Anew algorithm is proposed for the numerical solution of threshold problems in epidemics and population dynamics. These problems aremodeled by the delay-differential equations, where the delay function is unknown and has to be determined from the threshold conditions. The new algorithm is based on embedded pair of continuous Runge–Kutta method of order p = 4 and discrete Runge–Kutta method of o...

Usually the straightforward generalization of explicit Runge{Kutta methods for ordinary diierential equations to half-explicit methods for diierential-algebraic systems of index 2 results in methods of order q 2 ((8]). The construction of higher order methods is simpliied substantially by a slight modiication of the method combined with an improved strategy for the computation of the algebraic ...

Sugiura, H. and T. Torii, A method for constructing generalized Runge-Kutta methods, Journal of Computational and Applied Mathematics 38 (1991) 399-410. In the implementation of an implicit Runge-Kutta formula, we need to solve systems of nonlinear equations. In this paper, we analyze the Newton iteration process and a modified Newton iteration process for solving these equations. Then we propo...

A new diagonally implicit Runge-Kutta-Nyström (RKN) method is developed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. Presented is a method which is three-stage fourth-order with dispersive order six and 'small' principal local truncation error terms and dissipation constant. The analysis of phase-lag, dissipatio...

In this article, we report results of an experimental study on six iterative methods to compute the transient probabilities of large Markov models: full matrix exponentiation, forward Euler method, explicit Runge-Kutta methods of order 2 and 4 and Adams-Bashforth multi-steps methods of order 2 and 4. We suggest a simple but efficient implementation of these algorithms. We discuss how to tune th...

The 4-th order Runge-Kutta method in the complex plane is proposed for numerically advancing the solutions of a system of first order differential equations in one external invariant satisfied by the master integrals related to a Feynman graph. Some results obtained for the 2-loop self-mass MI are reviewed. The method offers a reliable and robust approach to the direct and precise numerical eva...

The 4-th order Runge-Kutta method in the complex plane is proposed for numerically advancing the solutions of a system of first order differential equations in one external invariant satisfied by the master integrals related to a Feynman graph. The particular case of the general massive 2-loop sunrise self-mass diagram is analyzed. The method offers a reliable and robust approach to the direct ...