Abstract
A novel parameter-uniform finite difference scheme on a Shishkin-type mesh for a singularly perturbed Volterra integro-differential equation is studied. The problem is discretized by the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is carried out. It is shown from the convergence analysis that our presented method is almost second-order uniformly convergent with respect to the perturbation parameter \(\varepsilon \) in the discrete maximum norm. Numerical results are given to support our theoretical result.
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References
Abdul I (1999) Introduction to integral equations with application. Wiley, New York
Bouchra A (2018) Qualitative analysis and simulation of a nonlinear integro-differential system modelling tumor-immune cells competition. Int. J. Biomath. 11(08):1850104
Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge
Cheng Y, Yan L, Mei Y (2022) Balanced-norm error estimate of the local discontinuous Galerkin method on layer-adapted meshes for reaction-diffusion problems. Numer Algorithm. https://doi.org/10.1007/s11075-022-01316-9
De Gaetano A, Arino O (2000) Mathematical modelling of the intravenous glucose tolerance test. J. Math. Biol. 40:136–168
Emmrich E (2005) Stability and error of the variable two-step BDF for semilinear parabolic problems. J. Appl. Math. Comput. 19:33–55
Huang J, Cen Z, Xu A et al (2020) A posteriori error estimation for a singularly perturbed Volterra integro-differential equation. Numer. Algorithms 83:549–563
Iragi BC, Munyakazi JB (2020) A uniformly convergent numerical method for a singularly perturbed volterra integro-differential equation. Int. J. Comput. Math. 97(4):759–771
Kauthen JP (1993) Implicit Runge-Kutta methods for some singularly perturbed volterra integro-differential-algebraic equation. Appl. Numer. Math. 13:125–134
Kauthen JP (1995) Implicit Runge-Kutta methods for singularly perturbed integro-differential systems. Appl. Numer. Math. 18:201–210
Li C, Yi Q, Chen A (2016) Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316:614–631
Liao HL, Zhang Z (2021) Analysis of adaptive BDF2 scheme for diffusion equations. Math. Comput. 90(329):1207–1226
Liao HL, Ji B, Zhang L (2022) An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA J. Numer. Anal. 42(1):649–679
Linß T (1998) Layer-adapted meshes for reaction-convection-diffusion problems. Springer, Berlin
Linß T (2004) Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem. IMA J. Numer. Anal. 24(2):239–253
Long G, Liu L-B, Huang Z (2021) Richardson extrapolation method on an adaptive grid for singularly perturbed volterra integro-differential equations. Numer. Funct. Anal. Optim. 42:739–757
Panda A, Mohapatra J, Amirali I (2021) A second-order post-processing technique for singularly perturbed volterra integro-differential equations. Mediterr. J. Math. 18:231
Ramos JI (2008) Exponential techniques and implicit Runge-Kutta method for singularly-perturbed Volterra integro-differential equations. Neural Parallel Sci. Comput. 16:387–404
Rudenko OV (2014) Nonlinear integro-differential models for intense waves in media like biological tissues and geostructures with complex internal relaxation-type dynamics. Acouset. Phys. 60:398–404
Salama AA, Bakr SA (2007) Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems. Appl. Math. Model. 31:866–879
Şevgin S (2014) Numerical solution of a singularly perturbed volterra integro-differential equation. Adv. Differ. Equ. 2014:171–196
Sumit S, Vigo-Aguiar Kumar J (2021) Analysis of a nonlinear singularly perturbed volterra integro-differential equation. J. Comput. Appl. Math. 404:113410
Tao X, Zhang Y (2019) The coupled method for singularly perturbed volterra integro-differential equations. Adv Differ Equ. https://doi.org/10.1186/s13662-019-2139-8
Yanman M, Amiraliyev GM (2020) A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation. Int. J. Comput. Math. 97(6):1293–1302
Acknowledgements
This work is supported by the key project of Natural Science Foundation of Guangxi Province (AD20238065), the Natural Science Foundation of Guangxi Province (2020GXNSFAA159010), the National Science Foundation of China (12261062).
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Liu, LB., Liao, Y. & Long, G. A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation. Comp. Appl. Math. 42, 12 (2023). https://doi.org/10.1007/s40314-022-02142-4
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DOI: https://doi.org/10.1007/s40314-022-02142-4
Keywords
- Volterra integro-differential equation
- Singularly perturbed
- Shishkin mesh
- Orthogonal convolution kernels
- BDF2