Spectral Method For Some Classes Of Fractional Differential Equations On Unbounded Domain

Fractional differential equations are widely used in modern life,they are often used to simulate natural phenomena in real life.Compared with integer order differential equation,fractional differential equation is more suitable to simulate the problem.Therefore,it is more and more favored by people.In recent years,with the rapid development of computer technology,fractional differential equations have been widely used in atmospheric monitoring,ocean exploration,space exploration,medical images,engineering construction and other aspects.With the further development of science and technology,fractional differential equations will have a broader application prospect.In this paper,according to the properties of two kinds of generalized Associated Laguerre functions,the specific expressions of fractional derivatives of two kinds of basis functions are derived respectively,so as to construct the spectral method for solving fractional differential equations on half line.Furthermore,a Laguerre spectral method for solving time fractional subdiffusion equations on half line is presented.Corresponding error analysis results are given for different numerical formats.At the same time,according to the nature of the Scale Hermite function,the problem is converted to Fourier space,and then two spectral methods are given to solve two kinds of fractional Laplacian equations.Using the same method,this method can be extended to the higher dimensional problems.The main contents can be summarized as followsFirstly,the generalized Associated Laguerre function is selected as the basis function,according to its properties,the fractional derivative of the basis function can be expressed as another kind of generalized Associated Laguerre function.According to this property,two kinds of spectral methods(Galerkin form and Petrov-Galerkin form)are given to solve fractional differential equations on half line,and corresponding error analysis results are given.After that,the Laguerre spectral method for solving the time fractional subdiffusion equation on half line is given by selecting the appropriate function as the basis function in space,the error analysis result is given,and the validity of the method is verified by numerical examples.Secondly,the generalized Associated Laguerre function with parameters is selected as the basis function.According to the properties of its Laplacian transformation and inverse transformation,the fractional derivative of the basis function can be expressed as some forms of the first confluence hypergeometric functions.According to these properties,a numerical scheme is constructed,two spectral methods(Galerkin spectral method and spectral collocation method)are given to solve fractional differential equations on half line,and error analysis results are given.Then,by selecting the appropriate function as the basis function in space,the modified Laguerre spectral method for solving the time fractional subdiffusion equation on half line is given,the error analysis result is given,and the effectiveness of the method is verified by specific numerical examples.Finally,the Scaling Hermite function is selected as the basis function to process the fractional Laplacian equation on unbounded domain,and the problem is converted to Fourier space by Fourier transform.Because the basis function is still a certain type of Scale Hermite function after transformation,the algorithm is constructed to solve the problem and the corresponding error analysis results are given.The method is also used to solve fractional Laplacian equations with one derivative term.Furthermore,using the same method,this method can be extended to high dimensional problems.