A Fibonacci Galerkin Method for Solving Certain Types of Boundary Value Problems

Annual Meeting in Mathematics 2023

Abstract

In the Fibonacci sequence, the first two numbers are 0 and 1, and the next numbers are equal to the sum of the previous two numbers. In this paper, the concept of the Fibonacci sequence is extended to polynomial functions which can be effectively applied in various function approximations. Based on these Fibonacci-based polynomials and the Galerkin method, we develop a Fibonacci Galerkin method (FGM) to solve some types of boundary value problems (BVPs) such as a linear singular two-point BVP and a nonlinear multi-point BVP. The FGM process constructs a residual function for a BVP by utilizing an approximate solution formed by the method and then evaluating the integral of the product between residual functions and weight functions over a domain. Equating the value of the integral close to zero, one obtains an analytical solution of the BVP. As examples, we apply the method to certain types of BVPs including a linear singular two-point BVP, a nonlinear multi-point BVP and a regular two-point BVP whose exact solutions are given. Their semi-analytical solutions are obtained. By comparing the solutions of the proposed boundary value problems obtained by this technique with their exact solutions, we believe that the technique is highly accurate and effective.