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In this article, an unconditionally stable compact high-order iterative finite difference scheme is developed on solving the two-dimensional fractional Rayleigh–Stokes equation. A relationship between the Riemann–Liouville (R–L) and Grunwald–Letnikov (G–L) fractional derivatives is used for the time-fractional derivative, and a fourth-order compact Crank–Nicolson approximation is applied for the space derivative to produce a high-order compact scheme. The stability and convergence for the proposed method will be proven; the proposed method will be shown to have the order of convergence O (τ + h 4) . Finally, numerical examples are provided to show the high accuracy solutions of the proposed scheme. [ABSTRACT FROM AUTHOR]

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, new group iterative numerical schemes based on the centred and rotated (skewed) seven-point finite difference discretisations are proposed for the solution of a three dimensional second order hyperbolic telegraph equation, subject to specific initial and Dirichlet boundary conditions. Both schemes are shown to be of second order accuracies and unconditionally stable. The scheme derived from the rotated grid stencil results in a reduced linear system with lower computational complexity compared to the scheme derived from the centred approximation formula. A comparative study with other common point iterative methods based on the seven-point centred difference approximation together with their computational complexity analyses is also presented. [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]

Abstract: In this paper, we present the development of new explicit group relaxation methods which solve the two dimensional second order hyperbolic telegraph equation subject to specific initial and Dirichlet boundary conditions. The explicit group methods use small fixed group formulations derived from a combination of the rotated five-point finite difference approximation together with the centered five-point centered difference approximation on different grid spacings. The resulting schemes involve three levels finite difference approximations with second order accuracies. Analyses are presented to confirm the unconditional stability of the difference schemes. Numerical experimentations are also conducted to compare the new methods with some existing schemes. [Copyright &y& Elsevier]

Abstract: This paper presents the numerical solution of the two-dimensional convection–diffusion partial differential equation (pde) discretized by several group iterative schemes based on the centred and rotated (skewed) five-point finite difference discretizations, namely the explicit group (EG) and explicit decoupled group (EDG) methods, respectively [W.S. Yousif, D.J. Evans, Explicit group over-relaxation methods for solving elliptic partial differential equations, Mathematics and Computer in Simulation 28 (1986) 453–466; A.R. Abdullah, The four-point explicit decoupled group (EDG) method: a fast Poisson solver, International Journal of Computer Mathematics 38 (1991) 61–70]. The application of a modified 2×2 block factorization preconditioner applied to the linear systems that arise from these group iterative schemes is discussed. Several Krylov subspace methods, such as Bi-CGSTAB, GMRES and TFQMR will be used to solve the preconditioned systems. The experimental results show that the condition numbers of the coefficient matrices of the transformed preconditioned systems are substantially reduced and thus result in improved convergence rates. The comparative performance analysis between the group schemes also indicate that the preconditioned EDG with Bi-CGSTAB acceleration technique is able to gain the most efficiency amongst the schemes tested. [Copyright &y& Elsevier]

Abstract: Since their introduction, the four-point explicit group (EG) and explicit decoupled group (EDG) methods in solving elliptic PDE’s have been implemented on various parallel computing architectures such as shared memory parallel computer and distributed computer systems. However, no detailed study on the performance analysis of these algorithms was done in any of these implementations. In this paper we developed performance models for these explicit group methods and present detailed study of their hypothetical implementation on two distributed memory multicomputers with different computation speed and communication bandwidth. Detailed performance analysis based on these models predicted different theoretical performance if the methods were implemented on the clusters. This was confirmed by the experimental results performed on the two distinct clusters. Theoretical analysis and experimental results indicated that both explicit group methods are scalable with respect to number of processors and the problem size. [Copyright &y& Elsevier]

Abstract: In Martins et al. [M.M. Martins, W.S. Yousif, D.J. Evans, Explicit group AOR method for solving elliptic partial differential equations, Neural, Parallel and Science Computation 10(4) (2002) 411–422], a new explicit four-point group accelerated OverRelaxation (Group AOR) iterative method was presented where the computational superiority of this new technique was established when compared with the point AOR method developed by Evans and Martins [D.J. Evans, M.M. Martins, The AOR method For , International Journal of Computer Mathematics 52 (1994) 75–82] and Martins et al. [M.M. Martins, D.J. Evans, M.E. Trigo, The AOR iterative method for new preconditioned linear systems, Journal of Computational and Applied Mathematics 132 (2001) 461–466]. In this work, we formulate an alternative group scheme from the AOR family derived from the rotated (skewed) five point formula [G. Dahlquist, A. Bjorck, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974, p. 320; Y. Saad, Iterative Methods For Sparse Linear Systems, second ed., PWS Publishing Company, Boston, 2000, p. 50]. The derivation of this new group scheme is presented and its performance is compared with the existing explicit four-point Group AOR. We include the analysis of the convergence results for the new group scheme. Numerical experiments are also presented to illustrate our results. [Copyright &y& Elsevier]

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]

In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The (2 + 1) -dimensional Chaffee–Infante equation and the dimensionless form of the Zakharov equation have widespread scopes of function in science and engineering fields, such as in nonlinear fiber optics, the waves of electromagnetic field, plasma physics, the signal processing through optical fibers, fluid dynamics, coastal engineering and remarkable to model of the ion-acoustic waves in plasma, the sound waves. In this article, the first integral method has been assigned to search closed form solitary wave solutions to the previously proposed nonlinear evolution equations (NLEEs). We have constructed abundant soliton solutions and discussed the physical significance of the obtained solutions of its definite values of the included parameters through depicting figures and interpreted the physical phenomena. It has been shown that the first integral method is powerful, convenient, straightforward and provides further general wave solutions to diverse NLEEs in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1 / 2 < α < 1 . The iterative scheme, called the fractional skewed grid Crank–Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 450 from the standard grid. The skewed finite difference scheme combined with Crank–Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank–Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy. [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]