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Question: Finding an approximate function for a PDE numerical solution

Question:Finding an approximate function for a PDE numerical solution

Al86 150 Maple 2015
plot input pde numeric
October 31 2015
0

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
                )

diff(v(x, t), t) = diff(diff(v(x, t), x), x)

  

int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t)

  

(D[1](v))(0, t) = 0, (D[1](v))(1, t) = -0.65e-4*v(1, t)^4, v(x, 0) = 1

  

_m649569600

 (1)
> 

#
# 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;
> 

 

t = 0., v(x, t) = Float(undefined)

  

t = .1, v(x, t) = .999977377613528229

  

t = .2, v(x, t) = .999967577518313666

  

t = .3, v(x, t) = .999959874331053822

  

t = .4, v(x, t) = .999952927885405241

  

t = .5, v(x, t) = .999946262964885979

  

t = .6, v(x, t) = .999939702966688881

  

t = .7, v(x, t) = .999933182128311282

  

t = .8, v(x, t) = .999926675964661227

  

t = .9, v(x, t) = .999920175361791563

  

t = 1.0, v(x, t) = .999913676928735229

 (2)
> 

 

> 

 

 

Download PDEprob2_(2).mw

 

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.

 
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