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

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 orderh² is guaranteed under the assumption of square-integrability of the third derivative of the solution of problem (1)–(3).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 268–276, 1979.