High Order Numerical Methods For Weakly Singular Volterra Integral And Integro-differential Equations

Volterra integral and integro-differential equations with weakly singular kernels are mathematical models that play an important role in biology,physics,finance and other scientific and engineering fields.Because it is difficult to solve these equations analytically,the study of numerical methods of these equations has attracted continuous attention of many scholars.An important characteristic of these equations is that the integral operators contain kernel functions with weak singularities,which makes the solutions usually have low regularity.This property brings special difficulties to the construction and analysis of numerical methods.In this dissertation,some high order numerical methods for Volterra integral and integro-differential equations with weakly singular kernels are studied,and the convergence is analyzed.The whole dissertation is divided into the following parts:(1)A piecewise fractional polynomial collocation method is proposed for solving the second-kind Volterra integral equations with weakly singular kernels.The interval is divided by graded meshes.In each subinterval,the collocation scheme is constructed by using fractional polynomials.For the proposed method,hp-version error estimates are obtained under the assumption that the solution is not required to have high regularity.For some problems with weakly singular solutions,the results can guarantee both exponential convergence of fractional polynomial degree and optimal convergence of mesh parameter(average step size).(2)The fractional polynomial collocation method for the second-kind Volterra integrodifferential equations with weakly singular kernels are studied.The hp-version error estimates of the method is proved on arbitrary meshes.The results show that for any given mesh partition,exponential rates of p-version convergence can be achieved for certain weakly singular solutions.The results also imply that in the case of uniform mesh,the method has no(h-version)order barrier for weakly singular solutions,which is different from classical polynomial collocation methods,and in the case of graded meshes,the optimal convergence rates for the h-version of the method can be obtained.(3)The numerical method for third-kind Volterra integral equations are studied.A modified graded mesh is presented,and a fractional polynomial collocation scheme is constructed based on it.The solvability of the collocation scheme is proved when the integral operator is noncompact.The numerical results show that,for some problems with weakly singular solutions,the h-version convergence rate of the method is higher than that of the existing methods in the literature and p-version exponential rates of convergence can be achieved under properly selected parameters.(4)The fractional polynomial collocation method for the third-kind Volterra integrodifferential equations are studied.For such equations with weakly singular solutions,the solvability of the numerical scheme and the hp-version convergence on general mesh are proved.The theoretical and numerical results show that,for this kind of equations with noncompact integral operator,the method can achieve high order convergence when the solution has weak singularity at the initial point.In short,several classes of weakly singular Volterra equations are studied in this dissertation.Collocation methods are constructed by using fractional polynomials and the hpversion error estimates of the methods are analyzed.The results show that,when the solution has weak singularity of fractional power form,the method can achieve high order convergence.