The method of lines for nonlinear parabolic differential equations with mixed derivatives

Summary By the so-called longitudinal method of lines the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper, for a wide class of nonlinear parabolic differential equations the spatial derivatives occuring in the original problem are replaced by suitable differences such that monotonicity methods become applicable. A convergence theorem is proved. Special interest is devoted to the equationu _t=f(x,t,u,u _x,u _xx), if the matrix of first order derivatives off(x,t,z,p,r) with respect tor may be estimated by a suitable Minkowski matrix.