Method of straight 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) ∈ C^4,−0(Q), Q={(x,y)∈Ω, 0⩽t⩽T} we construct a scheme of the method of straight lines of accuracy 0(h²) for the cases when ώ is a rectangle or a trapezoid. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 132–142, 1987.