Kapil K. Sharma

A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers' equation

In this work, a numerical scheme based on weighted average differential quadrature method is proposed to solve time dependent Burgers equation with appropriate initial and boundary conditions. In first step, time derivative is discretized by forward difference method. Then, quasilinearization process is used to tackle the non-linearity in the equation. The fully discretization leads to a system of linear equations which is solved by Gauss-elimination method. The method is analyzed for stability and convergence. Finally, the adaptability of proposed scheme is demonstrated by numerical experiments and compared with some existing numerical methods in literature. It is found that the proposed numerical scheme produce accurate results and quite easy to implement.

A review on singularly perturbed differential equations with turning points and interior layers

Singular perturbation problems with turning points arise as mathematical models for various physical phenomena. The problem with interior turning point represent one-dimensional version of stationary convection-diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. Boundary turning point problems, on the other hand, arise in geophysics and in modeling thermal boundary layers in laminar flow. In this paper, we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems. The purpose is to find out what problems are treated and what numerical/asymptotic methods are employed, with an eye towards the goal of developing general methods to solve such problems. Since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011.

Numerical method for solving fractional coupled Burgers equations

In this paper, we use the fractional variational iteration method (FVIM) to solve a time- and space-fractional coupled Burgers equations. Some numerical examples are presented to show the efficiency of considered method. A comparison of the proposed method is made with the exact solution, adomain decomposition method (ADM), generalized differential transformation method (GDTM) and homotopy perturbation method (HPM).

An O(N^{-1}lnN)^4 Parameter Uniform Difference Method for Singularly Perturbed Differential-Difference Equations

In this paper, we propose an O((N^{-1}lnN)^4) parameter uniform numerical scheme for singularly perturbed differential-difference equations (SPDDE) having both delay and advance arguments in reaction term. These types of problems are ubiquitous in many mathematical models of physical and biological phenomena. Piecewise uniform fitted mesh with improved fourth-order numerov method is used. A parameter uniform error estimate of order (O(N^{-1}lnN)^{4}) is proved. We calculate numerical solution of some examples using the proposed method to establish the higher order parameter uniform estimates.

Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings.

Cost-effectiveness of classification ensembles

Ensemble pruning is an important task in supervised learning because of the performance and efficiency advantage it begets to predictive modelling. Performance based empirical comparison (primarily on accuracy) is the most common modus operandi for critical evaluation of ensembles pruned by different algorithms. Deep analysis of existing literature reveals that ensemble size is an ignored attribute while judging the quality of ensembles.In this paper, we argue that the cost-effectiveness of an ensemble is a function of both performance and size. Hence, equitable comparison of two ensembles must take into account both these attributes to judge their relative merits. Following this argument, we propose an objective function called accrual function which quantifies the difference in performance and size of two ensembles, to gauge their relative cost-effectiveness. The function can be parameterized and has nice mathematical properties. Semantic interpretations of these properties are delineated in the paper. Finally, we apply the accrual function on published results from selected publications and demonstrate its ability to beget clarity while comparing ensembles. HighlightsProposed accrual function compares two ensembles on the basis of performance and size.It returns a single numerical value as the result of comparison.It can be parameterized to bias towards smaller size/high performance ensembles.Its utility is demonstrated by applying the function to select published results.Applications with memory constraints can bias the choice to smaller ensembles.

Singularly Perturbed Convection-Diffusion Turning Point Problem with Shifts

In this paper, a class of singularly perturbed turning point problem with shifts (i.e., delay as well as advance) is considered. Presence of turning point results into twin boundary layers in the solution of the problem under consideration. For the numerical approximation of the problem, a finite difference scheme is proposed on a uniform mesh. Interpolation is used to tackle the terms containing shifts and to deal with the difficulty arising due to presence of the turning point a combination of backward and forward difference is used in the first derivative term. Convergence analysis is given for the proposed numerical scheme. Numerical results are presented which illustrate the theoretical results and depict the effect of shifts on the layer behavior of the solution.