Convergence of the method of lines for the first periodic boundary-value problem for a second-order nonlinear parabolic equation

Abstract

For solving the first generalized periodic boundary-value problem in the case of a second-order quasilinear parabolic equation of form

with periodic condition

and boundary conditions

there is examined a longitudinal variant of the method of lines, reducing the solving of problem (1)–(3) to the solving of a two-point problem for a system ofN ^-1 first-order ordinary differential equations of form

with the two-point conditions

An error estimate is established. The convergence of the solutions of problem (4)–(5) to the generalized solution of problem (1)–(3) is established for two methods of choosing the functions

. Convergence with orderh ² is guaranteed under the assumption of square-integrability of the third derivative

of the solution of problem (1)–(3).