Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method

Abstract

Abstract:In this paper, the Galerkin method is applied to second order ordinary differential equation with mixed boundary after converting the given linear second order ordinary differential equation into equivalent boundary value problem by considering a valid assumption for the independent variable and also converting mixed boundary condition in to Neumann type by using secant and Runge-Kutta methods. The resulting system of equation is solved by direct method. In order to check to what extent the method approximates the exact solution, a test example with known exact solution is solved and compared with the exact solution graphically as well as numerically.

Useful Ideas on the Numerical Techniques Used for the Solution of the Two-Point Boundary Value Problems of Ordinary Differential Equations

“…Extensive work has been done on the analysis of the method, (Pan et al, 2005;Cicelia, 2014;Zavalani, 2015;Anulo. et al, 2017;Paffuti, 2019;Wang & Zhao, 2019), but it still is an attractive and challenging area of research.…”

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

“…To construct the exact schema defined in Theorem 2, the local green functions guarantee continuity boundary conditions in the connection of the subintervals. In fact, given the pair of test functions ψ i−1 (x) and ϕ i (x) with finite support on the interval [x i−1 , x i+1 ], their zeros and special properties provide the flux elimination necessary to produce the recursive formula shown in Equation (10). Moreover, the continuity boundary conditions in the connection points of the subintervals, for the equality of the values of u(x), arise from the special properties of the derivatives of the local green functions (Equations (3c) and (4c)), which will be exploited in the proof of Theorem 2.…”

“…Among the classical numerical methods for solving second-order differential equations, the finite element method (FEM) is presently the most widely used [9]. Nothing better illustrates the challenges of BVP defined by Equation (1) than the fact that, even in recent years, we can find publications where the authors researched methods for solving the non-constant coefficient problem [10][11][12].…”

Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy

“…Numerous methods in the literature have been introduced to obtain the solution of two point BVPs associated with the mixed conditions between Dirichlet and Neumann conditions that are prescribed at the boundary. Among them, some implemented the finite difference method, as in Cuomo and Marasco [2], the modified Adomian decomposition method, as in Duan et al [3], and the Galerkin method, as in Anulo et al [4]. However, as far as the authors are aware, not many researchers have given attention to solving Equation (1) directly with the Robin boundary condition that exist on one side of the conditions in Equation (2) and combining with either the Dirichlet or Neumann boundary conditions.…”