I am trying to solve a following Partial Differential Equation:
u_t(x,y,t)= u_xx(x,y,t)+u_yy(x,y,t) + 7 u(x,x,t)
which causes troubles due to the last term
u(x,x,t) not being
For example Mathematica complains that order of variables is bad, and I did not find anything appropriate to try in matlab.
One of methods to solve this problem is to discretize the equation and to get the time dependent ODE:
x'(t) = A(t) x(t), (2)
A(t) the discretization matrix. The matrix
A(t) is just a discrete laplacian plus some simple sparse matrix.
So in principle one could solve (2), but this seems inefficient (I know that NDsolve basically does just that but in practise NDSolve is much faster then the method implemented by hand). Is there something obvious that I am missing ?