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Use options in the Actions Preferences dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics3D-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, RenderingOptions->{"RenderThickness"->False}]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Example 2.1"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " can be used to show directly that the expression in Equation (2.37) \ solves the differential equation in (2.30) for any differentiable function ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[". We first define the function ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" according to the general solution in (2.37):\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData["f=F[x/y,x/u];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "\nThe general solution is now substituted into the differential equation \ in (2.30) and ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Simplify", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is used to show that an identity results for any function ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[":\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{Cell[TextData["Simplify[x D[f,x] + y D[f,y] + u D[f,u] == 0]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "\nAnalogous steps can be carried out to verify that the expression ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u=xG[x/y]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" solves Equation (2.27) for any function ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " that can be differentiated.\n\nThe transformation of (2.27) carried out \ after Equation (2.38) can also be performed within ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[". The transformed function ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is defined as\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u[x,y]=xG[x/y,y]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " and substituted into the original differential equation in (2.27). ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Simplify", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " is then used to combine terms to obtain the transformed differential \ equation.", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->14, FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData["u=x G[x/y,y];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Simplify[x D[u,x] + y D[u,y] - u == 0]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ x*y*Derivative[0, 1][G][x/y, y] == 0\ \>", "\<\ (0,1) x x y G [-, y] == 0 y\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["Clear[u]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Example 2.3"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The time development of the solution in (2.60) from a particular initial \ condition can be constructed with remarkable ease using ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[". Suppose the initial condition in (2.54) is given by", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[" f[x]=1-x^2", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", if ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["|x|", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is less than or equal to 1, and ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["f[x]=0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", if ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["|x|", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " is greater than 1. This initial function has been chosen because the \ solution in (2.60) can be solved for u as a function of ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " (Problem 18). However, the procedure described below for constructing \ the solution using ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " can be applied to initial functions that do not permit such an explicit \ solution.\n\nThe evolution of the point ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x=x0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u=f[x0]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" on the initial function at ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t=0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" can be determined from (2.56) and (2.57). At ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t=0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", we have ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["c2=x0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", so at time ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" the\npoint initially at ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" has moved to\n\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x = x0 + f[x0] t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ "\n\nThese are the characteristics of Equation (2.53).\n\nTo construct the \ characteristics and the evolution of the solution in ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ ", the initial profile (2.61) and the characteristics (2.62) are first \ entered as", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData["f[x_]:=If[Abs[x]<1,1-x^2,0]"], "Input", AspectRatioFixed->True], Cell[TextData["x[t_,x0_]:=x0+f[x0]t"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "\nBoth the characteristics and the evolution of the solution can now be \ displayed\nwith ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["ParametricPlot", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ ". The trajectory of the characteristic emanating from the\npoint ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x0=0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" up to time ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t=1", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is displayed with\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ StyleBox["\t\t\t\t\t\t\t\t\t", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["ParametricPlot[{x[t,0],t},{t,0,1}]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\nSimilarly, the solution at time t over the region", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[" -2", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox["<", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox["x0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox["<", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is displayed with the command\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["u[t_]:=ParametricPlot[{x[t,x0],f[x0]},{x0,-2,2}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["\nNotice that in generating the solution with ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["ParametricPlot", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" the quantity being\nplotted is the pair of points ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["{x[x0,t],f[x0]}", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", which corresponds to the position at time ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" of the point on the profile ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u[x,t]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" that was initially located at ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ ". Thus, this construction represents a point-by-point assembly of the \ solution at time ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " in terms of the initial profile and the characteristics.\n\nThe \ trajectories of several characteristics projected on to the ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["x-t", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " plane and the early stages of evolution of the solution from the initial \ profile (2.61) are shown in Figure 2.7. For short times the initial solution \ is deformed because the characteristics move with different speeds for \ different values of ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ ". At the point where two characteristics first cross, the solution \ becomes double-valued, and the slope in the ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["u-x", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " plane becomes infinite. From this point onward, the solution is no \ longer a continuous function of position, and corresponds to the onset of a \ 'shock' wave. This type of behavior is due to the form of the \ characteristics and not to a specific choice of initial condition", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14]}, Open]], Cell[CellGroupData[{Cell[TextData["Figure 2.7"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The top panel of Figure 2.7 can be generated by using the command ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Show:\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "u[t_]:=ParametricPlot[{x[t,x0],f[x0]},{x0,-2,2},\n \ DisplayFunction->Identity]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Show[{u[0],u[1],u[2]},AxesLabel->{x,u},\n \ DisplayFunction->$DisplayFunction]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ 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