When solving a [imath]2[/imath]D Heat Equation, suppose I separate the solution into time and space, i.e., [imath]f_1(T(t)) = f_2(Z(x,\ y)) = \lambda[/imath], and then separate space into its dimensions, i.e., [imath]f_3(X(x)) = f_4(Y(y),\ \lambda) = r[/imath]. The problem of this sort I worked seems to have two nontrivial paths, one in the case that [imath]\lambda = r \neq 0[/imath] and another in the case that [imath]\lambda \neq r, \lambda \neq 0, r \neq 0[/imath]. Usually in other problems I have encountered only one nontrivial path.
After I have solved for [imath]u[/imath] in each of the paths, the former being a Fourier Series solution and the latter being a double Fourier Series solution, am I supposed to combine the answers into a single particular solution to the problem somehow, or are these separate particular solutions which I would choose between based on some physical measurement to determine whether or not [imath]\lambda = r[/imath]?
After I have solved for [imath]u[/imath] in each of the paths, the former being a Fourier Series solution and the latter being a double Fourier Series solution, am I supposed to combine the answers into a single particular solution to the problem somehow, or are these separate particular solutions which I would choose between based on some physical measurement to determine whether or not [imath]\lambda = r[/imath]?