Al86

60 Reputation

5 Badges

4 years, 103 days

MaplePrimes Activity


These are replies submitted by Al86

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

 

 

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

 

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

 

@tomleslie 

Thanks, I appreciate your time you spent on this.

Can you please provide me with the links to the algorithms you talked about? or what did you search in google?

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

 

@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

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);
 

Error, missing operator or `;`

 

 

 

 

NULL

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?

 

1 2 3 4 Page 4 of 4