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NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID

Libin Liu1, Yanping Chen2, Ying Liang3   

  1. 1. School of Mathematics and Statistics, Nanning Normal University, Nanning 530023, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
    3. School of Mathematics and Statistics, Nanning Normal University, Nanning 530023, China
  • Received:2020-03-16 Revised:2020-08-18 Online:2022-03-15 Published:2022-03-29
  • Contact: Yanping Chen,Email: yanpingchen@scnu.edu.cn
  • Supported by:
    This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Science Foundation of China (41974133,11761015,11971410),the Natural Science Foundation of Guangxi (2020GXNSFAA159010)

Libin Liu, Yanping Chen, Ying Liang. 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.

In this paper, we study a nonlinear first-order singularly perturbed Volterra integrodifferential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.

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[1] R.D. Driver, Ordinary and Delay Differential Equations, Springer, Berlin, Heidelberg, New York, 1977.
[2] S. Marino, E. Beretta and D.E. Kirschner, The role of delays in innate and adaptive immunity to intracellular bacterial infection, Math. Biosci. Eng., 4(2007), 261-288.
[3] G.A. Bocharov and F.A. Rihan, Numerical modelling in biosciences with delay differential equations, J. Comput. Appl. Math., 125(2000), 183-199.
[4] A. De Gaetano and O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol., 40(2000) 136-168.
[5] Y. Wei and Y. Chen, Legendre spectral collocation methods for pantograph Volterra delay integrodifferential equations, J. Sci. Comput., 53(2012), 672-688.
[6] J. Zhao, Y. Cao and Y. Xu, Legendre spectral collocation methods for Volterra delay integrodifferential equations, J. Sci. Comput., 67(2016), 1110-1133.
[7] E.R. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
[8] P.A. Farrel, A.F. Hegarty, J.J.H. Miller, et.al., Robust Computational Techniques for Boundary Layers, ChapmanHall/CRC, NewYork, 2000.
[9] J.J.H. Miller, E. ORiordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
[10] N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39(2001), 423-441.
[11] N. Kopteva and M. Stynes, A robust adaptive method for quasi-linear one-dimensional convectiondiffusion problem, SIAM J. Numer. Anal., 39(2001), 1446-1467.
[12] G.M. Amiraliyev and S. Şevgin, Uniform difference method for singularly perturbed Volterra integro-differential equations, Appl. Math. Comput., 179(2005), 731-741.
[13] J.I. Ramos, Exponential techniques and implicit Runge Kutta method for singularly perturbed Volterra integro differential equations, Neural Parallel Sci. Comput., 16(2008), 387-404.
[14] A.A. Salama and S.A. Bakr, Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems, Appl. Math. Model., 31(2007), 866-879.
[15] O. Yapman and G.M. Amiraliyev, A novel second-order fitted computational method for a singularly perturbed Volterra integro differential equation, Int. J. Comput. Math., 97(2020), 1293- 1302.
[16] S. Şevgin, Numerical solution of a singularly perturbed Volterra integro-differential equation, Adv. Differential Equations, 2014(2014), 171.
[17] J. Huang, Z. Cen, A. Xu and L.B. Liu, A posteriori error estimation for a singularly perturbed Volterra integro-differential equation, Numer. Algor., 83(2020), 549-563.
[18] S. Wu and S. Gan, Errors of linear multistep methods for singularly perturbed Volterra delayintegro-differential equations, Math. Comput. Simu., 79(2009), 3148-3159.
[19] M. Kudu, I. Amirali and G.M. Amiraliyev, A finite-difference method for a singularly perturbed delay integro-differential equation, J. Comput. Appl. Math., 308(2016), 379-390.
[20] O. Yapman, G.M. Amiraliyev and I. Amirali, Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay, J. Comput. Appl. Math., 355(2019), 301-309.
[21] I.G. Amiraliyeva and G.M. Amiraliyev, Uniform difference method for parameterized singularly perturbed delay differential equations, Numer. Algor., 52(2009), 509-521.
[22] P. Das, An a posterior based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numer. Algor., 81(2019), 465-487.
[23] Y. Qiu, D.M. Sloan and T. Tang, Numerical solution of a singularly perturbed two point boundary value problem using equidistribution:analysis of convergence, J. Comput. Appl. Math. 116(2000), 121-143.
[24] Y. Chen, Uniform convergence analysis of finite difference approximation for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24(2006), 197-212.
[25] L.B. Liu and Y. Chen, A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equation with weak coupling, J. Sci. Comput., 61(2014), 1-16.
[26] L.B. Liu and Y. Chen, A-posteriori error estimation in maximum norm for a strong coupled system of two singularly perturbed convection-diffusion problems, J. Comput. Appl. Math., 313(2017), 152-167.
[27] T. Linß, Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM J. Numer. Anal., 47(2009), 1847-1862.
[28] T. Linß, A posteriori error estimation for a singularly perturbed problem with two small parameters, Int. J. Numer. Anal. Model., 7(2010), 491-506.
[29] F. Erdogan and G.M. Amiraliyev, Fitted finite difference method for singularly perturbed delay differential equations, Numer. Algor., 59(2012), 131-145.
[30] I.G. Amiraliyeva, F. Erdogan and G.M. Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay initial value problem, Appl. Math. Lett., 23(2010), 1221-1225.
[31] F. Erdogan and Z. Cen, A uniformly almost second order convergent numerical method for singularly perturbed delay differential equations, J. Comput. Appl. Math., 333(2018), 382-394.
[32] G.M. Amiraliyev and E. Climen, Numerical method for a singularly perturbed convection-diffusion problem with delay, Appl. Math. Comput., 216(2010), 2351-2359.
[33] P.P. Chakravarthy, T. Gupta and R. N. Rao, A numerical scheme for singularly perturbed delay differential equations of convection-diffusion type an an adaptive grid, Math. Model. Anal., 23(2018), 686-698.
[34] H. Zarin, On discontinuous Galerkin finite element method for singularly perturbed delay differential equations, Appl. Math. Lett., 38(2014), 27-32.
[35] J.A. Nelder and R. Mead, A simplex method for function minimization, Comput. J., 7(1965), 308-313.
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