(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 366440, 7337] NotebookOptionsPosition[ 360013, 7137] NotebookOutlinePosition[ 360392, 7154] CellTagsIndexPosition[ 360349, 7151] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Iterative Solution of Highly Nonlinear Boundary Value Problems\ \>", "Title", CellChangeTimes->{{3.5872092022457123`*^9, 3.5872092180106144`*^9}}, TextAlignment->Center], Cell["\<\ Frank J. Kampas Physicist at Large Consulting\ \>", "Subtitle", CellChangeTimes->{{3.5868880995057197`*^9, 3.5868881339136877`*^9}}, TextAlignment->Center], Cell[" Frank @ Physicistatlarge.com", "Subsubtitle", CellChangeTimes->{{3.586888147978492*^9, 3.5868881638824015`*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell[" Introduction", "Subtitle", TextAlignment->Center, FontSlant->"Italic"], Cell[TextData[{ "Nonlinear boundary value differential equations are usually solved with the \ \"shooting method\". In this technique, the initial conditions are adjusted \ until the boundary conditions at the other boundary are satisified. In \ situations in which the shooting method fails, the iterative \"relaxation\" \ method can be used. Initial guesses at the solution are improved repeatedly. \ The relaxation method can be implement using \"quasi-linearization\". The \ differential equation is linearized about the guessed solution and the \ linearized boundary value problem is solved. This technique can be \ implemented in a very straight-forward fashion in ", StyleBox["Mathematica", FontSlant->"Italic"], " because the numerical solutions to differential equations are \ interpolating functions.\n\nQuasi-linearization is first demonstrated with a \ single differential equation and then with the 5 coupled differential \ equations which describe a p-n junction. " }], "Text", CellChangeTimes->{3.5868882091489906`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Troesch's Equation", "Subtitle", TextAlignment->Center, FontSlant->"Italic"], Cell["\<\ The shooting method is problematic for Troesch's equation, y'' = 5 Sinh(5 y), \ y(0)=0, y(1) =1, as the solution has a singularity at approximately at x = \ 1.03, unless the initial estimate of the value of the derivative at x = 0 is \ very good. In the following calculation, shootsoln is the parametric \ solution to Troesch's equation, as a function of d, which is equal to y'(0). \ \>", "Text", CellChangeTimes->{{3.58688823991475*^9, 3.586888279123993*^9}, { 3.586888322303463*^9, 3.586888372482333*^9}, {3.5869502280565395`*^9, 3.586950258691291*^9}, {3.5869503815213165`*^9, 3.586950388465714*^9}, { 3.586950425900855*^9, 3.586950427184929*^9}, 3.5869506436173077`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"shootsoln", "=", RowBox[{ RowBox[{"ParametricNDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "'"}], "'"}], "[", "x", "]"}], "-", RowBox[{"5", " ", RowBox[{"Sinh", "[", RowBox[{"5", " ", RowBox[{"y", "[", "x", "]"}]}], "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{"y", "[", "0", "]"}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "0", "]"}], "==", "d"}]}], "}"}], ",", "y", ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", "d", "}"}]}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}]], "Input", CellChangeTimes->{{3.586884417916145*^9, 3.5868844673779736`*^9}, { 3.5869502768463297`*^9, 3.5869502906781206`*^9}}], Cell[BoxData[ RowBox[{"y", "\[Rule]", TagBox[ RowBox[{"ParametricFunction", "[", InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", ">"], Editable->False], "]"}], False, Editable->False]}]], "Output", CellChangeTimes->{{3.5868844637017636`*^9, 3.5868844688610587`*^9}, 3.586887985108176*^9, 3.5869502913661604`*^9, 3.5869504437538767`*^9, 3.586950735551566*^9, 3.586951434454541*^9, 3.5870358520140295`*^9, 3.587036446383025*^9, 3.58703744390308*^9, 3.58704110411861*^9, 3.5871530599723167`*^9, 3.5872061801258574`*^9, 3.587208212305091*^9, 3.5872083824258213`*^9, 3.587208450496715*^9, 3.587208571528638*^9, 3.587208674912551*^9, 3.5872087378781524`*^9, 3.587209042089552*^9, 3.5872903504707413`*^9, 3.5872957336536417`*^9, 3.58729596633395*^9, 3.5872960272724357`*^9, 3.5872962013693933`*^9}] }, Open ]], Cell["\<\ FindRoot is used to find the value of d for which the solution satisfies the \ boundry condition at x = 1. The error messages from FindRoot come from \ values of d which result in solutions which are singular in the range x = 0 \ to x = 1. These messages show the problems associated with the shooting \ method for this type of boundary value problem.\ \>", "Text", CellChangeTimes->{{3.5869502173849287`*^9, 3.586950235344956*^9}, { 3.586950647625537*^9, 3.5869507169225006`*^9}, {3.587209165051585*^9, 3.5872091918221164`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"soln", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"y", "[", "d", "]"}], "/.", " ", RowBox[{ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"y", "[", "d", "]"}], "[", "1", "]"}], "/.", " ", "shootsoln"}], ")"}], "\[Equal]", "1"}], ",", RowBox[{"{", RowBox[{"d", ",", "0", ",", ".01"}], "}"}]}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "/.", " ", "shootsoln"}]}]], "Input", CellChangeTimes->{{3.586884488483181*^9, 3.586884534304802*^9}, { 3.5868845996405387`*^9, 3.5868846123432655`*^9}, {3.5868847538773603`*^9, 3.586884785780185*^9}}], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"ParametricNDSolve", "::", "ndsz"}], "MessageName"], RowBox[{ ":", " "}], "\<\"At \[NoBreak]\\!\\(x$334\\)\[NoBreak] == \ \[NoBreak]\\!\\(0.9578978323627015`\\)\[NoBreak], step size is effectively \ zero; singularity or stiff system suspected. \\!\\(\\*ButtonBox[\\\"\ \[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/ParametricNDSolve\\\", ButtonNote -> \ \\\"ParametricNDSolve::ndsz\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{{3.5868845223061156`*^9, 3.5868845355768747`*^9}, 3.586884614196371*^9, {3.5868847725434284`*^9, 3.5868847868692474`*^9}, 3.5868879857752147`*^9, 3.586950453766449*^9, 3.586950739247778*^9, 3.586951434749558*^9, 3.5870358522770443`*^9, 3.5870364466590405`*^9, 3.587037444196097*^9, 3.5870411054446125`*^9, 3.587153061015376*^9, 3.587206180957905*^9, 3.5872082126231093`*^9, 3.5872083827278385`*^9, 3.587208451204756*^9, 3.5872085718436556`*^9, 3.587208675193567*^9, 3.5872087381811695`*^9, 3.58720904240357*^9, 3.587290351193783*^9, 3.587295733846653*^9, 3.5872959664709578`*^9, 3.5872960274794474`*^9, 3.5872962015864058`*^9}], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"InterpolatingFunction", "::", "dmval"}], "MessageName"], RowBox[{ ":", " "}], "\<\"Input value \[NoBreak]\\!\\({1}\\)\[NoBreak] lies outside \ the range of data in the interpolating function. 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]], Cell[CellGroupData[{ Cell[" P-N Junction Transport Equations", "Subtitle", TextAlignment->Center, FontSlant->"Italic"], Cell["\<\ Modeling of p-n junctions is typically performed by dividing the junction \ into neutral regions near the contacts and a depletion region in the center. \ The transition between these regions is neglected. 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In \ this section, those approximations are not made.\ \>", "Text", CellChangeTimes->{{3.587037859043825*^9, 3.5870379035353694`*^9}, { 3.5872063756970434`*^9, 3.58720648997958*^9}, {3.587206568048045*^9, 3.5872065820218444`*^9}, {3.587206732601457*^9, 3.587206760347044*^9}}], Cell["\<\ The equations describing the electron and hole transport in a p-n junction \ cannot be solved by the shooting method, but can be solved with \ quasi-linearization, as the dependence of the carrier densities on the \ quasi-Fermi levels is exponential.\ \>", "Text", CellChangeTimes->{{3.58720678300334*^9, 3.587206802346446*^9}}], Cell[TextData[{ "The two following equations give the electron density n as a function of \ potential v and the electron quasi-Fermi level \[Phi]n and the hole density p \ as a function of the potential and the hole quasi-Fermi level \[Phi]p. ni is \ the intrinsic carrier density, \[Beta] = ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"k", " ", "T"}]], TraditionalForm]]], " where k is Boltzmann's constant and T is the absolute temperature. 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