Mathematica Plotting of complex and convolution integrals
I was trying to plot the following:

    j = 10; 
    a = 0; b = 0; s = 0; 
    
    B[n_] = Integrate[2*Sin[n*Pi*x]*x, {x, 0, 1}];
    u[x_, t_] = Sum[B[n]*Sin[n*Pi*x]*Exp[-(n*Pi)^2*t], {n, 1, j}];
    
    K[x_, t_] = (1/(2*Pi))*
       Integrate[
        Exp[I*x*psi]*(Exp[-I*b*psi]*Exp[-I*a*psi^2]*
            Exp[-I*t*psi^2])/((1 + psi^2)^s), {psi, -10, 10}];
    
    S = (Total[u[x, t]]/Length[u[x, t]]);
    
    T[x_, t_] = Integrate[K[x - y, t]*y*S, {y, -10, 10}]
    
    Plot3D[T[x, t], {x, 0, 1}, {t, 0.01, 1}, AxesLabel -> {"x", "t", "y"}, Boxed -> False, Mesh -> False]

The problem is that this is a really complicated solution and I would like to get the results numerically, but it ran for a long time and nothing plotted -- just empty axis. In my definition of all the variables -- I have also used the := statement, but it didn't help!

If anyone can please help me with this then I'll greatly appreciate it.

Thanks!
Mathematica Plotting of complex and convolution integrals
===

I was trying to plot the following:

    j = 10; 
    a = 0; b = 0; s = 0; 
    
    B[n_] = Integrate[2*Sin[n*Pi*x]*x, {x, 0, 1}];
    u[x_, t_] = Sum[B[n]*Sin[n*Pi*x]*Exp[-(n*Pi)^2*t], {n, 1, j}];
    
    K[x_, t_] = (1/(2*Pi))*
       Integrate[
        Exp[I*x*psi]*(Exp[-I*b*psi]*Exp[-I*a*psi^2]*
            Exp[-I*t*psi^2])/((1 + psi^2)^s), {psi, -10, 10}];
    
    S = (Total[u[x, t]]/Length[u[x, t]]);
    
    T[x_, t_] = Integrate[K[x - y, t]*y*S, {y, -10, 10}]
    
    Plot3D[T[x, t], {x, 0, 1}, {t, 0.01, 1}, AxesLabel -> {"x", "t", "y"}, Boxed -> False, Mesh -> False]

The problem is that this is a really complicated solution and I would like to get the results numerically, but it ran for a long time and nothing plotted -- just empty axis. In my definition of all the variables -- I have also used the := statement, but it didn't help!

If anyone can please help me with this then I'll greatly appreciate it.

Thanks!