Numerical Solution Of RIEMANN-LIOUVILLE Fractional Differential Equation With Boundary Value Problem

Fractional differential equations appear in many applied models such as electronic,physics,etc.But due to the fractional order of differential equation,its solution is affected by various boundary value conditions and the fractional order value,which brings some difficulties to the study of the problem.For that purpose,many numerical methods are used to solve the problem of fractional differential equations,for example Taylor method,Euler method,Runge Okuta method,spline collocation method,Adomian decomposition method and Homotopy perturbation methods,etc.In our case,the numerical solution of Riemann-Liouville fractional differential equation is studied by using the quasi-Newton simplified reproducing kernel method.The Newton’s method is often used to solve nonlinear problems because of his fast convergence advantage.The method requires that the first derivative of function exists and at each step needs to compute the value of the derivative function.It’s not convenient to implement on computer.The quasi-Newton’s method is introduced to address this shortcoming and offer to us the best way to resolve the problem.Firstly,the quasiNewton’s method is used to linearize the nonlinear equation’s problem.According to the characteristics of the model,the satisfied reproducing kernel space is established.Then,the equation system is established according to the projection operator constructed.Next,we write out the matrix form of the system and finally we found the coefficients.Finally,the effectiveness of the presented technique is demonstrated by comparison with existing experiments.