A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Abstract

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.