Abstract
In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).
Highlights
Differential equations depend both on past and future values called functional differential equations
E functional differential equation has been multiplied by small parameter (0 < ε < 1) in the highest order derivative term called the singularly perturbed mixed delay differential equations. e main determination for such a problem is the study of biological science, epidemics, and population [5,6,7,8,9,10]
Where the values x x1 and x x2 correspond to inhibitory reversal potential and the threshold value of membrane potential for action potential generation. is biological problem motivates the investigation of boundary-value problems for differential-difference equations with mixed shifts. In this biological model, using the Taylor series for the small delay term, provided the delay is of order ε, the small delay problem has oscillatory solution that has been discussed in [12]. e same authors discussed the signal transmission problem in [13]
Summary
Differential equations depend both on past and future values (mixed delay) called functional differential equations. Is biological problem motivates the investigation of boundary-value problems for differential-difference equations with mixed shifts In this biological model, using the Taylor series for the small delay term, provided the delay is of order ε, the small delay problem has oscillatory solution that has been discussed in [12]. E authors in [14, 15] have considered the singularly perturbed problem with derivative depending on small delay term such as εy′′(t) + a(t)y′(t) + b(t)y(t) (3). E authors in [18, 19] investigated various concepts of singularly perturbed differential equation with derivative depending on both past and future small variables, εy′′(t) + a(t)y′(t) + b(t)y(t) + c(t)y(t − τ) (4). E study in [23] proposed solving singularly perturbed delay differential equation with integral boundary condition using finite difference method.
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