A Chebyshev polynomial approach to approximate solution of differential equations using differential evolution

Abstract

Most of the problems associated with physical, biological and engineering systems are complex and are usually modeled with the help of linear as well as non-linear differential equations. In general, the exact solutions of non-linear differential equations may not be obtained by analytical methods and thus, in this paper, a numerical method is proposed to find approximate solutions of differential equations using Chebyshev polynomials and metaheuristic optimization algorithms. In this paper, Ordinary differential equations (ODEs) are modeled as optimization problems to determine their approximate solutions by considering Chebyshev polynomials as the base approximation function, taking initial and/or boundary conditions as constraints. The unknown coefficients of the polynomials are calculated using Differential Evolution (DE) in order to get the approximate solution with minimal error. The proposed method is applied to find the approximate solution to both Initial and Boundary Value Problems. For all test problems considered in this paper, the algorithms have been programmed using MATLAB software to run on computer. The effectiveness of the method is demonstrated by graphical comparison of the computed approximate solutions with that of the exact ones. Further, the performance of the method proposed in this paper is shown by finding out the Root Mean Square Error (RMSE) between the exact and approximate solution, which is found to be better than that of the related existing papers.