Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficient...

The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of rst-order diierential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In ...

The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient tha...

This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to cheap practical implementation. Some of the stage values calculated in a step of the numerical int...

in this article, we present a fractional order hiv-1 infectionmodel of cd4+ t-cell. we analyze the effect of the changing the averagenumber of the viral particle n with initial conditions of the presentedmodel. the laplace adomian decomposition method is applying to checkthe analytical solution of the problem. we obtain the solutions of thefractional order hiv-1 model in the form of infinite se...

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combines adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual of the equation, and the Runge-Kutta method is supplied via its Butcher’s table. Ar...

In this paper, a hybrid technique of differential quadrature method and Runge-Kutta fourth order method is employed to analyze reaction-diffusion problems. The obtained results are compared with the available analytical ones. Further, a parametric study is introduced to investigate the influence of reaction and diffusion characteristics on behavior of the obtained results. Index Term-Reaction-d...

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear alg...