Comparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations 求解非线性分数阶微分方程的数值方法比较.pdf

Comparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations Farhad Farokhi, Mohammad Haeri, and Mohammad Saleh Tavazoei Abstract This paper is a result of comparison of some available numerical methods for solving nonlinear fractional order ordinary differential equations. These meth- ods are compared according to their computational complexity, convergence rate, and approximation error. The present study shows that when these methods are applied to nonlinear differential equations of fractional order, they have different convergence rate and approximation error. 1 Introduction Differentialequations of fractional order have been the focus of many studies due to their frequent appearance in various applications in physics, fluid mechanics, biology, and engineering. Consequently, considerable attention has been given to the solutions of fractional order ordinary differential equations, integral equations and fractional order partial differential equations of physical interest. Number of literatures concerning the application of fractional order differential equations in nonlinear dynamics has been grown rapidly in the recent years [2, 3, 5, 12– 14, 20]. Most fractional differential equations do not have exact analytic solutions and therefore, approximating or numerical techniques are generally applied. There are many different numerical methods such as Predictor Corrector Method (PCM) [8], Quadrature Methods (QM) [22], Kumar-Agrawal Method (KAM) [15], and Lubich Method [ 17] which have been developed to solve the fractional differential equa- tions. Many new ideas which try to solve these kinds of problems faster and in more convenient way are Nested Memory Principle (NMP) and Fixed Length Integral Principle (FLIP) [7, 10]. These methods are relatively new and provide an approxi- mated solution both for linear and nonlinear equations. There are several papers in F. Farok