In this paper, we introduce the numerical solution of the system of SEIR nonlinear ordinary differential equations, which are studied the effect of vaccine on the HIV (Human Immunology virus). We obtained the numerical solutions on stable manifolds by Runge-Kutta fourth order method.

In this short note we show how to obtain Lagrangian integrators from known sym-plectic partitioned Runge-Kutta methods.

This is a survey on discretizing delay equations from a geometric–qualitative view– point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center–unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various C j to...

We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form ẏ = J∇H(y) . Such a class of methods contains as particular cases the methods of Miesbach and Pesch [13], and also the family of symplectic Runge-Kutta methods. As in the case of the methods introduced in [13], the new integration methods are constructed by defining a generating functi...

Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by m-dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical B-convergence theory to...

A criterion is established for the convergence of sequences of a general type of Runge-Kutta method. The criterion is expressed in terms of "weak convergence", a property defined in the paper.

This paper presents new fifth-order diagonally implicit Runge-Kutta integration formulas for stiff initial value problems, designed to be Lstable method. The stability of the method is analyzed and numerical results are shown to verify the conclusions. Mathematics Subject Classifications: 51N20, 62J05, 70F99

The present paper develops a theory of Natural Continuous Extensions (NCEs) for the discrete approximate solution of an ODE given by a Runge-Kutta process. These NCEs are defined in such a way that the continuous solutions furnished by the one-step collocation methods are included.

In this paper, we consider the parallel solution of nonstii ordinary diierential equations with two diierent classes of Runge-Kutta (RK) methods providing embedded solutions: classical embedded RK methods and iterated RK methods which were constructed especially for parallel execution. For embedded Runge-Kutta methods, mainly the potential system parallelism is exploited. Iterated RK methods pr...