675 .30306 L .675 .30312 L .675 .30313 L .675 .30312 L .675 .3031 L .675 .30303 L .675 .30298 L .675 .30291 L .675 .30272 L .675 .30249 L .675 .30224 L .67501 .30161 L .67502 .30085 L .67504 .29904 L .67509 .29652 L .67516 .29311 L .67539 .28537 L .67572 .27642 L .67693 .25247 L .67926 .21956 L .68258 .18453 L .68748 .145 L .69351 .10794 L .70115 .07282 L .705 .05896 L .70935 .04592 L .71328 .03628 L .71768 .02772 L .71967 .02459 L .72179 .02174 L .72378 .01952 L .72565 .01782 L .7274 .01656 L .72831 .01604 L .7293 .01557 L .72986 .01535 L .73038 .01517 L .73092 .01502 L .73141 .01491 L .73189 .01482 L .73234 .01477 L .73282 .01473 L .73333 .01472 L Mfstroke 0 0 .996 r .5 .01472 m .49942 .01473 L .4989 .01478 L .4983 .01487 L .49772 .01499 L .49671 .01528 L .49561 .01573 L .4944 .01636 L .49312 .0172 L .49083 .01913 L .48849 .0217 L .48632 .02461 L .48147 .03305 L .47671 .04409 L .47253 .05616 L .46396 .08915 L .45759 .12261 L .45192 .16165 L .44781 .19856 L .44475 .23494 L .44369 .25082 L .44285 .26609 L .44234 .27743 L .44198 .28769 L .44187 .29168 L .44178 .29531 L .44173 .2979 L .44169 .30014 L .44168 .30117 L .44167 .30193 L .44167 .30248 L .44167 .3027 L .44167 .30288 L .44167 .30301 L .44167 .30308 L .44167 .30313 L .44167 .30313 L .44167 .30312 L .44167 .30308 L .44167 .303 L .44167 .30288 L .44167 .30274 L .44166 .30234 L .44166 .30176 L .44165 .30098 L .44164 .30009 L .44161 .29812 L .44156 .29548 L .44135 .28745 L .441 .27781 L Mistroke .43982 .25381 L .43752 .22094 L .43418 .18537 L .42927 .14561 L .4228 .10603 L .41546 .07259 L .4115 .05843 L .40709 .04531 L .40276 .03493 L .39869 .02723 L .39418 .02091 L .39166 .01836 L .3905 .01741 L .38926 .01656 L .3882 .01596 L .38703 .01543 L .38652 .01525 L .38597 .01508 L .38545 .01495 L .38497 .01485 L .3844 .01477 L .38377 .01473 L .38317 .01472 L .38252 .01475 L .38192 .01482 L .38135 .01492 L .38084 .01504 L .38029 .0152 L .37913 .01564 L .37789 .01627 L .37566 .0178 L .37303 .0203 L .37065 .02321 L .36558 .03151 L .36094 .04178 L .35253 .0676 L .34561 .09714 L .33961 .13072 L .33426 .16971 L .33047 .20563 L .32766 .24094 L .32604 .26906 L .32556 .28042 L .32523 .29039 L .32514 .2942 L .32507 .29719 L .32501 .30117 L .32501 .30188 L .325 .30221 L .325 .30247 L Mistroke .325 .30268 L .325 .30286 L .325 .30298 L .325 .30306 L .325 .30312 L .325 .30313 L .325 .30312 L .325 .3031 L .325 .30303 L .325 .30298 L .325 .30291 L .325 .30272 L .325 .30249 L .325 .30224 L .32499 .30161 L .32498 .30085 L .32496 .29904 L .32491 .29652 L .32484 .29311 L .32461 .28537 L .32428 .27642 L .32307 .25247 L .32074 .21956 L .31742 .18453 L .31252 .145 L .30649 .10794 L .29885 .07282 L .295 .05896 L .29065 .04592 L .28672 .03628 L .28232 .02772 L .28033 .02459 L .27821 .02174 L .27622 .01952 L .27435 .01782 L .2726 .01656 L .27169 .01604 L .2707 .01557 L .27014 .01535 L .26962 .01517 L .26908 .01502 L .26859 .01491 L .26811 .01482 L .26766 .01477 L .26718 .01473 L .26667 .01472 L Mfstroke .996 0 0 r .5 .01472 m .50075 .01474 L .50144 .0148 L .50223 .01491 L .50297 .01507 L .5043 .01546 L .50574 .01603 L .50732 .01686 L .50899 .01796 L .51198 .02049 L .51503 .02383 L .51787 .02764 L .5242 .03866 L .53043 .05308 L .53587 .06885 L .54707 .11193 L .55539 .15564 L .5628 .20663 L .56817 .25484 L .57217 .30235 L .57354 .32309 L .57464 .34304 L .57531 .35786 L .57578 .37126 L .57593 .37646 L .57604 .3812 L .57611 .3846 L .57616 .38752 L .57617 .38886 L .57618 .38985 L .57619 .39057 L .57619 .39085 L .57619 .3911 L .57619 .39126 L .57619 .39136 L .57619 .39142 L .57619 .39142 L .57619 .39141 L .57619 .39136 L .57619 .39125 L .57619 .39109 L .57619 .3909 L .5762 .39039 L .5762 .38963 L .57621 .38861 L .57623 .38745 L .57626 .38488 L .57633 .38143 L .57661 .37095 L .57705 .35835 L Mistroke .57861 .327 L .58161 .28407 L .58597 .23761 L .59238 .18568 L .60083 .13399 L .61042 .09031 L .61559 .07182 L .62135 .05468 L .627 .04112 L .63232 .03106 L .63821 .02281 L .6415 .01947 L .64302 .01823 L .64464 .01712 L .64602 .01634 L .64755 .01565 L .64822 .01541 L .64894 .01519 L .64962 .01502 L .65025 .0149 L .65099 .01479 L .6518 .01473 L .6526 .01472 L .65344 .01476 L .65423 .01485 L .65497 .01498 L .65564 .01514 L .65636 .01535 L .65787 .01592 L .65949 .01674 L .6624 .01875 L .66584 .02201 L .66894 .0258 L .67556 .03665 L .68163 .05007 L .69262 .08379 L .70165 .12237 L .70949 .16623 L .71648 .21716 L .72142 .26408 L .7251 .31019 L .72721 .34692 L .72784 .36175 L .72827 .37478 L .72839 .37975 L .72848 .38366 L .72855 .38886 L .72856 .38979 L .72857 .39022 L .72857 .39056 L Mistroke .72857 .39083 L .72857 .39106 L .72857 .39122 L .72857 .39133 L .72857 .3914 L .72857 .39142 L .72857 .39141 L .72857 .39138 L .72857 .39129 L .72857 .39122 L .72857 .39113 L .72857 .39088 L .72857 .39059 L .72858 .39026 L .72858 .38943 L .72859 .38845 L .72863 .38608 L .72868 .38279 L .72878 .37833 L .72907 .36823 L .72951 .35654 L .73109 .32526 L .73414 .28227 L .73847 .23652 L .74487 .18488 L .75274 .13648 L .76273 .09061 L .76776 .07251 L .77344 .05547 L .77857 .04288 L .78432 .0317 L .78691 .02761 L .78969 .02388 L .79228 .02099 L .79472 .01876 L .79702 .01712 L .7982 .01644 L .79949 .01583 L .80022 .01554 L .8009 .01531 L .80161 .01511 L .80224 .01497 L .80287 .01486 L .80346 .01478 L .80409 .01473 L .80476 .01472 L Mfstroke 0 0 .996 r .5 .01472 m .49925 .01474 L .49856 .0148 L .49777 .01491 L .49703 .01507 L .4957 .01546 L .49426 .01603 L .49268 .01686 L .49101 .01796 L .48802 .02049 L .48497 .02383 L .48213 .02764 L .4758 .03866 L .46957 .05308 L .46413 .06885 L .45293 .11193 L .44461 .15564 L .4372 .20663 L .43183 .25484 L .42783 .30235 L .42646 .32309 L .42536 .34304 L .42469 .35786 L .42422 .37126 L .42407 .37646 L .42396 .3812 L .42389 .3846 L .42384 .38752 L .42383 .38886 L .42382 .38985 L .42381 .39057 L .42381 .39085 L .42381 .3911 L .42381 .39126 L .42381 .39136 L .42381 .39142 L .42381 .39142 L .42381 .39141 L .42381 .39136 L .42381 .39125 L .42381 .39109 L .42381 .3909 L .4238 .39039 L .4238 .38963 L .42379 .38861 L .42377 .38745 L .42374 .38488 L .42367 .38143 L .42339 .37095 L .42295 .35835 L Mistroke .42139 .327 L .41839 .28407 L .41403 .23761 L .40762 .18568 L .39917 .13399 L .38958 .09031 L .38441 .07182 L .37865 .05468 L .373 .04112 L .36768 .03106 L .36179 .02281 L .3585 .01947 L .35698 .01823 L .35536 .01712 L .35398 .01634 L .35245 .01565 L .35178 .01541 L .35106 .01519 L .35038 .01502 L .34975 .0149 L .34901 .01479 L .3482 .01473 L .3474 .01472 L .34656 .01476 L .34577 .01485 L .34503 .01498 L .34436 .01514 L .34364 .01535 L .34213 .01592 L .34051 .01674 L .3376 .01875 L .33416 .02201 L .33106 .0258 L .32444 .03665 L .31837 .05007 L .30738 .08379 L .29835 .12237 L .29051 .16623 L .28352 .21716 L .27858 .26408 L .2749 .31019 L .27279 .34692 L .27216 .36175 L .27173 .37478 L .27161 .37975 L .27152 .38366 L .27145 .38886 L .27144 .38979 L .27143 .39022 L .27143 .39056 L Mistroke .27143 .39083 L .27143 .39106 L .27143 .39122 L .27143 .39133 L .27143 .3914 L .27143 .39142 L .27143 .39141 L .27143 .39138 L .27143 .39129 L .27143 .39122 L .27143 .39113 L .27143 .39088 L .27143 .39059 L .27142 .39026 L .27142 .38943 L .27141 .38845 L .27137 .38608 L .27132 .38279 L .27122 .37833 L .27093 .36823 L .27049 .35654 L .26891 .32526 L .26586 .28227 L .26153 .23652 L .25513 .18488 L .24726 .13648 L .23727 .09061 L .23224 .07251 L .22656 .05547 L .22143 .04288 L .21568 .0317 L .21309 .02761 L .21031 .02388 L .20772 .02099 L .20528 .01876 L .20298 .01712 L .2018 .01644 L .20051 .01583 L .19978 .01554 L .1991 .01531 L .19839 .01511 L .19776 .01497 L .19713 .01486 L .19654 .01478 L .19591 .01473 L .19524 .01472 L Mfstroke .996 0 0 r .5 .01472 m .50095 .01474 L .50182 .01482 L .50282 .01497 L .50376 .01516 L .50544 .01565 L .50726 .01639 L .50926 .01743 L .51138 .01882 L .51516 .02202 L .51902 .02626 L .52261 .03108 L .53063 .04503 L .53851 .06328 L .5454 .08323 L .55957 .13776 L .57011 .19307 L .57948 .2576 L .58627 .31863 L .59134 .37875 L .59308 .40501 L .59447 .43025 L .59531 .44901 L .59591 .46596 L .5961 .47255 L .59624 .47855 L .59633 .48285 L .59639 .48655 L .59641 .48824 L .59642 .4895 L .59642 .49041 L .59643 .49077 L .59643 .49108 L .59643 .49128 L .59643 .49141 L .59643 .49148 L .59643 .49148 L .59643 .49147 L .59643 .4914 L .59643 .49126 L .59643 .49106 L .59643 .49083 L .59643 .49018 L .59644 .48922 L .59646 .48792 L .59647 .48645 L .59652 .4832 L .59661 .47884 L .59696 .46557 L .59752 .44963 L Mistroke .59949 .40996 L .60329 .35561 L .60881 .29682 L .61692 .23109 L .62761 .16567 L .63976 .11039 L .64629 .08698 L .65359 .0653 L .66074 .04813 L .66747 .0354 L .67492 .02496 L .67909 .02074 L .68101 .01917 L .68306 .01776 L .68481 .01677 L .68674 .0159 L .68759 .01559 L .6885 .01531 L .68936 .0151 L .69016 .01495 L .6911 .01481 L .69213 .01473 L .69313 .01472 L .6942 .01477 L .6952 .01489 L .69614 .01506 L .69698 .01525 L .69789 .01552 L .69981 .01624 L .70185 .01728 L .70554 .01982 L .70989 .02395 L .71382 .02875 L .7222 .04248 L .72987 .05946 L .74378 .10214 L .75521 .15096 L .76513 .20648 L .77398 .27094 L .78024 .33031 L .78489 .38867 L .78756 .43516 L .78836 .45394 L .7889 .47043 L .78906 .47671 L .78916 .48166 L .78926 .48825 L .78927 .48942 L .78928 .48996 L .78928 .49039 L Mistroke .78928 .49073 L .78928 .49103 L .78929 .49123 L .78929 .49137 L .78929 .49146 L .78929 .49148 L .78929 .49147 L .78929 .49144 L .78929 .49132 L .78929 .49123 L .78929 .49111 L .78929 .4908 L .78929 .49043 L .78929 .49001 L .7893 .48896 L .78931 .48772 L .78936 .48473 L .78943 .48055 L .78955 .47491 L .78992 .46213 L .79047 .44733 L .79247 .40775 L 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0}} -> {-14.6552, -0.167575, 0.108181, 0.0139293}}], Cell[TextData[ "The figure shows the two solutions of the tautochrone problem for different \ values of \[Tau]\[Element][0.5,1] in steps of 0.1. The right hand side of \ the plot represents the first solution, the left side of the plot shows \ the second one."], "Caption", TextJustification->1], Cell["\<\ At the end, we recognize that the tautochrone problem is solved by \ a cycloid. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["A fractional relaxation equation", "Subsection", CellDingbat->None, CellTags->"Liouville's problem of potential theory"], Cell["\<\ The second example discussed is related to the fractional calculus \ by Weyl. This example is connected with the problem of a fractional \ relaxation process. The standard Maxwell\[Dash]Debye relaxation process, for \ example, is modeled by the initial value problem\ \>", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"\[Tau]", " ", FormBox[\(\(d\/\(d\ t\)\) \(\[Phi](t)\)\ = \ \(-\(\[Phi](t)\)\)\), "TraditionalForm"]}], TraditionalForm]]], ", with ", Cell[BoxData[ \(TraditionalForm\`t > 0\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\[Phi](0) = \[Phi]\_0\)]] }], "NumberedEquation", CellTags->"eq-3"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_0\)]], " is a constant and ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)]], " denotes the relaxation time. The solution is given by an exponential" }], "Text"], Cell[TextData[StyleBox[Cell[BoxData[ FormBox[ RowBox[{\(\[Phi](t)\), " ", "=", RowBox[{ StyleBox["{", "Text"], GridBox[{ { \(\(\[Phi]\_0\) \[ExponentialE]\^\(\(-t\)/\[Tau]\)\ \ \ \ \ if \ t \[GreaterEqual] \ 0\)}, {\(0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ t < 0. \)} }], " "}]}], TraditionalForm]]]]], "NumberedEquation", CellTags->"eq-4"], Cell[TextData[{ "The initial value problem (", ButtonBox["23", ButtonData:>"eq-3", ButtonStyle->"Hyperlink"], ") reads in an integral representation" }], "Text"], Cell[TextData[{ StyleBox[Cell[BoxData[ \(TraditionalForm \`\[Phi](t)\ - \[Phi]\_0\ = \ \(\(-\[Tau]\^\(-1\)\) \(\[Integral]\_0\%t\( \[Phi](t')\)\ \[DifferentialD]t'\)\ = \( : \ \(-\[Tau]\^\(-1\)\)\ \(\(d\^\(-1\)\) \(\[Phi](t)\)\)\/dt\^\(-1\)\)\)\)]]], StyleBox["."] }], "NumberedEquation", CellTags->"eq-5"], Cell[TextData[{ "The integral equation (", ButtonBox["25", ButtonData:>"eq-5", ButtonStyle->"Hyperlink"], ") incorporates the initial value ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_0\)]], " right from the beginning. This representation of a relaxation process \ contains the total information on the process in a nutshell. Relaxation \ processes deviating from the classical Maxwell\[Dash]Debye behavior are \ referred to as non-Debye, nonexponential or anomalous relaxation processes. \ Anomalous relaxation processes have been observed in dielectrics, in \ diffusion-controlled relaxations, in liquid crystals, polymer melts, \ amorphous polymers, rubber, biopolymers, and other disordered systems. The \ deviation from the exponential decay is part of current problems in physics \ which have not been completely resolved. Typical non-Debye relaxation \ processes are described empirically either by a Kohlrausch\[Dash]Williams\ \[Dash]Watts (KWW) decay" }], "Text"], Cell[TextData[StyleBox[Cell[BoxData[ \(TraditionalForm \`\[Phi](t)\ = \ \(\[Phi]\_0\) \[ExponentialE]\^\(-\((t/\[Tau])\)\^\[Alpha]\)\)]]]], "NumberedEquation", CellTags->"eq-6"], Cell[TextData[{ "with ", Cell[BoxData[ \(TraditionalForm\`0 < \[Alpha] < 1\)]], ", or by an asymptotic power-law (Nutting law)" }], "Text"], Cell[TextData[{ StyleBox[Cell[BoxData[ \(TraditionalForm \`\[Phi](t)\ = \ \[Phi]\_0\ 1\/\((1 + t/\[Tau])\)\^\[Gamma]\ \[Tilde] \ t\^\(-\[Gamma]\)\)]]], StyleBox[" if "], StyleBox[Cell[BoxData[ \(TraditionalForm\`t/\[Tau] \[Rule] \[Infinity]\)]]] }], "NumberedEquation", CellTags->"eq-7"], Cell[TextData[{ "with ", Cell[BoxData[ \(TraditionalForm\`0 < \[Gamma] < 1\)]], ". Both behaviors are connected to each other. In specific experiments, the \ transition between the KWW and the Nutting law can be observed, provided the \ range of the time domain extends over several decades. In fact, it is \ possible to find an analytic function interpolating between the KWW and the \ Nutting law. The interpolating function is a Mittag-Leffler function which is \ contained in the generalized function class of Fox's H-functions [", ButtonBox["7", ButtonData:>"Fox-1961", ButtonStyle->"Hyperlink"], "]. " }], "Text"], Cell[TextData[{ "The Mittag-Leffler function, as a special member of the class of Fox's \ H-functions, describes anomalous relaxation processes. These occur as \ solutions of a generalized relaxation equation which we now discuss. The \ basis of the generalization is the integral representation (", ButtonBox["25", ButtonData:>"eq-5", ButtonStyle->"Hyperlink"], ") of a standard relaxation process. The generalization consists in \ introducing a fractional integral order in (", ButtonBox["25", ButtonData:>"eq-5", ButtonStyle->"Hyperlink"], ") by " }], "Text"], Cell[TextData[StyleBox[Cell[BoxData[ \(TraditionalForm \`\[Phi](t)\ + \[Phi]\_0\ = \ \(\((\(-\[Tau]\))\)\^\(-q\)\ \(\(d\^\(-q\)\) \(\[Phi](t)\)\)\/dt\^\(-q\) = \( : \ \(\((\(-\[Tau]\))\)\^\(-q\)\) \(\(\[ScriptCapitalW]\&+\)\_t\%q\) \(\[Phi](t)\)\)\)\)]]]], "NumberedEquation", CellTags->"eq-8"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`q\)]], ", the fractional differentiation order, is a real constant ", Cell[BoxData[ \(TraditionalForm\`\( > 0\)\)]], ". The symbol ", Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalW]\&+\)\_t\%\(-q\)\)]], " denotes the Weyl integral operator defined by ", Cell[BoxData[ \(TraditionalForm \`\(\(\[ScriptCapitalW]\&+\)\_t\%\(-q\)\) \(\[Phi](t)\)\ = \[Integral]\_t\%\[Infinity]\( \[Phi](t')\)/\((t - t')\)\^\(1 - q\) \[DifferentialD]t'/\(\[CapitalGamma](q)\)\)]], ". This generalization of a standard relaxation process assumes that \ anomalous relaxation processes are deeply connected with memory integrals. " }], "Text"], Cell[TextData[{ "Solution of such an equation is gained either by a power series ansatz or \ by Laplace\[Dash]Mellin techniques. We prefer the second procedure because it \ directly connects the solution to Fox's H-functions. The first step to solve \ equation (", ButtonBox["28", ButtonData:>"eq-8", ButtonStyle->"Hyperlink"], ") is to apply the Laplace transform (", ButtonBox["28", ButtonData:>"eq-8", ButtonStyle->"Hyperlink"], ") and gain an algebraic solution in Laplace space. The Laplace transform \ of equation (", ButtonBox["28", ButtonData:>"eq-8", ButtonStyle->"Hyperlink"], ") in ", StyleBox["Mathematica", FontSlant->"Italic"], " follows from " }], "Text"], Cell[BoxData[ \(\(Assume[q > 0]; \)\)], "Input", CellLabel->"In[73]:="], Cell[CellGroupData[{ Cell[BoxData[ \(eq1 = \(\((\(-\[Tau]\))\)\^q\) \(\[ScriptCapitalW]\&+\)\_t\%q[\[Phi][t]] == \[Phi]0 + \[Phi][t]\)], "Input", CellLabel->"In[74]:="], Cell[BoxData[ \(\((\(-\[Tau]\))\)\^q\ \(\[ScriptCapitalW]\&+\)\_t\%q[\[Phi][t]] == \[Phi]0 + \[Phi][t]\)], "Output", CellLabel->"Out[74]="] }, Open ]], Cell[TextData[{ "under the assumption ", Cell[BoxData[ \(TraditionalForm\`q > 0\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eql = \[ScriptCapitalL]\_t\%s[eq1] // PowerExpand\)], "Input", CellLabel->"In[75]:="], Cell[BoxData[ \(\((\(-1\))\)\^\(2\ q\)\ \[Tau]\^q\ \((\(-C1[]\) + s\^q\ \[ScriptCapitalL]\_t\%s[\[Phi][t]])\) == \[Phi]0\/s + \[ScriptCapitalL]\_t\%s[\[Phi][t]]\)], "Output", CellLabel->"Out[75]="] }, Open ]], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`C1[]\)]], " is a constant introduced in the Laplace transform of the W operator. In \ the following we assume that this constant vanishes. The algebraic solution \ of (", ButtonBox["28", ButtonData:>"eq-8", ButtonStyle->"Hyperlink"], ") in the Laplace space is given by" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sl = Solve[eql, \[ScriptCapitalL]\_t\%s[\[Phi][t]]] /. C1[] \[Rule] 0 // Simplify\)], "Input", CellLabel->"In[76]:="], Cell[BoxData[ \({{\[ScriptCapitalL]\_t\%s[\[Phi][t]] \[Rule] \[Phi]0\/\(s\ \((\(-1\) + \((\(-1\))\)\^\(2\ q\)\ s\^q\ \[Tau]\^q)\)\)}}\)], "Output", CellLabel->"Out[76]="] }, Open ]], Cell[TextData[{ "Since this expression contains terms with arbitrary powers ", Cell[BoxData[ \(TraditionalForm\`q\)]], ", we face the problem of calculating the inverse Laplace transform. At \ this point we note that Miller and Ross [", ButtonBox["3", ButtonData:>"Miller-1993", ButtonStyle->"Hyperlink"], "] developed a solution strategy for fractional differential equations that \ works well for the case if the fractional order ", Cell[BoxData[ \(TraditionalForm\`q\)]], " is a rational number. A way out of this restriction, applicable to \ arbitrary ", Cell[BoxData[ \(TraditionalForm\`q > 0\)]], ", is the additional transform to the Mellin space. The inclusion of the \ Mellin transform allows us to resolve the pole structure of the algebraic \ expression in the complex plane. The Mellin transform of the Laplace solution \ becomes" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sm\ = \[ScriptCapitalM]\_s\%p[sl]\)], "Input", CellLabel->"In[77]:="], Cell[BoxData[ InterpretationBox[ \("Conditions to solve the integral:\n"\[InvisibleSpace]\(Re[ \(-\(1\/q\)\) + p\/q] > 0 && \(-Re[p\/q]\) + 1\/Re[q] > \(-1\) && Arg[\((\(-1\))\)\^\(\(-2\)\ q\)\ \[Tau]\^\(-q\)] \[NotEqual] 0\)\), SequenceForm[ "Conditions to solve the integral:\n", And[ Greater[ Re[ Plus[ Times[ -1, Power[ q, -1]], Times[ p, Power[ q, -1]]]], 0], Greater[ Plus[ Times[ -1, Re[ Times[ p, Power[ q, -1]]]], Power[ Re[ q], -1]], -1], Unequal[ Arg[ Times[ Power[ -1, Times[ -2, q]], Power[ \[Tau], Times[ -1, q]]]], 0]]], Editable->False]], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ StyleBox["\[CapitalGamma]", FontSlant->"Italic"], "[", "p", "]"}], " ", \(\[ScriptCapitalM]\_s\%\(1 - p\)[\[Phi][s]]\)}], "\[Rule]", \(\(\[Pi]\ \((\(-\((\(-1\))\)\^\(2\ q\)\)\ \[Tau]\^q)\)\^\(- \(\(\(-1\) + p\)\/q\)\)\ \[Phi]0\ Csc[\[Pi]\/q - \(p\ \[Pi]\)\/q]\)\/q\)}], "}"}], "}"}]], "Output", CellLabel->"Out[77]="] }, Open ]], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalM]\_s\%p\)]], "denotes the Mellin transform. A shift of the Mellin variable ", Cell[BoxData[ \(TraditionalForm\`p\)]], " allows us to rewrite the result" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ssm = Solve[\(sm /. Rule \[Rule] Equal\) /. p \[Rule] 1 - p, \[ScriptCapitalM]\_s\%p[\[Phi][s]]]\)], "Input", CellLabel->"In[78]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{\(\[ScriptCapitalM]\_s\%p[\[Phi][s]]\), "\[Rule]", FractionBox[ \(\[Pi]\ \((\(-\((\(-1\))\)\^\(2\ q\)\)\ \[Tau]\^q)\)\^\(p/q\)\ \[Phi]0\ Csc[\[Pi]\/q - \(\((1 - p)\)\ \[Pi]\)\/q]\), RowBox[{"q", " ", RowBox[{ StyleBox["\[CapitalGamma]", FontSlant->"Italic"], "[", \((1 - p)\), "]"}]}]]}], "}"}], "}"}]], "Output", CellLabel->"Out[78]="] }, Open ]], Cell["\<\ Since the inverse Mellin transform is closely connected with the \ definition of a Fox's H-function, we can use this link to derive the solution \ as \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(shh = \(\((\[ScriptCapitalM]\^\(-1\))\)\_p\%t[ssm] // PowerExpand\) // Flatten \)], "Input", CellLabel->"In[79]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(\[Phi][t]\), "\[Rule]", RowBox[{"\[Phi]0", " ", RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Italic"], \(q, 1\)], "[", " ", \((\((\(-1\))\)\^\(\(-2\)\ q\)\ t\^q\ \[Tau]\^\(-q\))\), "]"}]}]}], "}"}]], "Output", CellLabel->"Out[79]="] }, Open ]], Cell[TextData[{ "The result is a Mittag-Leffler function ", Cell[BoxData[ \(TraditionalForm\`\(E\_\(q, 1\)\)( z)\ = \ \[Sum]\_\(k = 0\)\%\[Infinity] z\^k\/\(\[CapitalGamma](q\ \ k + 1)\)\)]], ". For ", Cell[BoxData[ \(TraditionalForm\`q \[Rule] 1\)]], " this series approaches the exponential function. Derivation of the \ solution was possible since the Mellin representation of the solution is \ represented by \[CapitalGamma]-functions. Whenever we can represent the \ algebraic solution in terms of \[CapitalGamma]-functions, we are able to \ transform the solution from the Mellin space to the time domain. The \ mathematical link is a Barnes-type integral [", ButtonBox["8", ButtonData:>"Braaksma-1964", ButtonStyle->"Hyperlink"], "] defining either the inverse Mellin transform of the Fox's H-functions. \ Knowing this connection it is straightforward to write down the Fox's \ H-function from the solution in Mellin space. 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The upper curve \ represents the solution of the fractional relaxation equation of Weyl type.\ \>", "Caption"], Cell["\<\ The figure above shows that an exponential (red) decays much faster than the \ solution of the Weyl equation (blue). The asymptotic behavior of the derived \ solution is given by\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[\[Phi][t] /. shh, {t, \[Infinity], 1}]\)], "Input", CellLabel->"In[81]:="], Cell[BoxData[ RowBox[{"\[Phi]0", " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ \(\((\(-1\))\)\^\(4\ q\)\ t\^\(\(-2\)\ q\)\ \[Tau]\^\(2\ q\)\), RowBox[{ StyleBox["\[CapitalGamma]", FontSlant->"Italic"], "[", \((1 - 2\ q)\), "]"}]]}], "-", FractionBox[\(\((\(-1\))\)\^\(2\ q\)\ t\^\(-q\)\ \[Tau]\^q\), RowBox[{ StyleBox["\[CapitalGamma]", FontSlant->"Italic"], "[", \((1 - q)\), "]"}]]}], ")"}]}]], "Output", CellLabel->"Out[81]="] }, Open ]], Cell["\<\ confirming the power law behavior of anomalous relaxation process.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions", "Section", CellTags->"Conclusions"], Cell[TextData[{ "We demonstrated that ", StyleBox["Mathematica", FontSlant->"Italic"], " is able to handle fractional derivatives if the operations are defined in \ an appropriate way. We implemented with a minimal number of functions the \ Riemann\[Dash]Liouville and the Weyl calculi. The achievement of both calculi \ allows us to calculate fractional integrals and fractional derivatives. We \ also demonstrated by several examples that the calculation of either \ fractional derivatives or integrals will result in special functions. This is \ true just for simple functions. We also discussed two examples applying the \ fractional calculi to solve physical problems. We solved the tautochrone \ problem originated by Abel and a fractional relaxation equation. Both \ physical problems enable us to solve the related integral equations." }], "Text"], Cell[CellGroupData[{ Cell["REFERENCES ", "Subsection", CellTags->"References"], Cell[TextData[{ "[1] S.F. Lacroix, ", StyleBox["Trait\[EAcute] du Calcul Diff\[EAcute]rentiel et du Calcul Int\ \[EAcute]gral", FontSlant->"Italic"], ", 2nd ed., Vol.3, 409-410. Courcier, Paris (1819)." }], "Reference", CellTags->"Lacroix-1819"], Cell[TextData[{ "[2] J. Liouville, ", StyleBox["M\[EAcute]moiresur le calcul des diff\[EAcute]rentielles \ \[AGrave] indices quelconques", FontSlant->"Italic"], ", J. \[CapitalEAcute]cole Polytech., ", StyleBox["13", FontWeight->"Bold"], ", 71-162, (1832)." }], "Reference", CellTags->"Liouville-1832a"], Cell[TextData[{ "[3] K.S. Miller and B. Ross, ", StyleBox["An Introduction to the Fractional Calculus and Fractional \ Differential Equations", FontSlant->"Italic"], ", John Wiley & Sons, Inc., New York, (1993)." }], "Reference", CellTags->"Miller-1993"], Cell[TextData[{ "[4] K.B. Oldham and J. Spanier, ", StyleBox["The Fractional Calculus", FontSlant->"Italic"], ", Academic Press, New York, (1974)." }], "Reference", CellTags->"Oldham-1974"], Cell[TextData[{ "[5] G.F.B. Riemann, ", StyleBox["Gesammelte Werke", FontSlant->"Italic"], ", 353-366, Teubner, Leipzig, (1892)." }], "Reference", CellTags->"Riemann-1892"], Cell[TextData[{ "[6] H. Weyl, ", StyleBox["Bemerkungen zum Begriff des Differentialquotienten gebrochener \ Ordnung", FontSlant->"Italic"], ", Vierteljahresschr. Naturforsch. Ges. Z\[UDoubleDot]rich, ", StyleBox["62", FontWeight->"Bold"], ", 296-302, (1917)." }], "Reference", CellTags->"Weyl-1917"], Cell[TextData[{ "[7] C. Fox, ", StyleBox["The G and H Functions as Symmetrical Fourier Kernels", FontSlant->"Italic"], ", Trans. Am. Math. Soc., ", StyleBox["98", FontWeight->"Bold"], ", 395-429, (1961)." }], "Reference", CellTags->"Fox-1961"], Cell[TextData[{ "[8] B.L.J. Braaksma, ", StyleBox["Asymptotic Expansions and Analytic Continuations for a Class of \ Barnes-Integrals", FontSlant->"Italic"], ", Compos. Math. ", StyleBox["15", FontWeight->"Bold"], ", 239-341, (1964)." }], "Reference", CellTags->"Braaksma-1964"] }, Open ]], Cell[CellGroupData[{ Cell["ABOUT THE AUTHOR", "Subsection"], Cell[TextData[{ "Gerd Baumann is a faculty member in the Department of Mathematical Physics \ at the University of Ulm, Germany. He is the author of ", StyleBox["Mathematica in Theoretical Physics", FontSlant->"Italic"], " (1995) and ", StyleBox["Symmetry Analysis of Differential Equations with Mathematica", FontSlant->"Italic"], " (in press), both published by Springer-Verlag. \n\nGerd Baumann\n\ Department of Mathematical Physics\nUniversity of Ulm\nAlbert-Einstein-Allee \ 11\nD-89069 Ulm\nGermany\ne-mail:Gerd.Baumann@physik.uni-ulm.de\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["ELECTRONIC SUBSCRIPTIONS", "Subsection"], Cell[TextData[{ "Included in the distribution for each electronic subscription is the file \ ", StyleBox["FractionalCalculus.nb", "Input", FontWeight->"Plain"], ",containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the material described in this article." }], "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->"EditBar", WindowSize->{913, 655}, WindowMargins->{{50, Automatic}, {-2, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. 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