Stability Analysis Of Collocation Runge-kutta Methods For Volterra Delay Integral Equation

The integral equation is an important part in modern mathematics. manydisciplines, such as differential equations, computational mathematics, stochasticanalysis, modern functional analys is are in close contact with it. Two-way linkingmathematics and physics, delay integral equations develop rapidly as a branch ofmathematics, and become an important discipline. Delay integral problem is of greatsignificance for scientific research, and worthy of further exploration. Volterra delayintegral equation as a special case of delay integral equation is also favored by theacademic experts. Because sometimes very small delay will produce a very largeimpact on the system analysis of the added delay, it is partic ularly important. Thispaper studies configure RK method for solving the stability problem of the twotypes of Volterra integral equations with delay.Content and results of this study are:Delay integral equations and numerical analysis of problems related to theintroduction of the background and significance, recalling the stability results ofdelay integral equations and numerical methods and focuses on the status of suchequation method.The results of the paper is to establish a single delay integral equation ofVolterra-Runge-Kutta method, given the stability of the test equation method andorder defined, and the vector in the form of the equation, and then focuses on theconfiguration of the stability of the RK method, error by the linear case is extendedto the nonlinear situation. Finally, we give the example of a single delay, and getbetter error results.The Volterra-Runge-Kutta method for double delay integral equations comesnext, the article also gives the vector form of the equation, and ana lyze its stability,and finally get a suffic ient condition of P-stable. It is also pointed out that a singledelay and dual delay situations have common ground. In the same, we give thedual-delay example and point out the reason why the method is not P-stable.In the end, the article gives a summary of the conclus ion and looks into thefuture about research directions and prospects for the development of integral delayequation.