PDE Special Solutions V4 -- Documentation

Willy Hereman (2005-present)  and Robert Kragler (February 13, 2020-present)

Code PDESpecialSolutions-Jul4-2020. m has been tested with Mathematica version 12.1

Last modified by Hereman on Saturday, July 4, 2020 at 14 : 20 in Boulder.
                                                  Based on version PDESpecialSolutionsV4 - March30 - 2020 - RK - WH.m

If you saved the software in, for example, the subdirectory demoPDESpecialSolutions on drive D, then set the location with the command 
	SetDirectory[“d:\\demoPDESpecialSolutions”] 
	Example: SetDirectory[“d:\\data\\demoPDESpecialSolutions”]

(* Load the package PDESpecialSolutionsV4-Jul4-2020.m with the command *)
$Version
Clear["Global`*"]
SetDirectory[NotebookDirectory[ ] ];
Get["PDESpecialSolutionsV4-Jul4-2020.m"]

To see information about the package execute the command below

?? PDESpecialSolutions

Example 1: Korteweg-de Vries Equation

PDESpecialSolutions[D[u[x,t], t] + α u[x,t]D[u[x,t],x] + D[u[x,t],{x,3}] == 0,u[x,t], {x,t}, {α}, Verbose -> True]//.cRule

Use the Form option to select either Tanh, Sech, SechTanh, Cn or Sn.

PDESpecialSolutions[D[u[x,t], t] +α*u[x,t]*D[u[x,t],x] + D[u[x,t],{x,3}] == 0, u[x,t], {x,t}, {α}, Form->sech]//.cRule

Example 2: Boussinesq Equation (single equation)

(* Use the option Verbose -> True to see a trace of the main steps of the algorithm. *)

PDESpecialSolutions[{D[u[x,t],{t,2}]-D[u[x,t],{x,2}]+3u[x,t]*D[u[x,t],{x,2}]+3(((D[u[x,t],x]))^2)+α D[u[x,t],{x,4}] == 0}, 
                                         {u[x,t]}, {x,t}, {α}, Verbose->True]//.cRule

Example 3: The Boussinesq System

PDESpecialSolutions[{D[u[x,t],t]+D[v[x,t],x] == 0, 
                                   D[v[x,t], t]+D[u[x,t], x]-3u[x,t]*D[u[x,t],x]-α D[u[x,t],{x,3}] == 0},
                                   {u[x,t],v[x,t]}, {x,t}, {α}]//.cRule //Flatten//ColumnForm

Use the SolveAlgebraicSystem -> False option to output the algebraic system for the coefficients.

PDESpecialSolutions[{D[u[x,t],t]+D[v[x,t],x] == 0, 
                                         D[v[x,t], t]+D[u[x,t], x]-3u[x,t]*D[u[x,t],x]-α D[u[x,t],{x,3}] == 0},             
                                        {u[x,t],v[x,t]}, {x,t}, {α},SolveAlgebraicSystem->False] (*//.cRule *)

Example 4: Coupled KdV Equations

PDESpecialSolutions[{D[u[x,t],t] -α (6  u[x,t]  D[u[x,t],x]  +  D[u[x,t], {x,3}]) +2 β  v[x,t]D[v[x,t], x]  ==0,    
                                        D[v[x,t],t] + 3u[x,t]D[v[x,t],x] +D[v[x,t],{x,3}] == 0},
                                           {u[x,t], v[x,t]},{x,t},{α, β}, Form->Sech]//.cRule

Example 5: A Modified 3-D KdV Equation

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}] == 0,
                                           u[x,y,z,t], {x,y,z,t}, {}, Form->Sech]//.cRule

Example 6:  Equation due to Gao and Tian

PDESpecialSolutions[{D[u[x,t], t] - D[u[x,t], x] - 2v[x,t] == 0, 
                                           D[v[x,t], t] + 2u[x,t]*w[x,t] == 0, 
                                           D[w[x,t], t] + 2*u[x,t]*v[x,t] == 0},
                                          {u[x,t], v[x,t], w[x,t]}, {x,t}, {}, Form->SechTanh]//.cRule//Flatten//ColumnForm

Example 7: Zakharov-Kuznetsov KdV-type Equations

PDESpecialSolutions[ D[u[x,y,z,t],t] + α 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] ==0,u[x,y,z,t],{x,y,z,t},{α}]//.cRule

PDESpecialSolutions[D[u[x,y,z,t],t] + α 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] ==0,u[x,y,z,t],{x,y,z,t},{α}, Form->Sech]//.cRule

Example 8: Modified KdV-ZK Equation (due to Das and Verheest)

PDESpecialSolutions[D[u[x,y,z,t], t] +α  ((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] ==0,u[x,y,z,t],{x,y,z,t},{α}]//.cRule

PDESpecialSolutions[D[u[x,y,z,t], t] +α * 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] ==0,u[x,y,z,t],{x,y,z,t},{α}, Form->Sech]//.cRule

Example 9: Generalized Kuramoto-Sivashinsky Equation

PDESpecialSolutions[D[u[x,t],t] + u[x,t] D[u[x,t],x] + D[u[x,t],{x,2}]+α  D[u[x,t],{x,3}] + D[u[x,t],{x,4}] == 0,
                                          u[x,t],{x,t},{α}]//.cRule

Example 10: Coupled KdV Equations

PDESpecialSolutions[{D[u[x,t],t] -α (6 * u[x,t] D[u[x,t],x]  +  D[u[x,t], {x,3}]) +2 β  v[x,t]* D[v[x,t], x]  ==0,
                                         D[v[x,t],t] + 3u[x,t]D[v[x,t],x] +D[v[x,t],{x,3}] == 0},
                                        {u[x,t], v[x,t]},{x,t},{α, β}, Form->Cn]//.cRule

Example 11: Another Coupled KdV Systems (due to Guha and Roy)

PDESpecialSolutions[{D[u[x,t],t] +  α  v[x,t]D[v[x,t],x] + β u[x,t]D[u[x,t],x] +γ D[u[x,t],{x,3}] ==0,     
                                           D[v[x,t],t] + δ D[u[x,t]v[x,t],x] +ϵ v[x,t] D[v[x,t],x] ==0},
                                            {u[x,t], v[x,t]},{x,t},{α, β, γ , δ,ϵ}, Form->Sech]//.cRule

Example 12: Fisher Equation

PDESpecialSolutions[{D[u[x,t],t] -D[u[x,t],{x,2}] -u[x,t](1-u[x,t]) == 0},{u[x,t]},{x,t},{}]//.cRule

Example 13: Gardner Equation (single equation, various signs for cubic terms)

(* Use the option Verbose -> True to see a trace of the main steps of the algorithm. *)

PDESpecialSolutions[{D[u[x,t], t] + u[x,t]D[u[x,t],x] - u[x,t]^2D[u[x,t],x] + D[u[x,t],{x,3}] == 0}, 
                                         {u[x,t]}, {x,t}, {}, Form -> Tanh,Verbose->True]//.cRule

PDESpecialSolutions[{D[u[x,t], t] + u[x,t]D[u[x,t],x] - u[x,t]^2D[u[x,t],x] + D[u[x,t],{x,3}] == 0}, 
                                         {u[x,t]}, {x,t}, {}, Form -> Sn,Verbose->True]//.cRule

(* switch sign of cubic nonlinearity from minus to plus *)

PDESpecialSolutions[{D[u[x,t], t] + u[x,t]D[u[x,t],x] + u[x,t]^2D[u[x,t],x] + D[u[x,t],{x,3}] == 0}, 
                                         {u[x,t]}, {x,t}, {}, Form -> Sech,Verbose->True]//.cRule

PDESpecialSolutions[{D[u[x,t], t] + u[x,t]D[u[x,t],x] + u[x,t]^2D[u[x,t],x] + D[u[x,t],{x,3}] == 0}, 
                                         {u[x,t]}, {x,t}, {}, Form -> Cn,Verbose->True]//.cRule

Example 14: FitzHugh-Nagumo Equation

PDESpecialSolutions[{β D[v[z],z] + (Sqrt[2])D[v[z], {z,2}] -(Sqrt[2])v[z](1-(Sqrt[2])v[z])(α-(Sqrt[2])v[z]) == 0},
                                         {v[z]},{z},{α, β}]//.cRule//ColumnForm

Example 15: A Degenerate Hamiltonian System

PDESpecialSolutions[{D[u[x,t],t] -D[u[x,t],x] -2v[x,t] == 0,D[v[x,t],t] -2ϵ u[x,t]*v[x,t] ==0},
                                           {u[x,t],v[x,t]},{x,t},{ϵ}, Form->SechTanh]//.cRule

Example 16: The Combined Burgers-KdV Equation

PDESpecialSolutions[D[u[x,t],t] +α*u[x,t]*D[u[x,t],x] +β*D[u[x,t],{x,2}] + D[u[x,t], {x,3}] ==0,
                                           u[x,t],{x,t},{α, β}]//.cRule

Example 17: Duffing Equation

PDESpecialSolutions[{D[u[x], {x, 2}] + u[x] - α ((u[x])^3) == 0}, u[x], {x}, {α}, Form->Cn]//.cRule

PDESpecialSolutions[{D[u[x], {x, 2}] + u[x] - α ((u[x])^3) == 0}, u[x], {x}, {α}, Form->Tanh]//.cRule

Example 18: Another Fisher-type Equation

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))

PDESpecialSolutions[{generalizedFisherEquation ==0}, u[x,t], {x,t}, {}, Form->Tanh, Verbose -> True]//.cRule

Transform the generalized Fisher equation by setting  u[x,t] = ((v[x,t])^((2/3)))

transformedGeneralizedFisherEquation = generalizedFisherEquation /.
{u[x,t]  -> ((v[x,t])^((2/3))), Derivative[m_,n_][u][x,t] -> D[((v[x,t])^((2/3))),{x,m},{t,n}]}

transformedFisherEquation = Factor[transformedGeneralizedFisherEquation]//Numerator

PDESpecialSolutions[{transformedFisherEquation ==0}, v[x,t], {x,t}, {}, Form->Tanh, Verbose -> True]//.cRule//
Flatten//ColumnForm