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Cell[BoxData[
    \( (*\ Last\ modified\ by\ Hereman\ on\ Sunday, \ March\ 21, \ 
      2010\ at\ 17 : 35\ *) \)], "Input"],

Cell[BoxData[
    \( (*\ If\ you\ saved\ the\ software\ in, \ for\ example, \ 
      the\ subdirectory\ demopdespecialsolutions\ on\ drive\ D, \ 
      then\ set\ the\ location\ with\ the\ command\ *) \)], "Input"],

Cell[BoxData[
    \(\(\(SetDirectory["\<d:\\demopdespecialsolutions\>"]\)\(\ \)\)\)], \
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Cell[BoxData[
    \( (*\ Example : \ 
        SetDirectory["\<c:\\data\\demoPDESpecialSolutions\>"]\ *) \)], "Input"],

Cell[BoxData[
    \(SetDirectory["\<e:\\data\\demoPDESpecialSolutions\\\
AnalyzeSolveMarch2010\>"]\)], "Input"],

Cell[BoxData[
    \( (*\ Load\ the\ package\ PDESpecialSolutionsV2 . 
          m\ with\ the\ command\ *) \)], "Input"],

Cell[BoxData[
    \(Get["\<PDESpecialSolutionsV3.m\>"]\)], "Input"],

Cell[BoxData[
    \( (*\ 
      To\ see\ information\ about\ the\ package\ execute\ the\ command\ below\
\ *) \)], "Input"],

Cell[BoxData[
    \(?? \ PDESpecialSolutions\)], "Input"],

Cell[BoxData[
    \( (*\ 
      Investigation\ of\ exact\ solutions\ of\ the\ Verhulst\ equation\ for\ \
logistic\ growth\ *) \)], "Input"],

Cell[BoxData[
    \( (*\ Verhulst\ equation\ 1, \ 
      in\ standard\ variables\ r\  = \ intrinsic\ growth\ rate, \ 
      k\  = \ carrying\ capacity\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x], \ x]\  - \((r/k)\)*\ 
              u[x]*\((\ k - u[x])\)\  \[Equal] \ 0}, \ 
      u[x], \ {x}, \ {r, k}, \ Form \[Rule] Tanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Now\ define\ the\ solution\ \(\(via\)\(:\)\)\ *) \)], "Input"],

Cell[BoxData[
    \(uu[x_] := 
      1\/2\ k\ \((1 + Tanh[1\/2\ \((2\ phase + r\ x)\)])\)\)], "Input"],

Cell[BoxData[
    \( (*\ Compute\ the\ derivative\ of\ the\ solution\ *) \)], "Input"],

Cell[BoxData[
    \(D[uu[x], x]\)], "Input"],

Cell[BoxData[
    \( (*\ Verhulst\ equation\ 2\ as\ given\ in\ Liner' 
        s\ article\ in\ ``The\ Leading\ Edge \*"\"\< \>"*) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x], \ x]\  - 
            a1*\ u[x]\  + \ \((a1/a2)\)*u[x]^2\  \[Equal] \ 0}, \ 
      u[x], \ {x}, \ {a1, a2}, \ Form \[Rule] Tanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 1 : \ Korteweg - de\ Vries\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], \ t]\  + \ alpha*u[x, t]*D[u[x, t], x]\  + \ 
          D[u[x, t], {x, 3}]\  \[Equal] \ 0, \ 
      u[x, t], \ {x, t}, \ {alpha}, \ Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Use\ the\ Form\ option\ to\ select\ either\ Tanh, \ Sech, \ 
      SechTanh, \ Cn\ or\ \(\(Sn\)\(.\)\)\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], \ t]\  + \ alpha*u[x, t]*D[u[x, t], x]\  + \ 
          D[u[x, t], {x, 3}]\  \[Equal] \ 0, \ 
      u[x, t], \ {x, t}, \ {alpha}, \ Form \[Rule] sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 2 : \ 
        Boussinesq\ Equation\ \((single\ equation)\)\ *) \)], "Input"],

Cell[BoxData[
    \( (*\ Use\ the\ option\ Verbose\  \[Rule] \ 
        True\ to\ see\ a\ trace\ of\ the\ main\ steps\ of\ the\ \(\(algorithm\
\)\(.\)\)\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] + 
            3*u[x, t]*D[u[x, t], {x, 2}] + 3*\((D[u[x, t], x])\)^2 + 
            alpha*D[u[x, t], {x, 4}]\  \[Equal] \ 0}, \ {u[x, t]}, \ {x, 
        t}, \ {alpha}, \ Verbose \[Rule] True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 3 : \ The\ Boussinesq\ System\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], t] + D[v[x, t], x]\  \[Equal] \ 0, \ 
        D[v[x, t], \ t] + D[u[x, t], \ x] - 3*u[x, t]*D[u[x, t], x] - 
            alpha*D[u[x, t], {x, 3}]\  \[Equal] \ 0}, {u[x, t], 
        v[x, t]}, \ {x, t}, \ {alpha}]\)], "Input"],

Cell[BoxData[
    \( (*\ Use\ the\ SolveAlgebraicSystem\  \[Rule] \ 
        False\ option\ to\ output\ the\ algebraic\ system\ for\ the\ \
\(\(coefficients\)\(.\)\)\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], t] + D[v[x, t], x]\  \[Equal] \ 0, \ 
        D[v[x, t], \ t] + D[u[x, t], \ x] - 3*u[x, t]*D[u[x, t], x] - 
            alpha*D[u[x, t], {x, 3}]\  \[Equal] \ 0}, \ {u[x, t], 
        v[x, t]}, \ {x, t}, \ {alpha}, 
      SolveAlgebraicSystem \[Rule] False]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 4 : \ Coupled\ KdV\ Equations\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], t]\  - 
            alpha\ *\ \((6\ *\ u[x, t]\ *\ D[u[x, t], x]\ \  + \ \ D[
                    u[x, t], \ {x, 3}])\)\  + 
            2\ *\ beta\ *\ v[x, t]*\ D[v[x, t], \ x]\ \  \[Equal] 0, 
        D[v[x, t], t]\  + \ 3*u[x, t]*D[v[x, t], x]\  + 
            D[v[x, t], {x, 3}]\  \[Equal] \ 0}, {u[x, t], \ v[x, t]}, {x, 
        t}, {alpha, \ beta}, \ Form \[Rule] Sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 5 : \ A\ Modified\ 3 - D\ KdV\ Equation\ *) \)], "Input"],

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]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 6 : \ \ Equation\ due\ to\ Gao\ and\ Tian\ *) \)], "Input"],

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[
    \( (*\ Example\ 7 : \ 
        Zakharov - Kuznetsov\ KdV - type\ Equations\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, y, z, t], t]\  + \ alpha*u[x, y, z, t]*D[u[x, y, z, t], x]\  + 
          D[u[x, y, z, t], \ {x, 3}] + D[u[x, y, z, t], x, \ y, y]\  + 
          D[u[x, y, z, t], \ x, \ z, \ z]\  \[Equal] 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha}]\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, y, z, t], t]\  + \ alpha*u[x, y, z, t]*D[u[x, y, z, t], x]\  + 
          D[u[x, y, z, t], \ {x, 3}] + D[u[x, y, z, t], \ x, \ y, y]\  + 
          D[u[x, y, z, t], \ x, \ z, \ z]\  \[Equal] 0, 
      u[x, y, z, t], {x, y, z, t}, \[IndentingNewLine]{alpha}, \ 
      Form \[Rule] Sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 8 : \ 
        Modified\ KdV - 
          ZK\ Equation\ \((due\ to\ Das\ and\ Verheest)\)\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, y, z, t], \ t]\  + 
          alpha\ *\ u[x, y, z, t]\ ^\ 2\ *\ D[u[x, y, z, t], x]\  + \ 
          D[u[x, y, z, t], \ {x, 3}] + D[u[x, y, z, t], \ x, \ y, y]\  + 
          D[u[x, y, z, t], \ x, \ z, \ z]\  \[Equal] 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha}]\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, y, z, t], \ t]\  + 
          alpha\ *\ u[x, y, z, t]\ ^\ 2\ *\ D[u[x, y, z, t], x]\  + \ 
          D[u[x, y, z, t], \ {x, 3}] + D[u[x, y, z, t], \ x, \ y, y]\  + 
          D[u[x, y, z, t], \ x, \ z, \ z]\  \[Equal] 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha}, \ Form \[Rule] Sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 9 : \ 
        Generalized\ Kuramoto - Sivashinsky\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + \ u[x, t]*\ D[u[x, t], x]\  + \ D[u[x, t], {x, 2}] + 
          alpha*\ D[u[x, t], {x, 3}]\  + \ D[u[x, t], {x, 4}]\  \[Equal] \ 0, 
      u[x, t], {x, t}, {alpha}]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 10 : \ Coupled\ KdV\ Equations\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], t]\  - 
            alpha\ *\ \((6\ *\ u[x, t]\ *\ D[u[x, t], x]\ \  + \ \ D[
                    u[x, t], \ {x, 3}])\)\  + 
            2\ *\ beta\ *\ v[x, t]*\ D[v[x, t], \ x]\ \  \[Equal] 0, 
        D[v[x, t], t]\  + \ 3*u[x, t]*D[v[x, t], x]\  + 
            D[v[x, t], {x, 3}]\  \[Equal] \ 0}, {u[x, t], \ v[x, t]}, {x, 
        t}, {alpha, \ beta}, \ Form \[Rule] Cn]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 11 : \ 
        Another\ Coupled\ KdV\ Systems\ \((due\ to\ Guha\ and\ Roy)\)\ *) \
\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], t]\  + \ \ \[Alpha]\ *\ v[x, t]*
              D[v[x, t], x]\  + \ \[Beta]*u[x, t]*
              D[u[x, t], x]\  + \[Gamma]\ *\ D[u[x, t], {x, 3}]\  \[Equal] 0, 
        D[v[x, t], t]\  + \ \[Delta]*D[u[x, t]*v[x, t], x]\  + \[Epsilon]*
              v[x, t]*D[v[x, t], x]\  \[Equal] 0}, {u[x, t], \ v[x, t]}, {x, 
        t}, {\[Alpha], \ \[Beta], \ \[Gamma]\ , \ \[Delta], \[Epsilon]}, \ 
      Form \[Rule] Sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 12 : \ Fisher\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(\(\(\ \)\(PDESpecialSolutions[{D[u[x, t], t]\  - D[u[x, t], {x, 2}]\  - 
            u[x, t]*\((1 - u[x, t])\)\  \[Equal] \ 0}, {u[x, t]}, {x, 
        t}, {}]\)\)\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 14 : \ FitzHugh - Nagumo\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{beta*D[v[z], z]\  + \ 
            Sqrt[2]*D[v[z], \ {z, 2}]\  - 
            Sqrt[2]*v[
                z]*\((1 - Sqrt[2]*v[z])\)*\((alpha - 
                  Sqrt[2]*v[z])\)\  \[Equal] \ 0}, {v[z]}, {z}, {alpha, \ 
        beta}]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 15 : \ A\ Degenerate\ Hamiltonian\ System\ *) \)], "Input"],

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*epsilon*u[x, t]*v[x, t]\  \[Equal] 0}, {u[x, t], 
        v[x, t]}, {x, t}, {epsilon}, \ Form \[Rule] SechTanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 16 : \ 
        The\ Combined\ KdV - mKdV\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + 6*alpha*u[x, t]*D[u[x, t], x]\  + 
          6*beta*u[x, t]^2*D[u[x, t], x]\  + 
          gamma*D[u[x, t], \ {x, 3}]\  \[Equal] 0, 
      u[x, t], {x, t}, {alpha, \ beta, \ gamma}]\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + 6*alpha*u[x, t]*D[u[x, t], x]\  + 
          6*beta*u[x, t]^2*D[u[x, t], x]\  + 
          gamma*D[u[x, t], \ {x, 3}]\  \[Equal] 0, 
      u[x, t], {x, t}, {alpha, \ beta, \ gamma}, \ Form \[Rule] SechTanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 17 : \ Duffing\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x], \ {x, \ 2}]\  + \ u[x]\  - \ 
            alpha*u[x]^3\  \[Equal] \ 0}, \ u[x], \ {x}, \ {alpha}, \ 
      Form \[Rule] Cn]\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x], \ {x, \ 2}]\  + \ u[x]\  - \ 
            alpha*u[x]^3\  \[Equal] \ 0}, \ u[x], \ {x}, \ {alpha}, \ 
      Form \[Rule] Tanh]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 18 : \ Another\ Fisher - type\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(generalizedfisherequation\  = \ \ u[x, t]*D[u[x, t], t]\  - 
        u[x, t]*D[u[x, t], {x, 2}]\  - 
        D[u[x, t], 
            x]^2 - \(\(u[x, t]^2\)\(*\)\((1 - u[x, t]^3)\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{generalizedfisherequation\  \[Equal] 0}, \ 
      u[x, t], \ {x, t}, \ {}, \ Form \[Rule] Tanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Transform\ the\ generalized\ Fisher\ equation\ by\ setting\ u[x, 
            t]\  = \ v[x, t]^\((2/3)\)\ *) \)], "Input"],

Cell[BoxData[
    \(transformedgeneralizedfisherequation\  = \ 
      generalizedfisherequation\  /. \[IndentingNewLine]{u[x, 
              t]\ \  \[Rule] \ 
            v[x, t]^\((2/3)\), \ \(\(Derivative[m_, n_]\)[u]\)[x, 
              t]\  \[Rule] \ D[v[x, t]^\((2/3)\), {x, m}, {t, n}]}\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(transformedfisherequation\  = 
      Numerator[\ 
        Factor[transformedgeneralizedfisherequation]]\)\)\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{transformedfisherequation\  \[Equal] 0}, \ 
      v[x, t], \ {x, t}, \ {}, \ Form \[Rule] Tanh, \ 
      Verbose\  \[Rule] \ True]\)], "Input"]
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