This study introduces a sixth order numerical method for solving Liénard second order nonlinear differential equations. First, the second order Liénard differential equation is transformed into a first order system of equations. Then, the given interval is discretized, and the method is formulated by using Newton's backward difference interpolation formula. The stability and convergence analysis is studied. Three model examples have been presented to demonstrate the reliability and efficiency of the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well.
This paper presents a uniform convergent numerical method for solving singularly perturbed delay reaction-diffusion equations. The stability and convergence analysis are investigated. Numerical results are tabulated and the effect of the layer on the solution is examined. In a nutshell, the present method improves the findings of some existing numerical methods reported in the literature.
Keywords: Singularly perturbed, Time delay, Reaction-diffusion equation, Layer
This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size
,
where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from
10
−
03
up to
10
−
10
, and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.
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