Semi-orthogonal B-spline Wavelet Method For Some Fractional Equations

Fractional calculus is a generalization of integral calculus.It has nonlocal property due to non-integral order,and then fractional calculus is accurate in describing dynamic systems with memory and heredity property.These systems frequently appear in the fields,such as mechanics,physics,medicine,environmental science,image processing and financial markets.However,most of fractional equations can not be analytically solved.Therefore,an important subject is to study the numerical solution of fractional equations.Based on semi-orthogonal B-spline wavelets and corresponding scaling functions,several classes of fractional equations are solved in this thesis by numerical method including linear multi-term fractional ordinary differential equation,nonlinear fractional integro-differential equation,variable-order fractional similar diffusion-wave equation and two-sided space fractional advection-diffusion equation.Meanwhile,numerical examples show that the numerical methods based on semi-orthogonal B-spline are feasible.In this dissertation,the global wavelet method is used to solve the nonlocal fractional equations.Because the continuous orthogonal wavelets have no the compact support and symmetry simultaneously,the semi-orthogonal wavelet with the above properties is considered.In addition,B-spline functions have the advantages of simple structure and compact support,a set of semi-orthogonal B-spline wavelets with explicit expressions is utilized to construct numerical methods for solving fractional equations.The specific research contents are as follows:Firstly,for a class of generalized linear multi-term fractional ordinary differential equations,a semi-orthogonal B-spline wavelet collocation method is constructed by using the theory of multi-resolution analysis.The numerical method can transform the fractional differential equation to a system of algebraic equations.Because both the semi-orthogonal B-spline wavelet and its scaling function have compact support,the associated operational matrix of the numerical scheme is relatively sparse,which resultantly saves the storage space and reduces computation.The computational efficiency are improved.The error is analysed theoretically,and numerical examples are given to verify the applicability and effectiveness of the numerical method.Meanwhile,numerical examples show that the wavelet collocation method based on semi-orthogonal B-spline has high accuracy.Secondly,linear fractional differential model is extended to nonlinear fractional integrodifferential model.A quasilinearized semi-orthogonal B-spline wavelet method is proposed for a class of generalized nonlinear fractional integro-differential equations.In order to simplify the computational complexity,a quasilinearization method is used to transform the nonlinear fractional integro-differential equation into a linear fractional integrodifferential equation.The semi-orthogonal B-spline wavelet collocation method is used to solve the linear fractional integro-differential equation.The error analysis of the numerical method is given and its convergence is obtained.The effectiveness of the method is verified by numerical examples.Thirdly,fixed order fractional order model is extended to variable order fractional order model.For a class of generalized nonlinear variable-order fractional similar diffusionwave equations,a quasilinearized semi-orthogonal B-spline wavelet method is constructed.The quasilinearization method is applied to linearize nonlinear fractional order equations.Semi-orthogonal B-spline wavelet collocation method is employed to discretize the spatial and temporal directions.The errors in time and space direction of the wavelet collocation method based on semi-orthogonal B-spline are theoretically given.In comparison with the existing Chebyshev wavelet method and the local discontinuous Galerkin method,the quasilinearized semi-orthogonal B-spline wavelet method has higher accuracy.Finally,unilateral fractional order model is extended to two-sided fractional order model.An implicit semi-orthogonal B-spline wavelet method is proposed for a class of two-sided spatial fractional advection-diffusion equations.Because of the symmetry of semi-orthogonal B-spline wavelets and their scaling functions,the fractional derivatives on the left and right sides of their basis functions have simple finite difference schemes.By using implicit Euler method in time and semi-orthogonal B-spline wavelet collocation method in space,an implicit semi-orthogonal B spline wavelet method is constructed.The unconditional stability and convergence of this numerical method are proved and the order of convergence is given.Moreover,the theoretical convergence order is verified by numerical examples.