A mesh-free method for the numerical solution of the KdV–Burgers equation

AbstractThis paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the nonlinear dispersive and dissipative KdV–Burgers’ (KdVB) equation. The computed results show implementation of the method to nonlinear partial differential equations. This method has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. Accuracy of the method is assessed in terms of error norms L2,L∞, number of nodes in the domain of influence, parameter dependent RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving nonlinear dispersive and dissipative problems

New algorithms for numerical assessment of nonlinear integro-differential equations of second-order using Haar wavelets

This paper deals with the extended design for Fredholm and Volterra integral equations and design for Fredholm and Volterra integro-differential equations of first-order to second-order nonlinear Fredholm and second-order nonlinear Volterra integro-differential equations having square integrable kernels. This approach utilizes the inherent dynamics of the Haar wavelet. The Haar wavelet is used to provide a single platform for the proposed method. The method is tested on problems from literature, and numerical results are compared with existing methods. The numerical results indicate that the accuracy of the method is reasonably high, even on a coarse grid

An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only $O({\cal N})$ operations where ${\cal N}$ is the number of unknowns. Moreover,it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties

Optimum initialization of South Asian seasonal forecast using climatological relevant singular vectors

Designing an efficient seasonal forecasting system is ensuring that the uncertainty in the initial conditions is sampled optimally. Perturbation in the initial condition and the methodology used for sampling perturbation optimally plays a key role in the improvement of the current seasonal climate forecast. In this study the error growth properties of initial perturbation are investigated using climatically relevant singular vectors (CSVs). The Community Climate System Model version 4 (CCSM4) is used as a simulation tool to examine the growth of optimal perturbations with different lead times over the South Asian Monsoon region. It is found that reliable set of CSVs can be estimated by running an ensemble of model forecasts. Amplification of the optimal perturbations occurs for more than 1 month and possibly up to 6 months. The results show the growth rates of the singular vectors are very sensitive to the variable of perturbation, number of perturbations and the error norm. When the SV is used as an initial perturbation, the forecast skill of key atmospheric variables over South Asian Monsoon region is significantly improved. Further, it is demonstrated that the predictions with the singular vector have a more reliable ensemble spread, suggesting a potential merit for a probabilistic forecast. The promising results reported here should hopefully encourage further investigation of the methodology at different timescales.Non UBCUnreviewedAuthor affiliation: University of Northern British ColumbiaOthe

Non Arabic Words in the QUR’AN (Part-1)

The Holy Quran is the everlasting miracle which was revealed by  Allah  Al-Mighty in word  and meaning on His last messenger Sayyedena  Muhammad Mustafa (S.A.W). This is the Book of knowledge wisdom. The language of the Holy Quran is Arabic although several words have been derived from other languages e.g., Roman, Hebrew and Persian etc. In this article such words & terminologies have been identified from their origins. More over these words have been analyzed with respect to their meanings. It is worth considering that these words lose their originality whatsoever after getting  the Arabic structure