Reproducing Kernel-shifted Legendre Basis Function Method For Solving A Number Of Fractional Differential Equations

The fractional differential theory is a theory of arbitrary derivatives,which is generalized on the basis of integer differential theory.With the continuous devel-opment of science and technology and the continuous hard study of experts and scholars,it is found that many complex phenomena in real life are difficult to be described by integer order differential theory.Because of its global correlation,the fractional differential theory can well describe materials and processes with memory and genetic characteristics,and can overcome some serious disadvantages of integer differential theory,making the fractional differential equations widely concerned.And many experts and scholars at home and abroad have made more in-depth re-search on it.The study finds that the fractional differential equations can well describe the problems of physics,chemistry,mechanics,engineering and other field-s,so it is widely used in signal processing,control engineering,electrochemistry,hydrodynamics,viscoelastic material dynamics,etc.However,due to the historical dependence of fractional derivatives.It is very complex and difficult to calculate the exact solutions of fractional differential equations,so it is necessary to give corresponding numerical algorithms to obtain their numerical solutions.Based on the theory of the reproducing kernel,this paper construct a new reproducing kernel space by using the shifted Legendre polynomials as the basis function and the reproducing kernel function of the space is given.It is different from the reproducing kernel function in the classical reproducing kernel space that this function is no longer a piecewise form.Therefore,in the process of solving,when the fractional operator acts on the reproducing kernel function,the amount of calculation can be greatly reduced,so the approximate solution is more accurate.In the meantime,the error estimation and convergence analysis are also given,and the strict proof is given theoretically.Finally,the numerical examples are given to illustrate the validity of the method.