Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution

AbstractIn this paper, we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection–diffusion two-point boundary value problem. The solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter. In addition, we prove the uniform first-order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain. Numerical results are presented that demonstrate the sharpness of our results

Perturbation Expansion to the Solution of Differential Equations

The main purpose of this chapter is to describe the application of perturbation expansion techniques to the solution of differential equations. Approximate expressions are generated in the form of asymptotic series. These may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. These analytical techniques provide an alternative to the direct computer solution. Before attempting to solve these problems numerically, one should have an awareness of the perturbation approach

Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion.

[EN]In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh

Numerical treatment for the solution of singularly perturbed pseudo-parabolic problem on an equidistributed grid

The initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a difference scheme which consists of the implicit-Euler method for the time derivative and a central difference method for the spatial derivative on uniform mesh. The time domain is discretized with a nonuniform grid generated by equidistributing a positive monitor function. The performance of the numerical scheme is tested which confirms the expected behavior of the method. The existing method is compared with other methods available in the recent literature

Numerical simulation for two species time fractional weakly singular model arising in population dynamics

In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order α∈(0,1). The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near t=0. Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme.</p

A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations

In this paper, a singularly perturbed Volterra integro- differential equation is being surveyed. On a piecewise-uniform Shishkin mesh, a fitted mesh finite difference approach is applied using a composite trapezoidal rule in the case of integral component and a finite difference operator for the derivative component. The proposed technique acquires a uniform convergence in accordance with the perturbation parameter. To improve the accuracy of the computed solution, an extrapolation, specifically Richardson extrapolation, is used measured in the discrete maximum norm and almost second-order convergence is attained. Further numerical results are provided to assist the theoretical estimates.WOS:0006970795000012-s2.0-8511514782

A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations

In this paper, a singularly perturbed Volterra integro- differential equation is being surveyed. On a piecewise-uniform Shishkin mesh, a fitted mesh finite difference approach is applied using a composite trapezoidal rule in the case of integral component and a finite difference operator for the derivative component. The proposed technique acquires a uniform convergence in accordance with the perturbation parameter. To improve the accuracy of the computed solution, an extrapolation, specifically Richardson extrapolation, is used measured in the discrete maximum norm and almost second-order convergence is attained. Further numerical results are provided to assist the theoretical estimates.WOS:0006970795000012-s2.0-8511514782

Asymmetric nucleosome PARylation at DNA breaks mediates directional nucleosome sliding by ALC1

Abstract The chromatin remodeler ALC1 is activated by DNA damage-induced poly(ADP-ribose) deposited by PARP1/PARP2 and their co-factor HPF1. ALC1 has emerged as a cancer drug target, but how it is recruited to ADP-ribosylated nucleosomes to affect their positioning near DNA breaks is unknown. Here we find that PARP1/HPF1 preferentially initiates ADP-ribosylation on the histone H2B tail closest to the DNA break. To dissect the consequences of such asymmetry, we generate nucleosomes with a defined ADP-ribosylated H2B tail on one side only. The cryo-electron microscopy structure of ALC1 bound to such an asymmetric nucleosome indicates preferential engagement on one side. Using single-molecule FRET, we demonstrate that this asymmetric recruitment gives rise to directed sliding away from the DNA linker closest to the ADP-ribosylation site. Our data suggest a mechanism by which ALC1 slides nucleosomes away from a DNA break to render it more accessible to repair factors