## 70 Reputation

4 years, 335 days

## @Preben Alsholm For my problem I ca...

@Preben Alsholm For my problem I can use the following discretization scheme:

If we take time steps x_i = ih, y_j = jh, then:

1/h^2(-u_{i,j-1}-u_{i-1,j}+4u_{i,j} - u_{i+1,j} - u_{i,j+1}) = \sqrt(u_{i,j}) + (u_{i+1,j}-u_{i-1,j})^2/(4h^2 * u_{i,j}^{3/2}

But how to implement this in maple?

Thanks.

## @Preben Alsholm , how can I plot th...

@Preben Alsholm , how can I plot the error of this approximated solution?

@Carl Love u(x1,x2,x3,x4) satisfies: d^2u/dx1^2+d^2u/dx2^2 = d^2u/dx3^2+d^2u/dx4^2

and u is twice differentiable.

## error estimate....

@Preben Alsholm , hi, do you happen to know how can I estimate the error in this numerical solution?

## response...

@Preben Alsholm , this is another recursive integral equation which I want to solve numerically, I thought I could do it by using maple.

## the recursion...

@Axel Vogt t>0, we can also truncate it to t<1.

## @tomleslie  Thanks, I appreciate yo...

Thanks, I appreciate your time you spent on this.

did you search: "methods of solving 2-D inhomogeneous Volterra equations"?

## hi...

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 attachme...

Ok, here's the attachment:

Solving_numerical_integral.mw

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

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.

## @tomleslie  v(x,t) does depend on x...

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 op...

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\...

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 | ...

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

## @tomleslie , how do I find the nume...

@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?

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