جستجوی مقالات مرتبط با کلیدواژه "singularly perturbed problem" در نشریات گروه "ریاضی"

This study presents a numerical approach for solving temporal fractionalorder singularly perturbed parabolic convection-diffusion differential equations with a delay using a uniformly convergent scheme. We use the asymptotic analysis of the problem to offer a priori bounds on the exact solution and its derivatives. To discretize the problem, we use the implicit Euler technique on a uniform mesh in time and the midpoint upwind finite difference approach on a piece-wise uniform mesh in space. The proposed technique has a nearly first-order uniform convergence order in both spatial and temporal dimensions. To validate the theoretical analysis of the scheme, two numerical test situations for various values of ε are explored.

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem’s asymp-totic analysis. The Euler’s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of μ (free param-eter, when the free parameter μ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.

We develop a robust uniformly convergent numerical scheme for singularly perturbed time dependent Burgers-Huxley partial differential equation. We first discretize the time derivative of the equation using the Crank-Nicolson finite difference method. Then, the resulting semi-discretized nonlinear ordinary differential equations are linearized using the quasilinearization technique, and finally, design a fitted operator upwind finite difference method to resolve the layer behavior of the solution in the spatial direction. Our analysis has shown that the presented method is second order parameter uniform convergent in time and first order in space. Numerical experiments are conducted to validate the theoretical results.

In this paper, singularly perturbed differential equations having delay on the convection and reaction terms are considered. The highest order derivative term in the equation is multiplied by a perturbation parameter epsilon taking arbitrary values in the interval (0; 1]. For small epsilon, the problem involves a boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convective term. The terms involving the delay are approximated using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted upwind finite difference method. The stability of the proposed scheme is analysed and investigated using maximum principle and barrier functions for solution bound. The formulated scheme converges independent of the perturbation parameter with rate of convergence O(N−1). Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme to order O(N−2). To validate the theoretical finding, three model examples having boundary layer behaviour are considered. The maximum absolute error and rate of convergence of the scheme are computed. The proposed scheme gives accurate and parameter uniformly convergent result.

We consider a class of singularly perturbed semilinear three-point boundary value problems. An accelerated uniformly convergent numerical method is constructed via the exponential fitted operator method using Richardson extrapolation techniques to solve the problem. To treat the semilinear term, we use quasi-linearization techniques. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε, where the classical numerical methods fail to give a good result. It also improves the results of the methods existing in the literature. The method is shown to be second-order convergent independent of perturbation parameter ε.

This paper deals with the numerical treatment of singularly perturbed delay differential equations having a delay on the first derivative term. The solution of the considered problem exhibits boundary layer behavior on the left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting in an asymptotically equivalent singularly perturbed boundary value problem. The uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.

‎In this study‎, ‎a robust computational method involving exponential cubic spline for solving singularly perturbed parabolic convection-diffusion equations arising in the modeling of neuronal variability has been presented‎. ‎Some numerical examples are considered to validate the theoretical findings‎. ‎The proposed scheme is shown to be an $varepsilon-$uniformly convergent accuracy of order $ Oleft( left( Delta tright)‎ +‎h^2 right) $‎.

In this article, singularly perturbed differential difference equations having delay and advance in the reaction terms are considered. The highest-order derivative term of the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval (0, 1]. For the small value of ε, the solution of the equation exhibits a boundary layer on the left or right side of the domain depending on the sign of the convective term. The terms with the shifts are approximated by using the Taylor series approximation.The resulting singularly perturbed boundary value problem is solved using an exponentially fitted tension spline method. The stability and uniform convergence of the scheme are discussed and proved. Numerical exam ples are considered for validating the theoretical analysis of the scheme. The developed scheme gives an accurate result with linear order uniform convergence.

Singularly perturbed robin type boundary value problems with discontinuous source terms applicable in geophysical fluid are considered. Due to the discontinuity, interior layers appear in the solution. To fit the interior and boundary layers, a fitted nonstandard numerical method is constructed. To treat the robin boundary condition, we use a finite difference formula. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε, and mesh size, h. The numerical result is tabulated, and it is observed that the present method is more accurate and uniformly convergent with order of convergence of O(h).

In this paper, a new algorithm is presented to approximate the solution of a singularly perturbed boundary value problem with leftlayer based on the homotopy perturbation technique and applying the Laplace transformation. The convergence theorem and the error bound of the proposed method are proved. The method is examined by solving two examples. The results demonstrate the reliability and efficiency of the proposed method.