Search

The main objective of this article is to present an efficient numerical method to solve the delay differential equation of fractional order. We use the Caputo's fractional derivative for solving the fractional differentiation. The properties of shifted Chebyshev polynomials are exploited to reduce the Delay Fractional Differential Equation (DFDE) to a linear or non-linear easily solvable system of algebraic equations. A comparison is given between the present method and Adomian Decomposition Method (ADM) with the help of solved numerical illustrative example. The results shows that proposed method is very effective and simple. Which reveals the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

In this paper, a new block interface domain decomposition method (BI-DDM) with non-overlapping subdomains for the numerical solution of a two-dimensional convection-diffusion equation is presented. The block interface formulation is derived from the idea of using small groups of a certain number of mesh points where this group is treated explicitly similar to the way a single point is treated in the point method. The BI-DDM is incorporated with a correction phase which is able to economize further on the computing cost. The performance analysis of this method on several recently developed group iterative schemes implemented on a message-passing architecture are presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, a group iterative scheme based on rotated (cross) live-point unite difference discretisation, i.e. the four-point explicit decoupled group (EDG) is considered in solving a second order elliptic partial differential equation (PDE). This method was firstly introduced by Abdullah ["The four point EDO method: a fast poisson solver", Int. J. Comput. Math., 38 (1991) 61-70], where the method was found to be more superior than the common existing methods based on the standard live-point unite difference discretisation, The method was further extended to different type of PEW's, where similar improved results were established [Ali N.H.M., Abdullah, A.R. Four Point EDG: A Fast Solver For The Navier-Stokes Equation, M.H.Hamza (ed.) Proceedings of the IASTED International Conference on Modelling Simulation And Optimization, Gold Coast, Australia, May 6-9(1996) (CD Rom-File 242-165.pdf), ISBN: 0-88986-197-8; Ali, N,H.M., Abdullah, A.R. New Parallel Point iterative Solutions For the Diffusion-Convection Equation Proceedings of the International Conference on Parallel and Distributed Computing and Networks Singapore, Aug. 11-13 (1997) 136-139; Ali, N.H.M., Abdullah, AR. "New rotated iterative algorithms for the solution of a coupled system of elliptic equations" Int. J. Comput. Math. 74 (1999) 223-251). These new iterative algorithms had been developed to run on the Sequent Balance, a shared memory parallel computer [A.R. Abdullah, N.M. Ali, The Comparative Study of Parallel Strategies For The Solution of Elliptic PDE's Parallel Algorithms and Applications Vol. 10 (1996) 93-103; Ali, N.H.M., Abdullah, A.R. "Parallel four point explicit decoupled group (EDG) method for elliptic PDE' s" Proceedings of the Seventh IASTED/ISMM international Conference on Parallel and Distributed Computing and Systems (1995) 302-304 (Washington DC); Ali. N.H.M., Abdullah, A.R. New Parallel Point Iterative Solutions For the Diffusion-Convection Equation Proceedings of the International Conference on Parallel and Distributed Computing and Networks, Singapore. Aug. 11-13 (1997) 136-139: Yousif, W.S., Evans, D.J. "Explicit decoupled group iterative methods and their parallel implementations" Parallel Algorithms and Applications 7 (1995) 53-71] where they were shown to be suitable to be implemented in parallel. In this work, the four-point group algorithm was ported to run on a cluster of Sun workstations using a parallel virtual machine (PVM) programming environment together with the four-point explicit group EG method [Evans. D.J., Yousif. W.S. "The implementation of the explicit block iterative methods on the balance 8000 parallel computer" Parallel Computing 16 (1990) 81 -97]. We describe the parallel implementations of these methods in solving the Poisson equation and the results of some computational experiments are compared and reported. [ABSTRACT FROM AUTHOR]

Due to their rapid convergence properties, recent focus on iterative methods in the solution of linear system has seen a flourish on the use of gradient techniques which are primarily based on global minimisation of the residual vectors. In this paper, we conduct an experimental study to investigate the performance of several preconditioned gradient or variational techniques to solve a system arising from the so-called rotated (skewed) finite difference discretisation in the solution of elliptic partial differential equations (PDEs). The preconditioned iterative methods consist of variational accelerators, namely the steepest descent and conjugate gradient methods, applied to a special matrix 'splitting' preconditioned system. Several numerical results are presented and discussed. † E-mail: dj.evans@ntu.ac.uk [ABSTRACT FROM AUTHOR]

A second-order finite difference scheme derived from rotated discretisation formula is employed in conjunction with a preconditioner to obtain highly accurate and fast numerical solution of the two-dimensional elliptic partial differential equation. The use of a 'splitting' preconditioning strategy will be shown to improve the spectral properties of the matrix of the linear system resulting from this discretisation by minimising the eigenvalue spectrum of the transformed matrix. The application of this technique to several acceleration iterative methods, such as Simultaneous displacement, Richardson's and Chebyshev accelerated methods, are presented and discussed. † E-mail: dj.evans@ntu.ac.uk [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.