New Numerical Method For Solving Two Kinds Of Fractional Integro-differential Equation With Singular Kernel

Today,the equations with singular kernel are still a popular branch of mathematics field.This paper studies two kinds of equations with singular kernel,among which the singular integral equation with cauchy kernel studied is a classical method to solve various elastic problems,and also the earliest and most complete class of singular equations.This kind of equation in solid mechanics,contact mechanics,fracture mechanics and quantum theory,potential theory,has been widely used in electromagnetic scattering problems.This makes the singular integral equations with cauchy kernel has strong correlation with the actual problem.And the other with a weakly singular kernel fractional integro-differential equation though is a relatively new discipline,but this kind of equation has a broad application prospect in real life,it's the mathematical model of radiation balance,heat conduction problem,elastic fracture mechanics,materials science and engineering,heat and solid state physics problems.Since the equations with singular kernel have strong conditions when seeking approximate solutions,only a few of the numerical methods suitable for solving general equations are suitable for solving such equations.Therefore,the research on the algorithm has a substantial physical background and a very important practical application prospect.Based on the theory of collocation method,this paper use the Chebyshev polynomial as basis functions to construct the expression of approximate solution and fractional order differential term.Finally the original equation is transformed into a linear algebraic equations to solve the approximate solution of the equations.According to this method for Chebyshev collocation method.Using the Chebyshev collocation method into the equation of the form is more concise,greatly reducing the amount of calculation and save the time,so the calculation results more accurate and effective.Numerical examples are given to validate the effectiveness of the method.Finally,the convergence analysis and other related theories are given.