Collocation Methods For Second Kind Of Fractional Fredholm Integral Equations

Integral equations,an important mathematical tool,has been widely used in mechanics,weather forecasting,vibration theory,game theory,particle physics,and so on.In recent years,the study of the fractional calculus has been paid more and more attention of researchers,and has achieved good results in fluid mechanics,model design,geological exploration,dispersion theory and so on.In the calculation of fractional integral,several fractional integral equations are often involved in the calculation.However,it is difficult to obtain the analytic solution of fractional integral equation,the numerical method is usually used to obtain its numerical solution.Therefore,it is very important to further improve the numerical solution of fractional integral equation.This theis first studies the numerical solution of the first kind weakly singular Volterra integral equations,by introducing the generalized Jacobi function,the unknown function in the equations is replaced,and the equations is transformed into the form of linear equations,so as to obtain the numerical solution of the equation.The uniqueness of the solution are proved,and the error of the method is analyzed.In this paper,the numerical solution of a class of integro-differential equations is obtained by using the generalized Jacobi function.Next,we study the numerical solution of the second kinds of fractional Fredholm integral equations.By introducing the general Jacobi function,the unknown function in the integral equation is replaced,by the collocation method,the integral equation is transformed into linear equations,therefore,the numerical solution of the equations is obtained.A sufficient condition for the existence of the solution of the equation is given,and the error of the method is analyzed.Finally,several numerical examples are given to illustrate the effectiveness of the method.