The Cubic B-spline Operational Matrix Based on Haar Scaling Functions for Solving Varieties of the Fractional Integro-differential Equations

摘要

Nowadays, the operational matrix plays an important role insolving problems with partial, ordinary or fractional derivatives. In thecurrent study, we construct the operational matrix of the fractional integralfor cubic B-spline scaling function and wavelets and it applies to solve va-rieties of the fractional integro-differential equations. To do this, firstly,the operational matrix of fractional integral for Haar scaling functions isconstructed by using the definition of the Riemann-Liouville fractionalintegral operator and the orthogonal projection of polynomial on space ofHaar scaling functions. Afterward, we obtain the operational matrix ofcubic B-spline functions from fractional order using approximation cubicB-spline functions with Haar scaling functions and collocation method.The principal characteristics of this method are as follows: The opera-tional matrix of cubic B-spline functions is obtained simply because of theuseful properties of the Haar scaling functions, reducing the time, the lessoccupation of the computer memory which converts to a system of linearand nonlinear equations. Finally, we will show the validity and efficiencyof the new method by numerical examples and convergence analysis.