9 years, 14 days

## Finding an approximate function for a PD...

Maple 2015

Are there any commands in maple that will help me find a suitable function that approximates the numerical solution of:

 > restart;   PDE := diff(v(x, t), t) = diff(v(x, t), x, x);   JACOBIINTEGRAL := int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t);   IBC:= D[1](v)(0,t)=0,         D[1](v)(1,t)=-0.000065*v(1, t)^4,         v(x,0)=1; # # For x=0..1, t=0..1, the solution varies only very slowly # so I have increased the timestep/spacestep, just to speed # up results generation for diagnostic purposes #   pds := pdsolve( PDE, [IBC], numeric, time = t, range = 0 .. 1,                   spacestep = 0.1e-1, timestep = 0.1e-1,                   errorest=true                 )
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 > # # Plot the solution over the ranges x=0..1, # time=0..1. Not a lot happens! #   pds:-plot(x=1, t=0..1);
 > # # Plot the estimated error over the ranges x=0..1, # time=0..1 #   pds:-plot( err(v(x,t)), x=1,t=0..1);
 > # # Get some numerical solution values #   pVal:=pds:-value(v(x,t), output=procedurelist):   for k from 0 by 0.1 to 1 do       pVal(1, k)[2], pVal(1, k)[3];   od;
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I am refering to the first graph, is there a way in maple to find an explicit suitable approximating function?

I.e, I want the function to have the same first graph obviously, it seems like addition of exponent and a line function, I tried plotting exp(-t)-0.3*t, it doesn't look like it approximates it very well. Any suggestion on how to implement this task in maple?

Thanks.

## Problem in plotting ...

Maple

I have the solution to a heat PDE, v and the error esitmate u + cos(x+t) = v

I want to plot log v(1,t) as function of log u(1,t) in maple, but I seem to get an error:

Error, (in plot) unexpected option: ln(u(1, t))

I am attaching my code below.

How to fix this problem?

PDE+cos.mw

## Solving a nonlinear hyperbolic PDE...

Maple 2015

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:

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## Wave PDE equation...

Maple

I have the following PDE:

u_xx = u_tt + (2^{1/2}u_x-u)^{1/2}

Do you have a proposed algorithm to solve in maple for this PDE? I mean pdsolve won't solve it because it's a nonlinear PDE.

## PDE to solve numerically...

Maple 2015

I want to solve numerically the PDE:

u_xx + u_yy= = u^{1/2}+(u_x)^2/(u)^{3/2}

My assumptions are that  |sqrt(2)u_x/u|<<1 (but I cannot neglect the first term since its in my first order approximation of another PDE.

So I tried solving by using pdsolve in maple, but to no cigar.

Here's the maple file:

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