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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 39753, 1010]*) (*NotebookOutlinePosition[ 40645, 1040]*) (* CellTagsIndexPosition[ 40556, 1034]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Welcome to the PDESpecialSolutions Package for Mathematica 4.1: By Douglas Baldwin, Unal Goktas and Willy Hereman \[Copyright] 2001\ \>", "Subtitle"], Cell[BoxData[ \(SetDirectory["\"]\)], "Input"], Cell[BoxData[ \(<< PDESpecialSolutions.m\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(?? \ PDESpecialSolutions\)], "Input"], Cell[BoxData[ \("PDESpecialSolutions[eqns, funcs, vars, params, opts] solves a system \ of nonlinear partial differential equations (PDEs) for funcs, with \ independent variables vars and non-zero parameters params. \n\ PDESpecialSolutions takes the option Form with the default value Tanh. 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To test the \ solutions set either the NumericTest option to True, or set the SymbolicTest \ option to True, or both. \ \>", "Message"], Cell[BoxData[ InterpretationBox[\("Time Used:"\[InvisibleSpace]0.4299999999999926`\), SequenceForm[ "Time Used:", .42999999999999261], Editable->False]], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("The list of possible solutions:"\)], "Print"], Cell[BoxData[ \({{{u[x, t] \[Rule] \(-\(\(\(-c[1]\^2\) - 8\ alpha\ c[1]\^4 + c[2]\^2 + 12\ alpha\ c[1]\^4\ Tanh[phase + x\ c[1] + t\ \ c[2]]\^2\)\/\(3\ c[1]\^2\)\)\), v[x, t] \[Rule] a[2, 0] + 4\ alpha\ c[1]\ c[ 2]\ Tanh[phase + x\ c[1] + t\ c[2]]\^2}}}\)], "Output"] }, Open ]], Cell["\<\ Modified 3D KdV is an example of a single nonlinear PDE which yields two \ Sech solution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[ D[u[x, y, z, t], \ t]\ + \ 6*u[x, y, z, t]^2*D[u[x, y, z, t], x]\ + \ D[u[x, y, z, t], {x, 1}, {y, 1}, {z, 1}]\ \[Equal] \ 0, \ u[x, y, z, t], \ {x, y, z, t}, \ {}, \ Form \[Rule] Sech, \ Test \[Rule] True, \ SymbolicTest\ \[Rule] \ True]\)], "Input"], Cell[BoxData[ \(General::"spell" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(Test\)\" is similar to \ existing symbols \!\({Nest, Rest, Text}\)."\)], "Message"], Cell[BoxData[ \("Symbolically testing the solutions."\)], "Print"], Cell[BoxData[ \({{{u[x, y, z, t] \[Rule] \(-\@c[2]\)\ \@c[3]\ Sech[\(phase\ c[2]\ c[3] + y\ \ c[2]\^2\ c[3] + z\ c[2]\ c[3]\^2 - x\ c[4] + t\ c[2]\ c[3]\ c[4]\)\/\(c[2]\ \ c[3]\)]}}, {{u[x, y, z, t] \[Rule] \@c[2]\ \@c[3]\ Sech[\(phase\ c[2]\ c[3] + y\ \ c[2]\^2\ c[3] + z\ c[2]\ c[3]\^2 - x\ c[4] + t\ c[2]\ c[3]\ c[4]\)\/\(c[2]\ \ c[3]\)]}}}\)], "Output"] }, Open ]], Cell["\<\ Gao and Tian is a system of three equations which admit numerous mixed \ solutions of the mixed Sech-Tanh case. This one takes a while. In the mixed \ case, constant solutions are not removed.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{D[u[x, t], \ t]\ - \ D[u[x, t], \ x]\ - \ 2*v[x, t]\ \[Equal] \ 0, \ D[v[x, t], \ t]\ + \ 2*u[x, t]*w[x, t]\ \[Equal] \ 0, \ D[w[x, t], \ t]\ + \ 2*u[x, t]*v[x, t]\ \[Equal] \ 0}, \ {u[x, t], \ v[x, t], \ w[x, t]}, \ {x, t}, \ {}, \ Form\ \[Rule] \ SechTanh]\)], "Input"], Cell[BoxData[ \("Still building the algebraic system, please wait."\)], "Print"], Cell["\<\ These solutions are not being tested numerically or symbolically. To test the \ solutions set either the NumericTest option to True, or set the SymbolicTest \ option to True, or both. \ \>", "Message"], Cell["\<\ The following simplification rules are being used: {Sqrt[a^2]->a, \ Sqrt[-a^2]->I*a}\ \>", "Message"], Cell[BoxData[ \({{{u[x, t] \[Rule] \(-\(\(\@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ Sech[ phase + x\ c[1] + t\ c[2]]\)\/\@\(\(-c[1]\) + c[2]\)\)\), v[x, t] \[Rule] 1\/2\ \@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ \@\(\(-c[1]\) + c[2]\ \)\ Sech[phase + x\ c[1] + t\ c[2]]\ Tanh[phase + x\ c[1] + t\ c[2]], w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[ 2]\ \((\(-1\) + 2\ Sech[phase + x\ c[1] + t\ c[2]]\^2)\)}, {u[ x, t] \[Rule] \(\@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ \ Sech[phase + x\ c[1] + t\ c[2]]\)\/\@\(\(-c[1]\) + c[2]\), v[x, t] \[Rule] \(-\(1\/2\)\)\ \@c[2]\ \@\(\((c[1] - c[2])\)\ \ c[2]\)\ \@\(\(-c[1]\) + c[2]\)\ Sech[phase + x\ c[1] + t\ c[2]]\ Tanh[ phase + x\ c[1] + t\ c[2]], w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[ 2]\ \((\(-1\) + 2\ Sech[phase + x\ c[1] + t\ c[2]]\^2)\)}, {u[ x, t] \[Rule] \(-\(\(\((c[1] - c[2])\)\ c[ 2]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \@\(-\((c[1] - c[2])\)\^2\)\ \)\)\), v[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(\((c[1] - c[2])\)\ c[2]\ \((Sech[phase + x\ c[1] + \ t\ c[2]] - \[ImaginaryI]\ Tanh[phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \ \@\(-\((c[1] - c[2])\)\^2\)\), v[x, t] \[Rule] 1\/4\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(-\(\(\((c[1] - c[2])\)\ c[ 2]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \@\(-\((c[1] - c[2])\)\^2\)\ \)\)\), v[x, t] \[Rule] 1\/4\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(\((c[1] - c[2])\)\ c[2]\ \((Sech[phase + x\ c[1] + \ t\ c[2]] + \[ImaginaryI]\ Tanh[phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \ \@\(-\((c[1] - c[2])\)\^2\)\), v[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(-\(\(\@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ \ Tanh[phase + x\ c[1] + t\ c[2]]\)\/\(c[1] - c[2]\)\)\), v[x, t] \[Rule] 1\/2\ \@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ Sech[phase + x\ \ c[1] + t\ c[2]]\^2, w[x, t] \[Rule] \(-\(1\/2\)\)\ \((c[1] - c[2])\)\ c[ 2]\ Sech[phase + x\ c[1] + t\ c[2]]\^2}, {u[x, t] \[Rule] \(\@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ \ Tanh[phase + x\ c[1] + t\ c[2]]\)\/\(c[1] - c[2]\), v[x, t] \[Rule] \(-\(1\/2\)\)\ \@c[2]\ \@\(\((c[1] - c[2])\)\^2\ \ c[2]\)\ Sech[phase + x\ c[1] + t\ c[2]]\^2, w[x, t] \[Rule] \(-\(1\/2\)\)\ \((c[1] - c[2])\)\ c[ 2]\ Sech[phase + x\ c[1] + t\ c[2]]\^2}, {u[x, t] \[Rule] a[1, 0] + a[1, 1]\ Sech[phase + t\ c[2] + x\ c[2]] + a[1, 2]\ Sech[phase + t\ c[2] + x\ c[2]]\^2 + b[1, 0]\ Tanh[phase + t\ c[2] + x\ c[2]] + b[1, 1]\ Sech[phase + t\ c[2] + x\ c[2]]\ Tanh[ phase + t\ c[2] + x\ c[2]], v[x, t] \[Rule] 0, w[x, t] \[Rule] 0}}}\)], "Output"] }, Open ]], Cell["\<\ Duffing Equation (ODE) yields solutions of the Jacobi elliptic CN and SN \ functions. Both are shown. In the Jacobi elliptic function options, \ constant solutions are not removed.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{D[u[x], {x, 2}] + u[x] + alpha*u[x]^3 \[Equal] 0}, \ u[x], {x}, {alpha}, \ Form \[Rule] Cn, \ InputForm \[Rule] False]\)], "Input"], Cell["\<\ These solutions are not being tested numerically or symbolically. To test the \ solutions set either the NumericTest option to True, or set the SymbolicTest \ option to True, or both. \ \>", "Message"], Cell[BoxData[ \({{{mod \[Rule] \(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/\(2\ c[1]\^2\), u[x] \[Rule] \(-\(\(\@\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\ \ JacobiCN[phase + x\ c[1], \(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/\(2\ \ c[1]\^2\)]\)\/\@alpha\)\)}, {mod \[Rule] \(\((\(-1\) + c[1])\)\ \((1 + \ c[1])\)\)\/\(2\ c[1]\^2\), u[x] \[Rule] \(\@\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\ \ JacobiCN[phase + x\ c[1], \(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/\(2\ \ c[1]\^2\)]\)\/\@alpha}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{D[u[x], {x, 2}] + u[x] + alpha*u[x]^3 \[Equal] 0}, \ u[x], {x}, {alpha}, \ Form \[Rule] Sn, \ InputForm \[Rule] False, \ \[IndentingNewLine]NumericTest\ \[Rule] \ True, \ SymbolicTest\ \[Rule] \ True]\)], "Input"], Cell[BoxData[ \("Numerically testing the solutions."\)], "Print"], Cell[BoxData[ \("Symbolically testing the solutions."\)], "Print"], Cell[BoxData[ \({{{u[ x] \[Rule] \(-\(\(\@2\ \@\(\((\(-1\) + c[1])\)\ \((1 + \ c[1])\)\)\ JacobiSN[phase + x\ c[1], \(-\(\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/c[1]\^2\)\)]\)\/\@alpha\)\), mod \[Rule] \(-\(\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/c[1]\^2\)\)}}, {{u[ x] \[Rule] \(\@2\ \@\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\ \ JacobiSN[phase + x\ c[1], \(-\(\(\((\(-1\) + c[1])\)\ \((1 + \ c[1])\)\)\/c[1]\^2\)\)]\)\/\@alpha, mod \[Rule] \(-\(\(\((\(-1\) + c[1])\)\ \((1 + c[1])\)\)\/c[1]\^2\)\)}}}\)], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1152}, {0, 791}}, WindowSize->{1038, 755}, WindowMargins->{{1, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. 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