B-spline Wavelet Method For Solving Fractional Integro-differential Equation

Fractional calculus as known as a new mathematical tool with the real world and the perfect mathematical theory,has been widely applied in various fields,such as various viscoelastic mechanics,statistics and random processes,signal analysis and processing.Fractional integral-differential equation can describe many complex physical processes more accurately and objectively than integer order calculus equations.Therefore,the research on the theory and calculation method of fractional equations is very urgent,which plays an important role in the field of application.Unfortunately,most of the exact solutions of fractional equations are very complex,difficult to find and consume a lot of time.Moreover,not all fractional calculus equations can be solved exact solutions.Therefore,it is very necessary to develop a new numerical algorithm and establish the numerical method of fractional calculus equation,which has important theoretical significance and practical application value.In this paper,several kinds of Riemann-Liouville fractional calculus equations are solved.In Chapter 1,the research background,research significance and research status at home and abroad are briefly introduced.In Chapter 2,fractional integral operator matrix of semi-orthogonal B spline Wavelet is inferred.In Chapter 3,existence and uniqueness of solution for second kind of fractional Fredholm integral equation is proved,using semi-orthogonal B spline wavelet to solve the numerical solution of fractional Fredholm integral equations of the second kind.Meanwhile,numerical solution is given,when exact solution is unknown.In Chapter 4,existence and uniqueness of solutions for systems of fractional Fredholm integral equations are proved,using semi-orthogonal B spline wavelet to solve the systems of fractional Fredholm integral equations.Meanwhile,error analysis is given,when exact solution is unknown.In Chapter 5,nonlinear fractional Fredholm integro-differential equation is solved by B spline wavelet fractional operator matrix.discrete Integro differential equations into algebraic equations,numerical examples verify the feasibility and effectiveness of the proposed method.In Chapter 6,the work done are summarized in this paper and the prospects for future work is proposed.