Accelerated Numerical Method for Singularly Perturbed Differential Difference Equations

H. Debela, G. Duressa, Masho Jima Kebeto
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Abstract

—In this paper, accelerated finite difference method for solving singularly perturbed delay reaction-diffusion equations is presented. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is obtained, which can easily be solved by Thomas algorithm. The consistency, stability and convergence of the method have been established. To increase the accuracy of our established scheme we used Richard- son’s extrapolation techniques. To validate the applicability of the proposed method, four model examples have been considered and solved for different values of perturbation parameters and mesh sizes. The numerical results have been presented in tables and graphs to illustrate; the present method approximates the exact solution very well. Moreover, the present method gives better accuracy than the existing numerical methods mentioned in the literature.
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奇摄动微分差分方程的加速数值解法
本文给出了求解奇摄动时滞反应扩散方程的加速有限差分法。首先,将解域离散化。然后,将给定边值问题中的导数替换为有限差分近似,得到了提供代数方程组的数值格式,该格式易于用Thomas算法求解。证明了该方法的一致性、稳定性和收敛性。为了提高我们所建立的方案的准确性,我们使用了理查德-森的外推技术。为了验证所提方法的适用性,考虑了四个模型实例,并对不同的扰动参数值和网格尺寸进行了求解。数值结果以表格和图表的形式给出,以便说明;本方法很好地逼近了精确解。此外,该方法比现有文献中提到的数值方法具有更好的精度。
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