Numerical solutions of the generalized Burgers-Fisher equation are presented based on a polynomial-based differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a third-order strong stability preserving Runge-Kutta scheme in time have been used. The proposed technique successfully worked t...

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial-boundary value problems with time-dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order of convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved and computati...

This research demonstrates a solution for the DICE 2007 integrated assessment model in the continuous domain through the use of a Runge-Kutta sampling technique for solving differential transcendental equations. The use of a savings ratio helper constraint was not required. It is shown that the introduction of a savings ratio constraint leads to a 12% underestimation of maximum atmospheric temp...

in this paper, numerical spline-based differential quadrature is presented for solving the boundary and initial value problems, and its application is used to solve the fixed rectangular membrane vibration equation. for the time integration of the problem, the runge–kutta and spline-based differential quadrature methods have been applied. the runge–kutta method was unstable for solving the prob...

This paper presents a new simple technique to improve the order behaviour of Runge-Kutta methods when applied to index 2 DAEs. It is then shown how this can be incorporated into a more eecient version of the code RADAU5 developed by E. Hairer and G. Wanner.

Literature For a great deal of information on Runge-Kutta methods consult J.C. Butcher, Numerical Methods for Ordinary Differential Equations, second edition, Wiley and Sons, 2008, ISBN 9780470723357. That book also has a good introduction to linear multistep methods. In these notes we refer to this books simply as Butcher. The notes were written independently of the book which accounts for som...

In this paper, a numerical technique is applied to a five variable giving up smoking fractional mathematical model. This model is based on five types of smokers, i.e. potential, occasional, heavy, temporary quitters and permanent quitters. Efficacy of Perturbation Iteration Algorithm on fractional system of differential equations is shown graphically between standard Runge-Kutta method and PIA.

In this paper we consider the numerical solution of taxis–diffusion–reaction models. Such nonlinear partial differential equation models appear often in mathematical biology and we consider examples from tumour growth and invasion, and the aggregation of amoebae. These examples are characterised by a taxis term which is large in magnitude compared with the diffusion term and this can cause nume...

Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and 'solving for the error estimate' are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques...

In this work we consider Symplectic Runge Kutta Nyström methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nyström of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply the new methods on several Hamiltonian systems and on the computation of the eigenval...