Al86

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I have the following integral equation to solve numerically:

 

v(x,t)=1 - h*\int_0^t JacobiTheta0(1/2x , \pi i s) v^4(1,t-s)ds

where h is a numerical parameter, and v(1,t) = 1-h*\int_0^t \theta_3(r)v^4(1,t-r)dr (theta3 is Jacobi theta3 function).

 

So I want to use an iteration method that will converge numerically to the solution, where v(1,0)=1.

How to use maple for this?

I want also to find the rate of convergence to the numerical solution.

 edit: I should note that v(x,0)=1, even though it's implied from v(x,t) above.

 

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,0)=1

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

v_x(0,t)=0

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:

[code]

 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)

[/code]

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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