@tomleslie 

No need to be so harsh, I made some syntax mistakes, but it seems you got it right:

JT4:=JacobiTheta4((Pi/2)*x, exp((-Pi^2)*s));
JT3:=JacobiTheta3(0, exp((-Pi^2)*r));
h:=0.000065;
INT := v(x,t)=1-h*int(JT4*(1-h*int(JT3*v(1,t-r)^4,r=0..t-s))^4,s=0..t);

this is my problem, I want to solve for INT numerically, and I don't have an idea how, I also checked for this integral equation in a handbook of integral equations, but it's not mentioned there.

Do you have some suggestion how to solve this integral equation numerically?

@tomleslie 

Ok, here's the attachment:

Solving_numerical_integral.mw

P.S

I want to numerically solve this equation for v(x,t), I thought of using some numerical integration method, but I seem to first get the above error.

Download Solving_numerical_integral.mw

@tomleslie 

v(x,t) does depend on x, since JacobiTheta0(1/2 x , i\pi t) = \sum_{m=-\infty}^\infty (-1)^m e^{-m^2\pi^2 t} \cos(m\pi x) , where the cosine depends on x.

I am not sure how to solve this integral equation numerically, so I asked for assitance here, if you can do that superb, if you can't I'll try by myself.

@tomleslie 

I tried the errorest option, but it doesn't display the errors in time and space, or of what order are these errors are.

I also need the rate of convergence of this numerical solution.

@Carl Love 

JacobiTheta0(1/2 x , i\pi t) = \sum_{m=-\infty}^\infty (-1)^m e^{-m^2\pi^2 t} \cos(m\pi x)

\theta_3(t) = \sum_{m=-\infty}^\infty e^{-m^2\pi^2 t}

@Carl Love 

i is \sqrt{-1} and x | y mean JacobiTheta0(x,y); I'll edit my question.

@tomleslie , how do I find the numerical method that has been used in your code? and the spatial and temporal errors in the execution of the solver of this pde?

when I change the method to ForwardTimeCenteredSpace, I get the following errors:

Error, (in pdsolve/numeric/plot) unable to compute solution for t<INFO["failtime"]:
unable to store 10.*RootOf(.130000000000000e-1*_Z^4+300*_Z-30.) when datatype=float[8]
Error, (in pdsolve/numeric/plot) unable to compute solution for t>INFO["failtime"]:
unable to store 10.*RootOf(.130000000000000e-1*_Z^4+300*_Z-30.) when datatype=float[8]
Error, (in pdsolve/numeric/plot) unable to compute solution for t>INFO["failtime"]:
unable to store 10.*RootOf(.130000000000000e-1*_Z^4+300*_Z-30.) when datatype=float[8]
Error, (in plots:-display) expecting plot structures but received: {p1, p2, p3}

how to fix this?