JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (12): 84-93.doi: 10.6040/j.issn.1671-9352.0.2023.475

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Adaptive moving grid method for a second-order singularly perturbed delay boundary value problem

XU Yinuo1, LIU Libin1*, YANG Xiu2   

  1. 1. Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, Guangxi, China;
    2. School of Mathematics and Statistics, Shandong University, Weihai 264209, Shandong, China
  • Published:2025-12-10

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Abstract: An adaptive grid algorithm for a second-order singularly perturbed delay boundary value problem is studied. Firstly, by integral transformation, the problem is rewritten into a first-order singularly perturbed delay Volterra integral differential equation. Secondly, a first-order finite difference format is constructed on an arbitrary mesh by using the backward difference and left rectangle formula. Using the discrete Grönwall inequality, a prior error estimation of the proposed discretization scheme is derived and an adaptive moving grid algorithm is designed based on the mesh equidistribution principle. It is proved that our proposed adaptive moving grid method is first-order uniformly convergent. Finally, some numerical results are given to support our presented theoretical result.

Key words: singularly perturbed, delay differential equation, monitor function, error estimation

CLC Number: 

  • O241
[1] BELLMAN R, COOKE K L. Differential-difference equations[M]. New York: Academic Press, 1963:155-171.
[2] DRIVER R D. Ordinary and Delay Differential Equations[M]. New York: Springer, 1977.
[3] BELLEN A, ZENNARO M. Numerical methods for delay differential equations[M]. Oxford: Clarendon Press, 2003.
[4] (·overE)L'SGOL'TS L(·overE). Qualitative methods in mathematical analysis[M]. Providence: American Mathematical Society, 1964.
[5] STEIN R B. A theoretical analysis of neuronal variability[J]. Biophysical Journal, 1965, 5(2):173-194.
[6] TUCKWELL H C. On the first-exit time problem for temporally homogeneous Markov processes[J]. Journal of Applied Probability, 1976, 13(1):39-48.
[7] MCCARTIN B J. Exponential fitting of the delayed recruitment/renewal equation[J]. Journal of Computational and Applied Mathematics, 2001, 136(1/2):343-356.
[8] AMIRALIYEV G M, ERDOGAN F. Uniform numerical method for singularly perturbed delay differential equations[J]. Computers & Mathematics with Applications, 2007, 53(8):1251-1259.
[9] ERDOGAN F. An exponentially fitted method for singularly perturbed delay differential equations[J]. Advances in Difference Equations, 2009,(1):781579.
[10] AMIRALIYEVA I G, ERDOGAN F, AMIRALIYEV G M. A uniform numerical method for dealing with a singularly perturbed delay initial value problem[J]. Applied Mathematics Letters, 2010, 23(10):1221-1225.
[11] KADALBAJOO M K, SHARMA K K. Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations[J]. Comput Appl Math, 2005, 24(2):151-172.
[12] KADALBAJOO M K, SHARMA K K. A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations[J]. Applied Mathematics and Computation, 2008, 197(2):692-707.
[13] KADALBAJOO M K, GUPTA V. A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers[J]. International Journal of Computer Mathematics, 2010, 87(14):3218-3235.
[14] RAI P, SHARMA K K. Numerical analysis of singularly perturbed delay differential turning point problem[J]. Applied Mathematics and Computation, 2011, 218(7):3483-3498.
[15] PODILA R C, GUPTA T, RAO N. A numerical scheme for singularly perturbed delay differential equations of convection-diffusion type on an adaptive grid[J]. Mathematical Modelling and Analysis, 2018, 23(4):686-698.
[16] AMIRALIYEV G M, CIMEN E. Numerical method for a singularly perturbed convection-diffusion problem with delay[J]. Applied Mathematics and Computation, 2010, 216(8):2351-2359.
[17] DAS A, NATESAN S. Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh[J]. Applied Mathematics and Computation, 2015, 271:168-186.
[18] PRAMOD CHAKRAVARTHY P, DINESH KUMAR S, NAGESHWAR RAO R. Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension[J]. International Journal of Applied and Computational Mathematics, 2017, 3(3):1703-1717.
[19] VAID M K, ARORA G. Solution of second order singular perturbed delay differential equation using trigonometric B-spline[J]. International Journal of Mathematical, Engineering and Management Sciences, 2019, 4(2):349-360.
[20] PODILA P C, KUMAR K. A class of finite difference schemes for singularly perturbed delay differential equations of second order[J]. Turkish Journal of Mathematics, 2019, 43(3):1061-1079.
[21] MANIKANDAN M, SHIVARANJANI N, MILLER J J H, et al. A parameter-uniform numerical method for a boundary value problem for a singularly perturbed delay differential equation[M] //Springer Proceedings in Mathematics & Statistics. Cham: Springer, 2014:71-88.
[22] DOOLAN E R, MILLER J J H, SCHILDERS W H A. Uniform numerical methods for problems with initial and bundary layers[M]. Dublin: Boole Press, 1980.
[23] FARRELL P, HEGARTY A, MIUER J M, et al. Robust computational techniques for boundary layers[M]. New York: Chapman & Hall/CRC, 2000:37-55.
[24] MILLER J H, O'RIORDAN E, SHISHKIN G I. Fitted numerical methods for singular perturbation problems[M]. Singapore: World Scientific Press, 1996:11-24.
[25] KOPTEVA N. Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem[J]. SIAM Journal on Numerical Analysis, 2001, 39(2):423-441.
[26] KOPTEVA N, STYNES M. A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem[J]. SIAM Journal on Numerical Analysis, 2001, 39(4):1446-1467.
[27] LIU L B, CHEN Y P, LIANG Y. Numerical analysis of a nonlinear singularly perturbed delay Volterra integro-differential equation on an adaptive grid[J]. Journal of Computational Mathematics, 2022, 40(2):258-274.
[28] YAPMAN Ö, KUDU M, AMIRALIYEV G M. Method of exact difference schemes for the numerical solution of parameterized singularly perturbed problem[J]. Mediterranean Journal of Mathematics, 2023, 20(3):146.
[29] LONG G Q, LIU L B, HUANG Z T. Richardson extrapolation method on an adaptive grid for singularly perturbed Volterra integro-differential equations[J]. Numerical Functional Analysis and Optimization, 2021, 42(7):739-757.
[30] YAPMAN Ö, AMIRALIYEV G M, AMIRALI I. Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay[J]. Journal of Computational and Applied Mathematics, 2019, 355:301-309.
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