How can I integrate this function to give a symbolic expression?

I am trying to compute the above double integral to give a function C which depends on x. I attempted to use the following code to compute the double integral: (c,a,b,Da,Beta and sigma are known constants and C_s(x,t) is known to be exp(x*t) in this example. I am trying to plot the function which results with C on the y-axis and x on the x-axis. What is wrong with my integral?

c=0;
Da = .4
sigma = .1
a=1
b=1.3
L=Inf
t=1
Greens= @(x,zeta,t) (1/2*sqrt(pi*a*t))*exp(b(zeta-x)/2*a + c-b^2/4*a)*(exp(-(x-zeta)^2/4*a*t)+exp(-(x+zeta)^2/4*a*t))
fun = @(x,t,zeta,tau) Da.*sigma.*exp(x.*t).*(1/2.*sqrt(pi.*a.*t)).*exp(b(zeta-x)/2.*a + c-b^2/4.*a).*(exp(-(x-zeta)^2/4*a*t)+exp(-(x+zeta)^2/4*a*t))
upper=@(tau) t
q=integral2(fun,0,L,0,upper)

I am getting the following error: "Error using arrayfun All of the input arguments must be of the same size and shape. Previous inputs had size 45 in dimension 2. Input #4 has size 1"

Answer 1

[Star Strider]

It will not give you a symbolic expression because you are not using the Symbolic Math Toolbox to integrate it. (That would likely not work anyway.) You can have it produce an anonymous function:

c=0;
Da = .4;
sigma = 0.1;
a=1;
b=1.3;
L=Inf;
Greens= @(x,zeta,t) (1/2*sqrt(pi*a*t)).*exp(b*(zeta-x)/(2*a) + (c-b^2/(4*a))*t).*(exp(-(x-zeta).^2/(4*a*t))+exp(-(x+zeta).^2./(4*a*t)));
fun = @(x,t,zeta,tau) Da.*sigma.*exp(x.*t).*Greens(x,zeta,t-tau);
C = @(x,t) integral2(@(zeta,tau)fun(x,t,zeta,tau),0,t,0,L);

X = linspace(0,0.05,10);
T = linspace(0,0.08,20);
for k1 = 1:numel(X)
    for k2 = 1:numel(T)
        Z(k1,k2) = C(X(k1),T(k2));
    end
end

figure
surf(T, X, Z)

There were several errors in your code that I corrected. This runs, although it does not give any useful output (all are NaN). I leave you to solve that problem, and make any corrections that may be necessary, since I have no idea what you are doing or what the correct values for ‘x’ and ‘t’ are.