I am trying to input an equation with multiple integrals and infinite summations.

I am trying to input these three equations into MATLAB but I keep getting errors. This is the analytical solution to the 2D Wave Equation.

where

Here is my code:

a = 10; % Boundary of X Axis
b = 10; % Boundary of Y Axis
c = 1; % Speed
w = pi;
Alpha = 0.75;
Sigma = 0.1;
xNot = a/2;
yNot = b/2;

x = 1;
y = 1;
t = 1;

syms n m
P_c = symsum(symsum(4/(c*sqrt(n^2*pi^2/a^2 + m^2*pi^2/b^2)*sin(c*t*sqrt(n^2*pi^2/a^2 + m^2*pi^2/b^2))) + 4*cos(c*t*sqrt(n^2*pi^2/a^2 + m^2*pi^2/b^2))*sin(n*pi*x/a)*sin(m*pi*y/b),n,1,Inf),m,1,Inf);
P_p = integral(@(s) symsum(symsum(4/(a*b*c*sqrt(n^2*pi*2/a^2 + m^2*pi^2/b^2))*integral2(@(x,y) sin(w*s).*exp(-Alpha*s).*exp(-1/Sigma^2*((x-xNot).^2 + (y-yNot).^2))*sin(n*pi*x./a)*sin(m*pi*y./b),0,a,0,b)*sin(c*(t-s).*sqrt(n^2*pi^2/a^2 + m^2*pi^2/b^2))*sin(n*pi*x/a)*sin(m*pi*y/b),n,1,Inf),m,1,Inf),0,t);
P = P_c + P_p;

Answer 1

[Walter Roberson]

integral() is only for numeric integrals. For symbolic integrals you need int()

[Richard Batelaan]

How do I need to write my three equations in MATLAB so I can graph it using the surf() function?

[Walter Roberson]

Q = @(v) sym(v);
INT = @(expr,var,lb,ub) int(expr,var,lb,ub);

a = Q(10); % Boundary of X Axis
b = Q(10); % Boundary of Y Axis
c = Q(1); % Speed
Pi = Q(pi);
w = Pi;
Alpha = Q(0.75);
Sigma = Q(0.1);
xNot = a/2;
yNot = b/2;

syms x y t real

syms n m s real
P_c_n = symsum(4/(c*sqrt(n^2*Pi^2/a^2 + m^2*Pi^2/b^2)*sin(c*t*sqrt(n^2*Pi^2/a^2 + m^2*Pi^2/b^2))) + 4*cos(c*t*sqrt(n^2*Pi^2/a^2 + m^2*Pi^2/b^2))*sin(n*Pi*x/a)*sin(m*Pi*y/b),n,1,Inf);
P_c = symsum(P_c_n,m,1,Inf);

P_p_x = INT(sin(w*s).*exp(-Alpha*s).*exp(-1/Sigma^2*((x-xNot).^2 + (y-yNot).^2))*sin(n*Pi*x./a)*sin(m*Pi*y./b),x,0,a);
P_p_X = (rewrite(P_p_x, 'exp'));
P_p_y = INT(P_p_X,y,0,b);
P_p_n = symsum(4/(a*b*c*sqrt(n^2*Pi*2/a^2 + m^2*Pi^2/b^2))*P_p_y*sin(c*(t-s).*sqrt(n^2*Pi^2/a^2 + m^2*Pi^2/b^2))*sin(n*Pi*x/a)*sin(m*Pi*y/b),n,1,Inf);
P_p_m = symsum(P_p_n,m,1,Inf);
P_p = INT( P_p_m, s, 0, t);
P = P_c + P_p;
Ps = simplify(P,'steps',100);

P is now symbolic in x, y, and t. In order to surf() you need to subs() in a constant for one of those three variables, and substitute in grids of numbers for the other two variables. Then double() the resulting symbolic array, and use it as the dependent variable for surf()

Note that P will have two symsum(), so the calculation will not be all that efficient. The rewrite() step does manage to convert the int() into a closed form solution, so it avoids two levels of int() in the final expression.

[Richard Batelaan]

Thank you so much for your answer. I have been trying to make a surface in the way that you said, but I cannot figure it out. Here is what I have:

[X,Y] = meshgrid(1:10,1:10);
Ps = subs(Ps,x,X);
Ps = subs(Ps,y,Y);
Ps = subs(Ps,t,3);
Ps = vpa(Ps);
Ps = double(Ps);
surf(X,Y,Ps)

[Walter Roberson]

[X,Y] = meshgrid(1:10,1:10);
Pss = subs(Ps, {x,y,t}, {X, Y, 3});
Psd = double(Pss);
surf(X, Y, Psd, 'edgecolor', 'none');

That is, suppose you had the formula F = x + y . Suppose you subs(F, x, 1:3) . Then the result you get back would be [1+y, 2+y, 3+y], Suppose you then subs() y, [4 5 6] into that. Then the 4 5 6 has to go in for each location that y occurs. It will not be substituted for corresponding locations. So you would get [1+4, 1+5, 1+6, 2+4, 2+5, 2+6, 3+4, 3+5, 3+6]

However if you subs(F, {x, y}, {1:3, 4:6}) then MATLAB will do simultaneous substitution of corresponding elements, producing [1+4, 2+5, 3+6]

You were substituting in over multiple steps, so you were getting arrays larger than you expected.