# Fitted operator average finite difference method for singularly perturbed delay parabolic reaction diffusion problems with non-local boundary conditions

## Abstract

This paper deals with numerical solution of singularly perturbed delay parabolic reaction diffusion problem having large delay on the spatial variable with non-local boundary condition. The solution of the problem exhibits parabolic boundary layer on both sides of the spatial domain and interior layer is also created. Introducing a fitting parameter into asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem under consideration. To treat the non-local boundary condition, Simpson's rule is applied. The stability and $\varepsilon$ uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter $\varepsilon$ and mesh sizes. 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 shown to be second order Uniformly convergent in both direction, and it also improves the results of the methods existing in the literature.

## Article Details

How to Cite
Gobena, W. T., & Duressa, G. F. (2022). Fitted operator average finite difference method for singularly perturbed delay parabolic reaction diffusion problems with non-local boundary conditions. Tamkang Journal of Mathematics, 54(4), 293–312. https://doi.org/10.5556/j.tkjm.54.2023.4175
Issue
Section
Papers

## References

R.D.Driver. Ordinary and delay differnential equations, Springer, New York, 1977.

A.Bellen and M.Zennaro. Numerical methods for delay differential Equations, Oxford Science Publications, New York, 2003.

J.R.Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21(2),(1963),155--160.

R.E.Ewing and T.Lin$beta$, A class of parameter estimation techniques for fluid flow in porous media, Adv.Water Res. 14,(1991),89--97.

L.Formaggia, F.Nobile, A.Quarteroni and A.Veneziani, Multi scale modeling of the circulatory system: a preliminary analysis, Comput. Vis. Sci. 2,(1999),75--83.

T.A.Bullo, G.F.Duressa and G.A.Delga, Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems, Computational Methods for Differential Equations. 9(3),(2020), 886--898.

M.Kumar and C.S.Rao, High order parameter-robust numerical method for time dependent singularly perturbed reaction-diffusion problems, Computational Methods for Differential Equations. 90(1-2),(2010), 15--38.

C.Clavero, J.C.Jorge, F.Lisbona and G.I.Shishkin, An alternating direction scheme on a non-uniform mesh for reaction-diffusion parabolic problems, IMA journal of numerical analysis 20(2),(2000),263--280.

A.R.Ansari, S.A.Bakr and G.I.Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205,(2007),552--566.

J.J.H.Miller, E.O'Riordan, G.I.Shishkin and L.P.Shishkina, Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. Roy. Irish Acad.A 98,(1998),173--190.

O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs,vol. 23, American Mathematical Society, USA, 1968

M.M.Woldaregay, W.T.Aniley and G.F.Duressa, Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation, Advances in Mathematical Physics 2021(2021),1--13.

D.Bahuguna and J.Dabas, Existence and uniqueness of a solution to a semilinear partial delay differential equation with an integral condition, Nonlinear Dyn. Syst. Theory 8(1),(2008),7--19.

R.E.O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, SpringerVerlag,New York, 1991.

D.Kumar and M.K.Kadalbajoo, A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations, Appl. Math. Model. 35,(2011), 2805--2819.

J.Singh, S.Kumar and M.Kumar, A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay, Numerical Methods for Partial Differential Equations, 34 (5); (2018)

D.Kumar and P.Kumari, Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay, Appl. Numer. Math (2020)

E.B.M.Bashier and K.C.Patidar, A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation, Appl. Math. Comput. 217,(2011a), 4728--4739.

K.C.Patidar and K.K. Sharma, Uniformly convergent nonstandard finite difference methods for singularly perturbed differential difference equations with delay and advance, Int. J. Numer. Methods Eng. 66 (2),(2006),272--296.

K.Bansal and K.K.Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments, Numer, Algorithms 75 (1) (2017),113--145.

K.Bansal and K.K.Sharma, Parameter-robust numerical scheme for time-dependent singularly perturbed reaction-diffusion problem with large delay, Numer. Funct. Anal. Optim. 39,(2018),127--154.

S.Kumar and M.Kumar,High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay, Comput. Math. Appl. 68 (10) (2014),1355--1367.

G.Amiraliyev, I.Amiraliyeva and M. Kudu, A numerical treatment for singularly perturbed differential equations with integral boundary condition, Applied mathematics and computation 185 (1),(2017),574--582.

H.G.Debela and G.F.Duressa, Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition, Int J Numer Meth Fluids.(2020),1--13.

H.G.Debela and G.F.Duressa : Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition,journal of the Egyptian Mathematical Socity 28 :16,(2020).

E.Sekar and A.Tamilselvan, Singularly perturbed delay differential equations of convection-diffusion type with integral boundary condition, Journal of Applied Mathematics and Computing 59(1-2),(2019),701--722.

H.G.Debela and G.F.Duressa : Exponentially Fitted Finite Difference Method for Singularly perturbed Delay differential Equations with Integral Boundary condition, Int J Eng Appl Sci 11(4),(2019),476--493.

G.M.Amiraliyev and B.Ylmaz, Finite difference method for singularly perturbed differential equations with integral boundary condition, Int J Math Comput. 22(1),(2014),1--10.

M. Kudu and G.M. Amiraliyev, Finite difference method for singularly perturbed differential equations with integral boundary condition, Int.J.Math.Comput. 26(3) (2015),71--79.

P.A.Farrell, A.F.Hegarty, J.J.H.Miller,E.O'Riordan and G.I.Shishkin, Robust Computational Techniques for Boundary Layers, Charman and Hall/CRC, Boca Raton, 2002.

S.Elango, A.Tamilselvan, R.Vadivel, N.Gunasekaran, H.Zhu, J.Cao and X.Li, Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition, Advances in Difference Equations 2021:151 (2021).

W.T.Gobena and G.F.Duressa, Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition, International Journal of Differential Equations (2021); Article ID 9993644, 16 pages;https://doi.org/10.1155/2021/9993644

H.G.Roos, M.Stynes and L.Tobiska, Robust Numerical Methods for Singularly Perturbed Dierential Equations: Convection-Diusion-Reaction and Flow Problems, Springer Science and Business Media, vol. 24,(2008) Berlin Heidelberg.

M.Kumar and C.S.Rao, High order parameter-robust numerical method for time dependent singularly perturbed reaction-diffusion problems, Computing, 90 (1-2),(2010), 15--38.