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These are questions asked by Al86

I have the following PDE system to solve numerically and I am not sure how to use maple to solve it.


v_t = v_{xx} for 0<x<1 , t>0


v_x(1,t)=-hv^4(1,t) (where h is some numerical number);


To solve this pde numerically I need to use the following condition on v(1,t):

v(1,t) = 1-h*\int_{0}^t \theta_3(\tau)v(1,t-\tau)^4d\tau

this is the numerical boundary condition, where \theta_3 is Jacobi theta3 function.


I don't see how can I use maple for this numerical pde problem.


Here's my attempt at solution:


 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 := {`&PartialD;`(v(0, t))/`&PartialD;`(x) = 0, `&PartialD;`(v(1, t))/`&PartialD;`(x) = -0.65e-4*v(1, t)^4, v(x, 0) = 1};

pds := pdsolve(PDE, IBC, numeric, time = t, range = 0 .. 1, spacestep = 0.1e-2, timestep = 0.1e-2, numericalbcs = {v(1, t) = 1-0.65e-4*JACOBIINTEGRAL}, method = ForwardTimeCenteredSpace)


But I get the next error message:

Error, (in pdsolve/numeric/process_IBCs) improper op or subscript selector


How to fix this or suggest me a better way to solve this pde numerically?


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