A second order convergent initial value method for singularly perturbed system of differentialdifference equations of convection diffusion type
Authors
L. S. Senthilkumar
 Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and technology, Kattankulathur603 203, Tamilnadu, India.
R. Mahendran
 Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and technology, Kattankulathur603 203, Tamilnadu, India.
V. Subburayan
 Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and technology, Kattankulathur603 203, Tamilnadu, India.
Abstract
In this article, a system of second order singularly perturbed delay differential equations of convection diffusion type problem is considered. An asymptotic expansion approximation of the solution is constructed. Further the asymptotic expansion approximation is numerically approximated using the Runge Kutta methods and hybrid finite difference methods. The error estimate is obtained and it is of almost second order. Numerical examples are given to illustrate the present method.
Share and Cite
ISRP Style
L. S. Senthilkumar, R. Mahendran, V. Subburayan, A second order convergent initial value method for singularly perturbed system of differentialdifference equations of convection diffusion type, Journal of Mathematics and Computer Science, 25 (2022), no. 1, 7383
AMA Style
Senthilkumar L. S., Mahendran R., Subburayan V., A second order convergent initial value method for singularly perturbed system of differentialdifference equations of convection diffusion type. J Math Comput SCIJM. (2022); 25(1):7383
Chicago/Turabian Style
Senthilkumar, L. S., Mahendran, R., Subburayan, V.. "A second order convergent initial value method for singularly perturbed system of differentialdifference equations of convection diffusion type." Journal of Mathematics and Computer Science, 25, no. 1 (2022): 7383
Keywords
 Delay differential equations
 singularly perturbed problem
 asymptotic expansion approximation
 initial value method
 Shishkin mesh
MSC
References

[1]
R. P. Agarwal, Y. M. Chow, Finitedifference methods for boundaryvalue problems of differential equations with deviating arguments, Comput. Math. Appl. Ser. A, 12 (1986), 11431153

[2]
G. M. Amiraliyev, C. Cimen, Numerical method for a singularly perturbed convectiondiffusion problem with delay, Appl. Math. Comput., 216 (2010), 23512359

[3]
P. M. Basha, V. Shanthi, Fitted mesh method for a weakly coupled system ofsingularly perturbed reactionconvectiondiffusion problems with discontinuous source term, Ain Shams Eng. J., 9 (2018), 10891101

[4]
Z. Cen, Parameteruniform finite difference scheme for a system of coupled singularly perturbed convectiondiffusion equations, Int. J. Comput. Math, 82 (2005), 177192

[5]
Z. Cen, A hybrid finite difference scheme for a class of singularly perturbed delay differential equations, Neural Parallel Sci. Comput., 16 (2008), 303308

[6]
Z. Cen, A. Xu, A. Le, A secondorder hybrid finite difference scheme for a system of singularly perturbed initial value problems, J. Comput. Appl. Math., 234 (2010), 34453437

[7]
Z. Cen, A. Xu, A. Le, L.B. Liu, A uniformly convergent hybrid difference scheme for a system of singularly perturbed initial value problems, Int. J. Comput. Math., 97 (2020), 10581086

[8]
C. Clavero, J. L. Gracia, F. J. Lisbona, An almost third order finite difference scheme for singularly perturbed reactiondiffusion systems, J. Comput. Appl. Math., 234 (2010), 25012515

[9]
V. Y. Glizer, Asymptotic analysis and solution of a finitehorizon $H_\infty$ control problem for singularlyperturbed linear systems with small state delay, J. Optim. Theory Appl., 117 (2003), 295325

[10]
S. A. Gourley, Y. Kuang, A stage structured predatorprey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188200

[11]
M. K. Kadalbajoo, K. K. Sharma, Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations, Comput. Appl. Math., 24 (2005), 151172

[12]
C. G. Lange, R. M. Miura, Singular perturbation analysis of boundary value problems for differentialdifference equations. V. Small shifts with layer behavior, SIAM J. APPL. MATH., 54 (1994), 249272

[13]
A. Longtin, J. G. Milton, Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback, Math. Biosci., 90 (1988), 183199

[14]
N. Madden, M. Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction–diffusion problems, IMA J. Numer. Anal.,, 23 (2003), 627644

[15]
M. Mariappan, J. J. H. Miller, V. Sigamani, A parameteruniform first order convergent numerical method for a system of singularly perturbed second order delay differential equations, Boundary and interior layers, computational and asymptotic methods–BAIL 2014, Springer, Cham, 108 (2015), 183195

[16]
S. Matthews, E. O’Riordan, G. I. Shishkin, A numerical method for a system of singularly perturbed reaction–diffusion equations, J. Comput. Appl. Math., 145 (2002), 151166

[17]
W. G. Melesse, A. A. Tiruneh, G. A. Derese, Solving systems of singularly perturbed convection diffusion problems via initial value method, J. Appl. Math., 2020 (2020), 8 pages

[18]
J. Mohapatra, S. Natesan, Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid, Numer. Math. Theory Methods Appl., 3 (2010), 122

[19]
H. Ramos, J. VigoAguiar, A new algorithm appropriate for solving singular and singularly perturbed autonomous initialvalue problems, Int. J. Comput. Math., 85 (2008), 603611

[20]
L. S. Senthilkumar, V. Subburayan, An improved initial value method for singularly perturbed convection diffusion delay differential equations, Adv. Math., Sci. J., 10 (2021), 9911001

[21]
V. Subburayan, N. Ramanujam, Asymptotic initial value technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers, Appl. Math. Lett., 25 (2012), 22722278

[22]
V. Subburayan, N. Ramanujam, An initial value technique for singularly perturbed convectiondiffusion problems with a negative shift, J. Optim. Theory Appl., 158 (2013), 234250

[23]
V. Subburayan, N. Ramanujam, An asymptotic numerical method for singularly perturbed convectiondiffusion problems with a negative shift, Neural Parallel Sci. Comput., 21 (2013), 431446

[24]
V. Subburayan, N. Ramanujam, An asymptotic numerical method for singularly perturbed weakly coupled system of convectiondiffusion type differential equations, Novi Sad J. Math., 44 (2014), 5368

[25]
V. Subburayan, N. Ramanujam, Uniformly convergent finite difference schemes for singularly perturbed convection diffusion type delay differential equations, Differ. Equ. Dyn. Syst., 29 (2021), 139155

[26]
A. Tamilselvan, N. Ramanujam, V. Shanthi, A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term, J. Comput. Appl. Math., 202 (2007), 203216

[27]
H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 1, Cambridge University Press, Cambridge (1988)

[28]
J. VigoAguiar, S. Natesan, A parallel boundary value technique for singularly perturbed twopoint boundary value problems, J. Supercomput., 27 (2004), 195206