Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Abstract

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

Fitted Computational Method for Solving Singularly Perturbed Small Time Lag Problem

“…Computing the Exponential Fitting Factor In this part, we introduce the fitting factor σ and for the obtained scheme of ( 6)-( 7) at (i, n)th level. As the theory of singular perturbation given in [15,20], the zero order asymptotic solution of the problem of the form…”

“…Podila and Kumar [16] used a stable finite difference scheme, which works on a uniform mesh and on an adaptive mesh. An exponentially fitted finite difference scheme is used by Woldaregay et al [20]. An exponentially fitted spline method is discussed in [12].…”

Fitted Operator Average Finite Difference Method for Singularly Perturbed Delay Parabolic Reaction Diffusion Problems with Non-Local Boundary Conditions

“…[9], Woldaregay et.al. [12], Kumar and Kadalbajoo [15], Singh et.al. [16], Kumar and Kumari [17], Bashier and Patidar [18], Patidar and Sharma [19], Bansal and Sharma [20,21], S.Kumar and M.Kumar [22] and the references therein) have received considerable attention over the past few 2 W.T.Gobena and G.F.Duressa decades.…”

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

“…Kanth and Kumar in [9][10][11] used the tension spline method to solve singularly perturbed convection-dominated di erential equations. Woldaregay et al [3,[12][13][14][15][16] developed an exponentially tted numerical method for solving di erent singularly perturbed di erential-di erence equations. In [14,17], they proposed the nonstandard nite di erence techniques and the tted mesh methods, respectively.…”