The Research Of Numerical Solutions For Three Kinds Of Fractional Order Differential Equations Based On Fractional Order Legendre Polynomial

In this paper, a general formulation for the fractional-order Legendre functions isconstructed to obtain the numerical solutions of three kinds of fractional-order differentialequations. Fractional-order Legendre Functions whose order is the same to the order of thefractional differential is used to approximation the unknown function.Firstly, this paper introduces the research objects and the characteristics of numericalmethod, as well as the application status of numerical calculation method in fractionalorder calculus, and the historical background and current research status of orthogonalfunction. And then we give some basic knowledge.Secondly, in this study, a general formulation for the fractional-order Legendrefunctions is constructed to obtain the solution of fractional-order differential equations.Fractional-order Legendre Functions whose order is the same to the order of thefractional differential is used to approximation the unknown function. Also generalformulations for Fractional-order Legendre Functions fractional integral operational andproduct matrices are driven based on the characteristic of fractional order calculus and theorthogonality of Fractional-order Legendre Functions. These matrices together with theoperator matrices and function approximation ideas are utilized to reduce the solution ofthis problem to the solution of a system of algebraic equations. Therefore, we achieve thepurpose of solving.Thirdly, the numerical solutions of fractional partial differential equations areobtained by using fractional-order Legendre functions. The main characteristic behind thisapproach is that generalize the new orthogonal functions based on shifted Legendrepolynomials to the fractional calculus. Also the Fractional-order Legendre Functionsfractional differential operational matrix is driven. Then the matrix with the Tau method isused to transform the solution of this problem to the solution of a system of liner algebraicequations. Numerical examples will be used to demonstrate the good accuracy of thisalgorithm.Finally, a general formulation for the generalized fractional-order Legendre functionsis constructed to obtain the numerical solution of fractional partial differential equations with variable coefficients. The special feature of the proposed approach is that we definegeneralized fractional order Legendre functions over [0, h]based on fractional-orderLegendre functions. We use these functions to approximate the unknown function on thearbitrary interval. The error analysis shows that the algorithm is convergent. The method istested on examples. The results show that the generalized fractional-order Legendrefunctions yields better results.