DIRECT METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR SOLVING BACKWARD HEAT EQUATION

Abstract

A direct method of Lie-algebraic discrete approximation for numerical solving the Cauchy problem for the backward heat equation is proposed in this paper. The key idea of direct method of Lie-algebraic discrete approximations is using analytical approaches, in particular the method of small parameter or Taylor series expansion, to construct analytical approximation of the solution for the problem in the form of power series with respect to the time variable. The conditions for convergence of analytical series are studied in particular. By means of small parameter method the recurrence relation for evaluation of each member of a sequence is provided. This approach enables fast computation and signicant reduction of computational cost in compare to Generalized method of Lie-algebraic discrete approximations which performs complete discretization by all variables. Thereafter, the discrete match of recurrence relation is built using quasi-representations of the Lie-algebra basis elements, which means, that each dierential operator is replaced by its analogue matrix which is quasi-representation of dierential operator in nite dimensional space. It is proved that computational scheme has a factorial rate of convergence. The proposed approach is applied to model case and obtained results are compared with nite dierence method, classical method of Lie-algebraic discrete approximations and Generalized method of Lie-algebraic discrete approximation. The convergence rates for all of these methods are compared in dierent functional spaces. In addition, we study the count of arithmetical operations for equal set of nodes. Demonstrated a possibility for reusability of the numerical scheme for heat equation.