Numerical Solution for Kawahara Equation by Using Spectral Methods

In this paper, the dynamics of an incompressible fluid in a bounded connected domain, described by Navier-Stokes equations coupled with a heat transfer equation, is investigated by a method inspired from the non-commutative strategy developed by Bagarello, (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394).The solution of involved systems of partial differential equations is derived with the help of the unbounded self-adjoint densely defined Hamiltonian operator of the physical model and the Hankel transform.

Solving Navier-Stokes equations coupled with a heat transfer equation using Bagarello's approach and the Hankel transform

Some nonlinear wave equations are more difficult to solve analytically. Exponential Time Differencing (ETD) technique requires minimum stages to obtain the required accurateness, which suggests an efficient technique relating to computational duration that ensures remarkable stability characteristics upon resolving the nonlinear wave equations. This article solves the non-diagonal example of Fisher equation via the exponential time differencing Runge-Kutta 4 method (ETDRK4). Implementation of the method is demonstrated by short Matlab programs.

Numerical Solution of the Nonlinear Wave Equation via Fourth-Order Time Stepping

Nonlinear wave equations are more difficult to study mathematically, and no general analytical method exists for their solution. It is found that the Exponential Time Differencing (ETD) scheme requires the steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a nonlinear equation. This article solves the diagonal example of nonlinear heat equation via the exponential time difference Runge-Kutta 4 methods (ETDRK4). Implementation of the method is proposed by short Matlab programs.

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An efficient numerical technique for the solution of nonlinear heat equation via spectral method

In this note it is illustrated that the Exponential Time Differencing (ETD) scheme needs the least steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a stiff type problem. Nonetheless, the celebrated and well established method like RungeKutta is still being applied as the basis of many efficient codes. However, the stiff type problems seem cannot be solved efficiently via some of these methods. This note overcomes such stiff type problem via the exponential method. Furthermore, the exponential time differencing Runge-Kutta 4 method (ETDRK4) is used to solve the diagonal example of a well known nonlinear partial differential equation (PDE) in the form of Burgers’ equation. In addition, we use Fourier transformation for solving Burgers’ equation. Mathematics Subject Classification: 65M70, 65Z05

Hassan Panj Mini

Papers

A note on fourth-order time stepping for stiff PDE via spectral method

An efficient numerical technique for the solution of nonlinear heat equation via spectral method