Block Configuration Boundary Value Method For Volterra Integral Equations

Volterra integral equation(VIE)arises in many research fields,such as acoustic scattering problem,isochronous pendulum problem,population prediction model and so on.In general,it is difficult to obtain the analytical solution of VIE,so the study on the numerical solution of VIE has raised much attention.The purpose of this thesis is restricted to study a class of block collocation boundary value methods for solving the first-kind and the second-kind VIEs of convolution type,and it consists of four chapters.Firstly,research significance and development of numerical solutions of VIEs are introduced.Secondly,a kind of block collocation boundary value methods are constructed to solving the second-kind VIE.With the help of multi-step collocation methods,the numerical algorithm is devised by utilizing approximations to the exact solution in future steps.Then VIE is discretized into a special linear system,which can be efficiently solved by using the fast Fourier transform.Moreover,based on the Peano kernel theory,the convergence property of block collocation boundary value methods is also developed in this chapter.Numerical results indicate the high-performance of the proposed approach and verify given theoretical analysis.Finally,block collocation boundary value solutions of the first-kind VIE are studied.Due to the ill-posedness of the first-kind VIE,the solvability of boundary value method is not guaranteed,even for the uniform mesh.Therefore,this Chapter discusses its solvability by studying the special structure of the collocation equation,and gives the sufficient condition for the existence of the collocation solution.Furthermore,with the help of interpolation remainders,the convergence order is analyzed.The innovations of this thesis are the construction and the convergence analysis of block collocation boundary value methods,the solvability analysis of the collocation equation.