Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation

Abstract

This article presents a highly accurate and efficient method—a discrete version of the preintegration method—for numerically solving the Cauchy problem for a singularly perturbed third-order equation. The method is based on expanding the highest derivative of the differential equation into a finite series in Chebyshev polynomials of the first kind with unknown expansion coefficients. All lower derivatives and an approximate solution to the differential equation are determined by preintegrating the series for the highest derivative using a discrete integration formula that reduces the order of the highest derivative. This yields the fundamental algebraic equations of the proposed method. By adding additional equations derived from three initial conditions to these fundamental equations, a system of linear algebraic equations is obtained for determining the unknown expansion coefficients for the proposed solution. The number of equations and the number of unknowns in the resulting algebraic system coincide. This system is solved by a standard method; in this paper, it is solved by the Gaussian method. The calculation results show that for arbitrary values of the small problem parameter, a slight increase in the number of Chebyshev polynomials leads to a geometrically progressive reduction in absolute errors. Thus, the proposed discrete version of the pre-integration method is not only computationally efficient but also highly accurate and sufficiently versatile for solving a wide range of problems involving singularly perturbed equations.