Exp-Function Method for Duffing Equation and New Solutions of (2+1) Dimensional Dispersive Long Wave Equations

In this paper, the general solutions of the Duffing equation with third degree nonlinear term is obtain using the Exp-function method. Using the Duffing equation and its general solution, the new and general exact solution with free parameter and arbitrary functions of the (2+1) dimensional dispersive long wave equation are obtained. Setting free parameters as special values, hyperbolic as well as trigonometric function solutions are also derived. With the aid of symbolic computation, the Exp-function method serves as an effective tool in solving the nonlinear equations under study. Key words: Exp-function method; Duffing equation; Exact solutions; Nonlinear evolution equation

Improving pipelined time stepping algorithm for distributed memory multicomputers

Time stepping algorithm with spatial parallelisation is commonly used to solve time dependent partial differential equations. Computation in each time step is carried out using all processors available before sequentially advancing to the next time step. In cases where few spatial components are involved and there are relatively many processors available for use, this will result in fine granularity and decreased scalability. Naturally one alternative is to parallelise the temporal domain. Several time parallelisation algorithms have been suggested for the past two decades. One of them is the pipelined iterations across time steps. In this pipelined time stepping method, communication however is extensive between time steps during the pipelining process. This causes a decrease in performance on distributed memory environment which often has high message latency. We present a modified pipelined time stepping algorithm based on delayed pipelining and reduced communication strategies to improve overall execution time on a distributed memory environment using MPI. Our goal is to reduce the inter-time step communications while providing adequate information for the next time step to converge. Numerical result confirms that the improved algorithm is faster than the original pipelined algorithm and sequential time stepping algorithm with spatial parallelisation alone. The improved algorithm is most beneficial for fine granularity time dependent problems with limited spatial parallelisation

Generalized and Improved (G'/G)-Expansion Method for Nonlinear Evolution Equations

A generalized and improved (G'/G)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov- Benjamin-Bona-Mahony �ZKBBM� equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering

Modified Explicit Group AOR Methods in the Solution of Elliptic Equations

Abstract The recent convergence results of faster group iterative schemes from the Accelerated OverRelaxation (AOR) family has initiated considerable interest in exploring the ehavior of these methods in the solution of partial differential equations (pdes). 2466 Norhashidah Hj. Mohd Ali and Foo Kai Pin schemes. Numerical experimentations of this new modified AOR group method will show significant improvement in computational complexity and execution timings compared to the group AOR formulation presented i

The improved F-expansion method with Riccati equation and its applications in mathematical physics

The improved F-expansion method combined with Riccati equation is one of the most effective analytical methods in finding the exact traveling wave solutions to non-linear evolution equations in mathematical physics. In this article, this method is implemented to investigate new exact solutions to the Drinfel’d–Sokolov–Wilson (DSW) equation and the Burgers equation. The performance of this method is reliable, direct, and simple to execute compared to other existing methods. The obtained solutions in this work are imperative and significant for the explanation of some practical physical phenomena

Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results

Preconditioned Modified Explicit Decoupled Group for the Solution of Steady State Navier-Stokes Equation

Combining iterative schemes with suitable preconditioners may improve the rate of the convergence of the methods. However, the real difficulty lies in the construction of the correct preconditioners applied to the formulated schemes. In this paper, the Modified Explicit Decoupled Group Successive Over-Relaxation method is formulated to solve the two dimensional steady-state Navier-Stokes equations. A new block splitting preconditioned matrix is applied to the formulated scheme as an effort to accelerate the convergence rate. Numerical experiments are carried out to confirm the effectiveness of the preconditioner in terms of accuracy and execution timings. Comparison with its unpreconditioned counterpart will also be reported