The method of lines in application to some two-dimensional nonlinear parabolic equations

We consider the first mixed problem for nonlinear parabolic equation. Assuming that the exact solution of the problem is $u(t,x,y)\in C^{4,0}(Q)$, $Q=\{(x,y)\in\Omega,0\leq t\leq T\}$ we construct a scheme of the method of straight lines of accuracy $O(h^2)$ for the cases when $\Omega$ is a rectangle or a trapezoid.