Block Pulse Functions Method For Solving A Class Of Nonlinear Fredholm-volterra Integro-differential Equations

The integro-differential equations of fractional order are always an importantresearch topic in mathematic field. The Fredholm-Volterra integro-differential equations,as a classical equation in the math field, have attached enough attention by experts in thevarious fields in recent years. In this paper, we mainly study the application of Block Pulsefunctions as a segmented orthogonal basis functions to solve the nonlinear integro-differential equations of fractional order. The main means is to derive its correspondingfractional operational matrices via the related characteristics of Block Pulse functions.Furthermore, the nonlinear Fredholm-Volterra integro-differential equations aretransformed into the products of some matrices. Lastly, by solving the nonlinear system ofalgebraic equations, we can obtain the numerical solutions.Firstly, aiming at the nonlinear Fredholm-Volterra-Hammerstein integral equations,the thesis transformed the functions into some matrices through Block Pulse functions andtheir properties, then bring these matrices into original equations，solve these nonlinearequations system and get the numerical solution. Convergence analysis discusses theconvergence of the algorithm.Secondly, we add the equations to nonlinear Fredholm-Volterra integro-differentialequations with fractional derivative term. Furthermore, the derivative term of the obtainedequations are converted via the Block Pulse functions and the definition of fractionalcalculus. Subsequently, this article discrete the unknown variables of previous equationsand get the numerical solutions. Corresponding error analysis and numerical examples aregiven at last.Finaly, we also study the nonlinear mixed Volterra-Fredholm integral equations.Using two dimensional Block Pulse functions which have the characteristics of simplestructure, the vector is utilized to approximate the dual function in the subject equation,and then bring it into the original equation, so that the original equation is transformedinto a system of linear equations whose coefficients constitute lower triangular matrix, then we solve the equations to obtain the numerical solution.