Volume 22, Issue 4

- Local absorbing boundary conditions with fifth order of absorption to approximate the solution of an initial value problem, for the spatially discretized wave equation, are considered. For the one dimensional case, it is proved necessary conditions for well posedness. Numerical experiments confirming well posedness and showing good results of absorption are included.

-By use of an additive subspace preconditioning method it is shown how a set of small eigenvalues of a given system operator can be moved to the upper part of the spectrum, thereby significantly improving the spectral condition number and the rate of convergence of the iterative solution method used for the corresponding preconditioned system.

-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

- In this paper, a novel modification of the spectral-homotopy analysis method (SHAM) technique for solving highly nonlinear initial value problems that model systems with chaotic and hyper-chaotic behaviour is presented. The proposed method is based on implementing the SHAM on a sequence of multiple intervals thereby increasing it’s radius of convergence to yield highly accuratemethod which is referred to as the piece-wise spectral homotopy analysis method (PSHAM). We investigate the application of the PSHAM to the L¨u system [20] which is well known to display periodic, chaotic and hyper-chaotic behaviour under carefully selected values of it’s governing parameters. Existence and uniqueness of solution of SHAM that give a guarantee of convergence of SHAM, has been discussed in details. Comparisons are made between PSHAMgenerated results and results from literature and Runge-Kutta generated results and good agreement is observed.

- In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.