How to solve integral equation using integral2 matlab function

Dear all,

I would like to solve an integral equation using an iterative scheme which is described in the attached file.

I wrote the following matlab code (I also attach the m file of this code):

function Test3

clear all; close all; clc

R = 1;

alpha = 0.50;

x0 = 5;

y0 = 5;

vmin = @(u)-sqrt(R^2-u.^2);

vmax = @(u)sqrt(R^2-u.^2);

C = @(u,v)(R^2+1)./(u.^2+v.^2+1);

T1 = @(x,y,s)sqrt(-1)/(4*pi)*besselh(0,1,sqrt(-1)*sqrt(1+s^(2*alpha))*sqrt((x-x0).^2+(y-y0).^2));

phi = @(u,v,s)u+v+s;

q = [0.1:0.1:0.5;0.1:0.1:0.5];

iter = 10; % for example

history = zeros(size(q,2),iter); % to save the solution

for n=1:iter

n

val = @(u,v,x,y,s)(sqrt(-1)/(4*pi)^2)*besselh(0,1,sqrt(-1)*sqrt(1+s^(2*alpha)).* ...

sqrt((x-u).^2+(y-v).^2)).*(1-C(u,v)).*phi(u,v,s);

phi = @(x,y,s)T1(x,y,s) + integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax);

phi_value = zeros(size(q,2),1);

for i=1:size(q,2)

phi_value(i,1) = phi(q(1,i),q(2,i),0.5);

end

history(:,n) = phi_value; % to save the solution at nth iteration evaluated on the vector q

end

end

%=============================================================

Then, I got the following error:

%===================================================

Matrix dimensions must agree.

Error in

Test3>@(u,v,x,y,s)sqrt(-1)/(4*pi)*besselh(0,1,sqrt(-1)*sqrt(1+s^(2*alpha)).*sqrt((x-u).^2+(y-v).^2)).*(1-C(u,v)).*phi(u,v,s)

(line 19)

sqrt((x-u).^2+(y-v).^2)).*(1-C(u,v)).*phi(u,v,s);

Error in Test3>@(u,v)val(u,v,x,y,s) (line 20)

phi = @(x,y,s)T1(x,y,s) + integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax);

Error in integral2Calc>integral2t/tensor (line 237)

Z1 = FUN(X(VTSTIDX),Y(VTSTIDX)); NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)

[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)

[q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 106)

Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

Error in Test3>@(x,y,s)T1(x,y,s)+integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax) (line 20)

phi = @(x,y,s)T1(x,y,s) + integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax);

Error in

Test3>@(u,v,x,y,s)sqrt(-1)/(4*pi)*besselh(0,1,sqrt(-1)*sqrt(1+s^(2*alpha)).*sqrt((x-u).^2+(y-v).^2)).*(1-C(u,v)).*phi(u,v,s)

(line 19)

sqrt((x-u).^2+(y-v).^2)).*(1-C(u,v)).*phi(u,v,s);

Error in Test3>@(u,v)val(u,v,x,y,s) (line 20)

phi = @(x,y,s)T1(x,y,s) + integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax);

Error in integral2Calc>integral2t/tensor (line 228)

Z = FUN(X,Y); NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)

[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)

[q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 106)

Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

Error in Test3>@(x,y,s)T1(x,y,s)+integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax) (line 20)

phi = @(x,y,s)T1(x,y,s) + integral2(@(u,v)val(u,v,x,y,s),-1,1,vmin,vmax);

Error in Test3 (line 29)

phi_value(i,1) = phi(q(1,i),q(2,i),0.5);

%============================================

I appreciate any idea you might have.

Thanks in advance.

Moncef Mahjoub