| Delay integro-differential equations are important kind of equations in engineering,which have been widely used in mechanics,biology and other fields.However,the exact solutions of these kind of equations are not easy to obtain,so it is particularly important to explore the high-precision numerical solutions of these equations.Based on the one-leg methods for solving initial value problem of ordinary differential equations,a new class of oneleg methods were constructed to solve initial value problems of delay integro-differential equations,and the convergence,nonlinear stability and dissipativity of the methods were proved under the classical and one-sided Lipschitz conditions,.In the first chapter,we introduce the research background,history and present situation of delay integro-differential equations,as well as the main research contents of this paper.In chapter 2,for initial value problems of delay integro-differential equations,we construct a type of new one-leg methods that can solve the problem.In order to analyze the convergence of new one-leg methods,we construct a numerical scheme with disturbance term by adding the perturbation term to the method,and the error analysis of the method and the convergence results are obtained.Finally,theoretical results are verified by numerical experiments and then compared with other numerical methods.In chapter 3,we use the new one-leg method to solve initial value problems of delay integro-differential equations and their perturbed problems,and the nonlinear stability of the method is proved as well.Furthermore,we also prove the dissipativity of new one-leg methods for solving initial value problems of delay integro-differential equations.Finally,the numerical experimental results show that the nonlinear stability and nonlinear dissipativity of the method are consistent with the theoretical analysis.In the last chapter,we make a brief summary of the work of this paper and explain some problems to be further studied. |
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