The Analytical Methods For One Kind Of Volterra Intergro-differential Equations

Volterra integro-differential equations appear frequently in the mathematical modeling such as biology,physics,engineering and other practical problems.This kind of mathematical model has the integral term with the unknown kernel function,which can reflect the dynamics system of no-nlocal properties and memory feedback properties better,and the Volterra integrodifferential equations seem to be closer to the simulation of practical problems by comparison.Therefore,solving such models is also a hot topic of today's research.In this paper,a class of analytical solutions on the type of Volterra integro-differential equations with memory kernel function of generalized Mittag-Leffler type,power-law type,and exponential factor type are studied:(1)The analytical solution of the high-dimensional non-homogeneous(Parabolic Volterra Integro-Differential,PVI-D)equations with three kinds of memory kernel is discussed in the infinite domain respectively.Based on the method of integral transform as well as special function,the paper testifies that the infinite solutions are obtained,which contains infinite series of generalized Mittag-Leffler function,Fox-H function,integral operator and integral forms.Then the analytical solution of one-dimensional homogeneous PVI-D equations with power law memory kernel at the initial value for the Dirac-? function is obtained.Finally,the analytical solution of homogeneous PVI-D equations with power law memory kernel is simulated.The numerical simulation results show that the analytical solutions of curves(surfaces)reach the peak at x(28)0,and graphical representation present the Gaussian symmetrical form as well have the characteristics of slow Gaussian decay distribution.(2)The analytical solution of one-dimensional non-homogeneous PVI-D equations with three kinds of memory kernel is discussed in the semi-infinite domain.Based on the FourierSine transform,Fourier-Cosine transform,Laplace transform as well the properties of MittagLeffler function and Fox-H function,this thesis demonstrates that the semi-infinite solution expression can be expressed using an infinite series of the generalized Mittag-Leffler function and Fox-H function similar to the analytic forms under infinite boundary conditions.(3)The analytical solution of non-homogeneous(one,two,three)-dimensional PVI-D equations with three kinds of memory kernel is discussed in the finite interval,circular domain and spherical domain respectively.Based on the methods of the separation of variables,the integral transform as well the special function,this thesis works out the analytic expression of multiple infinite series with trigonometric function,integral operator,the generalized MittagLeffler function,Bessel function and Legendre function.Finally,the analytical solution of two-dimensional homogeneous PVI-D equations with power law type memory kernel under the circular domain is simulated.The numerical simulation results show that the images present calyptriform and have the characteristics of slow dissipation.Meanwhile,the given contour line variation diagrams of surfaces clearly show the dissipation process of the energy,the intensive and sparse distribution of contour line determine the dissipation degree of the energy.(4)A class of analytical solutions on the type of non-homogeneous Fokker-Planck equations with three kinds of memory kernel is discussed under the infinite(finite)domain.Based on the method of separation of variables and integral transform,the corresponding analytic expression forms are obtained.