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Question: Solving a nonlinear hyperbolic PDE

Question:Solving a nonlinear hyperbolic PDE

Al86 145 Maple 2015
pde pdsolve
September 04 2015
1

I am considering the following PDE and I am getting an error, please suggest a better numerical method than the default one used in maple:

 

the PDE is:

u_{xx}u^3 - sin(xt)u_{tt} = u(x,t)

u(x, 0) = sin(x), (D[2](u))(x, 0) = cos(x), u(0, t) = cos(t), (D[1](u))(0, t) = sin(t)

Please suggest me a method that will also work for the following PDEs:

u^m* u_{xx} - sin(xt)u_{tt} = u^n

for m,n =0,1,2,3,... for the cases m=n and m not equal n

Here's the code:

 

> 

pde := u(x, t)^3*(diff(u(x, t), x, x))-sin(x*t)*(diff(u(x, t), t, t)) = u(x, t);

u(x, t)^3*(diff(diff(u(x, t), x), x))-sin(x*t)*(diff(diff(u(x, t), t), t)) = u(x, t)

 (1)
> 

ibc := u(x, 0) = sin(x), (D[2](u))(x, 0) = cos(x), u(0, t) = cos(t), (D[1](u))(0, t) = sin(t);

u(x, 0) = sin(x), (D[2](u))(x, 0) = cos(x), u(0, t) = cos(t), (D[1](u))(0, t) = sin(t)

 (2)
> 

pds := pdsolve(pde, [ibc], numeric, time = t, range = 0 .. 1, spacestep = 0.1e-1)

module () local INFO; export plot, plot3d, animate, value, settings; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; end module

 (3)
> 

pds:-plot3d(u(x, t), t = 0 .. 1, x = 0 .. 1, labels = [t, x, u(x, t)], labelfont = [times, bold, 20], axesfont = [times, bold, 16])

Error, (in pdsolve/numeric/plot3d) unable to compute solution for t>HFloat(0.0):
Newton iteration is not converging

  
> 

``

 

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