| The fractional differential equation came into being when dealing with practical problems.Compared with the integer order differential equations,it can accurately describe the physical meaning of practical problems.Nonlinear fractional differential equations are widely used in many fields,such as thermal engineering,signal processing and viscoelasticity.But,it is usually difficult to find the exact solution of this equation.So scholars actively solve its approximate solution.In this paper,we will study the numerical solution of two kinds of nonlinear fractional differential equations.In the first chapter of this paper,a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed based on the piecewise Picard iteration method.The convergence proof and error estimations of the Picard and the piecewise Picard iteration method are obtained.Meanwhile,a sufficient condition for the stability of the proposed algorithm is also given.It's worth noting that our method is quite suited for solving linear,weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays.In the second chapter of this paper,a new reproducing kernel iterative method with convergence proof is proposed for solving the nonlinear multi-term time-fractional differential equation.In order to solve the equation,we apply the fractional integral operator to both sides of the differential equation to obtain an integral equation.Based on the reproducing kernel theory,we construct a new iterative method to solve the integral equation and give the convergence proof and numerical experiments.Then we obtain a numerical accuracy satisfying the application requirements by a small amount of calculation.For two different fractional differential equations,we propose new numerical algorithms with different innovative ideas.The new approach is easily implemented and yield numerical solutions with high precision. |
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