A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations
Abstract
In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution 
to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.
Keyword : delay parabolic equation, reproducing kernel method, collocation method, numerical solution
How to Cite
Xie, R., Zhang, J., Niu, J., Li, W., & Yao, G. (2023). A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations. Mathematical Modelling and Analysis, 28(3), 469–486. https://doi.org/10.3846/mma.2023.16852
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R.P. Agarwal, M. Bohner and W.-T.Li. Nonoscillation and oscillation theory for functional differential equations. CRC Press, 2004. https://doi.org/10.1201/9780203025741
M. Ahmadinia and Z. Safari. Numerical solution of singularly perturbed boundary value problems by improved least squares method. Journal of Computational and Applied Mathematics, 331:156–165, 2018. https://doi.org/10.1016/j.cam.2017.09.023
A.R. Ansari, S.A. Bakr and G.I. Shishkin. A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. Journal of computational and applied mathematics, 205(1):552–566, 2007. https://doi.org/10.1016/j.cam.2006.05.032
K. Bansal, P. Rai and K.K. Sharma. Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. Differential Equations and Dynamical Systems, 25(2):327–346, 2017. https://doi.org/10.1007/s12591-015-0265-7
A. Bellen and M. Zennaro. Numerical methods for delay differential equations. Oxford university press, 2013.
J.B. Burie, A. Calonnec and A. Ducrot. Singular perturbation analysis of travelling waves for a model in phytopathology. Mathematical Modelling of Natural Phenomena, 1(1):49–62, 2006. https://doi.org/10.1051/mmnp:2006003
P.P. Chakravarthy and K. Kumar. A novel method for singularly perturbed delay differential equations of reaction-diffusion type. Differential Equations and Dynamical Systems, 29(3):723–734, 2021.
M.G. Cui and Y.Z. Lin. Nonlinear numerical analysis in the RK-space. Oxford university press, 2013.
A. Das and S. Natesan. Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh. Journal of Applied Mathematics and Computing, 59(1):207–225, 2019. https://doi.org/10.1007/s12190-018-1175-y
H. Du and J.H. Shen. Reproducing kernel method of solving singular integral equation with cosecant kernel. Journal of Mathematical Analysis and Applications, 348(1):308–314, 2008. https://doi.org/10.1016/j.jmaa.2008.07.037
F. Geng and M. Cui. A reproducing kernel method for solving nonlocal fractional boundary value problems. Applied Mathematics Letters, 25(5):818–823, 2012. https://doi.org/10.1016/j.aml.2011.10.025
F. Geng, Z. Tang and Y. Zhou. Reproducing kernel method for singularly perturbed one-dimensional initial-boundary value problems with exponential initial layers. Qualitative Theory of Dynamical Systems, 17(1):177–187, 2018. https://doi.org/10.1007/s12346-017-0242-3
F.Z. Geng and X.Y. Wu. A novel kernel functions algorithm for solving impulsive boundary value problems. Applied Mathematics Letters, 134:108318, 2022. https://doi.org/10.1016/j.aml.2022.108318
V. Gupta, M. Kumar and S. Kumar. Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations. Numerical Methods for Partial Differential Equations, 34(1):357–380, 2018. https://doi.org/10.1002/num.22203
J. In’t Houtk. Stability analysis of Runge–Kutta methods for systems of delay differential equations. IMA journal of numerical analysis, 17(1):17–27, 1997. https://doi.org/10.1093/imanum/17.1.17
Y. Jia, M. Xu, Y. Lin and D. Jiang. An efficient technique based on leastsquares method for fractional integro-differential equations. Alexandria Engineering Journal, 64:97–105, 2023.
Y.F. Jin, J. Jiang, C.M. Hou and D.H. Guan. New difference scheme for general delay parabolic equations. Journal of Information &Computational Science, 9(18):5579–5586, 2012.
A. Kaushik and M. Sharma. A robust numerical approach for singularly perturbed time delayed parabolic partial differential equations. Computational Mathematics and Modeling, 23(1):96–106, 2012. https://doi.org/10.1007/s10598-012-9122-5
S. Kumar and B.V.R. Kumar. A domain decomposition Taylor Galerkin finite element approximation of a parabolic singularly perturbed differential equation. Applied Mathematics and Computation, 293:508–522, 2017. https://doi.org/10.1016/j.amc.2016.08.031
S. Kumar and S.C.S. Rao. A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction–diffusion problems. Journal of computational and applied mathematics, 261:127–138, 2014. https://doi.org/10.1016/j.cam.2013.10.053
X.Y. Li and B.Y. Wu. A kernel regression approach for identification of first order differential equations based on functional data. Applied Mathematics Letters, 127:107832, 2022. https://doi.org/10.1016/j.aml.2021.107832
A. Longtin and J.G. Milton. Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback. Mathematical Biosciences, 90(1-2):183–199, 1988. https://doi.org/10.1016/0025-5564(88)90064-8
J. Niu, Y. Jia and J. Sun. A new piecewise reproducing kernel function algorithm for solving nonlinear Hamiltonian systems. Applied Mathematics Letters, 136:108451, 2023. https://doi.org/10.1016/j.aml.2022.108451
J. Niu, L. Sun, M. Xu and J. Hou. A reproducing kernel method for solving heat conduction equations with delay. Applied Mathematics Letters, 100:106036, 2020. https://doi.org/10.1016/j.aml.2019.106036
J. Niu, M. Xu and G. Yao. An efficient reproducing kernel method for solving the Allen–Cahn equation. Applied mathematics letters, 89:78–84, 2019. https://doi.org/10.1016/j.aml.2018.09.013
R. Nageshwar Rao and P. Pramod Chakravarthy. Fitted numerical methods for singularly perturbed one-dimensional parabolic partial differential equations with small shifts arising in the modelling of neuronal variability. Differential Equations and Dynamical Systems, 27(1):1–18, 2019. https://doi.org/10.1007/s12591-017-0363-9
H.-G. Roos, M. Stynes and L. Tobiska. Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, volume 24. Springer Science & Business Media, 2008.
H. Sahihi, S. Abbasbandy and T. Allahviranloo. Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay. Applied Mathematics and Computation, 361:583–598, 2019. https://doi.org/10.1016/j.amc.2019.06.010
S.H. Saker. New oscillation criteria for second-order nonlinear neutral delay difference equations. Applied Mathematics and Computation, 142(1):99–111, 2003. https://doi.org/10.1016/S0096-3003(02)00286-2
G.I. Shishkin. Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems. Mathematical Modelling and Analysis, 10(4):393–412, 2005. https://doi.org/10.3846/13926292.2005.9637296
L. Sun, J. Niu and J. Hou. A high order convergence collocation method based on the reproducing kernel for general interface problems. Applied Mathematics Letters, 112:106718, 2021. https://doi.org/10.1016/j.aml.2020.106718
J. Wu. Theory and applications of partial functional differential equations, volume 119. Springer Science & Business Media, 1996.
M. Xu, R. Lin and Q. Zou. A C0 linear finite element method for a second order elliptic equation in non-divergence form with Cordes coefficients. Numerical Methods for Partial Differential Equations, 39(3):2244–2269, 2023.
M. Xu and C. Shi. A Hessian recovery-based finite difference method for biharmonic problems. Applied Mathematics Letters, 137:108503, 2023. https://doi.org/10.1016/j.aml.2022.108503
M. Xu, E. Tohidi, J. Niu and Y. Fang. A new reproducing kernelbased collocation method with optimal convergence rate for some classes of BVPs. Applied Mathematics and Computation, 432(1):127343, 2022. https://doi.org/10.1016/j.amc.2022.127343
M. Xu, L. Zhang and E. Tohidi. A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems. Applied Numerical Mathematics, 162:124–136, 2021. https://doi.org/10.1016/j.apnum.2020.12.015
M. Xu, L. Zhang and E. Tohidi. An efficient method based on least-squares technique for interface problems. Applied Mathematics Letters, 136:108475, 2022. https://doi.org/10.1016/j.aml.2022.108475
M.-Q. Xu and Y.-Z. Lin. Simplified reproducing kernel method for fractional differential equations with delay. Applied Mathematics Letters, 52:156–161, 2016. https://doi.org/10.1016/j.aml.2015.09.004
Y. Yu, J. Niu, J. Zhang and S. Ning. A reproducing kernel method for nonlinear C-q-fractional IVPS. Applied Mathematics Letters, 125:107751, 2022. https://doi.org/10.1016/j.aml.2021.107751
B.-G. Zhang and X. Deng. Oscillation of delay differential equations on time scales. Mathematical and Computer Modelling, 36(11-13):1307–1318, 2002. https://doi.org/10.1016/S0895-7177(02)00278-9
J. Zhang and J. Niu. Lobatto-reproducing kernel method for solving a linear system of second order boundary value problems. Journal of Applied Mathematics and Computing, 63:3631–3653, 2021. https://doi.org/10.1007/s12190-021-01685-9
M. Ahmadinia and Z. Safari. Numerical solution of singularly perturbed boundary value problems by improved least squares method. Journal of Computational and Applied Mathematics, 331:156–165, 2018. https://doi.org/10.1016/j.cam.2017.09.023
A.R. Ansari, S.A. Bakr and G.I. Shishkin. A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. Journal of computational and applied mathematics, 205(1):552–566, 2007. https://doi.org/10.1016/j.cam.2006.05.032
K. Bansal, P. Rai and K.K. Sharma. Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. Differential Equations and Dynamical Systems, 25(2):327–346, 2017. https://doi.org/10.1007/s12591-015-0265-7
A. Bellen and M. Zennaro. Numerical methods for delay differential equations. Oxford university press, 2013.
J.B. Burie, A. Calonnec and A. Ducrot. Singular perturbation analysis of travelling waves for a model in phytopathology. Mathematical Modelling of Natural Phenomena, 1(1):49–62, 2006. https://doi.org/10.1051/mmnp:2006003
P.P. Chakravarthy and K. Kumar. A novel method for singularly perturbed delay differential equations of reaction-diffusion type. Differential Equations and Dynamical Systems, 29(3):723–734, 2021.
M.G. Cui and Y.Z. Lin. Nonlinear numerical analysis in the RK-space. Oxford university press, 2013.
A. Das and S. Natesan. Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh. Journal of Applied Mathematics and Computing, 59(1):207–225, 2019. https://doi.org/10.1007/s12190-018-1175-y
H. Du and J.H. Shen. Reproducing kernel method of solving singular integral equation with cosecant kernel. Journal of Mathematical Analysis and Applications, 348(1):308–314, 2008. https://doi.org/10.1016/j.jmaa.2008.07.037
F. Geng and M. Cui. A reproducing kernel method for solving nonlocal fractional boundary value problems. Applied Mathematics Letters, 25(5):818–823, 2012. https://doi.org/10.1016/j.aml.2011.10.025
F. Geng, Z. Tang and Y. Zhou. Reproducing kernel method for singularly perturbed one-dimensional initial-boundary value problems with exponential initial layers. Qualitative Theory of Dynamical Systems, 17(1):177–187, 2018. https://doi.org/10.1007/s12346-017-0242-3
F.Z. Geng and X.Y. Wu. A novel kernel functions algorithm for solving impulsive boundary value problems. Applied Mathematics Letters, 134:108318, 2022. https://doi.org/10.1016/j.aml.2022.108318
V. Gupta, M. Kumar and S. Kumar. Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations. Numerical Methods for Partial Differential Equations, 34(1):357–380, 2018. https://doi.org/10.1002/num.22203
J. In’t Houtk. Stability analysis of Runge–Kutta methods for systems of delay differential equations. IMA journal of numerical analysis, 17(1):17–27, 1997. https://doi.org/10.1093/imanum/17.1.17
Y. Jia, M. Xu, Y. Lin and D. Jiang. An efficient technique based on leastsquares method for fractional integro-differential equations. Alexandria Engineering Journal, 64:97–105, 2023.
Y.F. Jin, J. Jiang, C.M. Hou and D.H. Guan. New difference scheme for general delay parabolic equations. Journal of Information &Computational Science, 9(18):5579–5586, 2012.
A. Kaushik and M. Sharma. A robust numerical approach for singularly perturbed time delayed parabolic partial differential equations. Computational Mathematics and Modeling, 23(1):96–106, 2012. https://doi.org/10.1007/s10598-012-9122-5
S. Kumar and B.V.R. Kumar. A domain decomposition Taylor Galerkin finite element approximation of a parabolic singularly perturbed differential equation. Applied Mathematics and Computation, 293:508–522, 2017. https://doi.org/10.1016/j.amc.2016.08.031
S. Kumar and S.C.S. Rao. A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction–diffusion problems. Journal of computational and applied mathematics, 261:127–138, 2014. https://doi.org/10.1016/j.cam.2013.10.053
X.Y. Li and B.Y. Wu. A kernel regression approach for identification of first order differential equations based on functional data. Applied Mathematics Letters, 127:107832, 2022. https://doi.org/10.1016/j.aml.2021.107832
A. Longtin and J.G. Milton. Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback. Mathematical Biosciences, 90(1-2):183–199, 1988. https://doi.org/10.1016/0025-5564(88)90064-8
J. Niu, Y. Jia and J. Sun. A new piecewise reproducing kernel function algorithm for solving nonlinear Hamiltonian systems. Applied Mathematics Letters, 136:108451, 2023. https://doi.org/10.1016/j.aml.2022.108451
J. Niu, L. Sun, M. Xu and J. Hou. A reproducing kernel method for solving heat conduction equations with delay. Applied Mathematics Letters, 100:106036, 2020. https://doi.org/10.1016/j.aml.2019.106036
J. Niu, M. Xu and G. Yao. An efficient reproducing kernel method for solving the Allen–Cahn equation. Applied mathematics letters, 89:78–84, 2019. https://doi.org/10.1016/j.aml.2018.09.013
R. Nageshwar Rao and P. Pramod Chakravarthy. Fitted numerical methods for singularly perturbed one-dimensional parabolic partial differential equations with small shifts arising in the modelling of neuronal variability. Differential Equations and Dynamical Systems, 27(1):1–18, 2019. https://doi.org/10.1007/s12591-017-0363-9
H.-G. Roos, M. Stynes and L. Tobiska. Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, volume 24. Springer Science & Business Media, 2008.
H. Sahihi, S. Abbasbandy and T. Allahviranloo. Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay. Applied Mathematics and Computation, 361:583–598, 2019. https://doi.org/10.1016/j.amc.2019.06.010
S.H. Saker. New oscillation criteria for second-order nonlinear neutral delay difference equations. Applied Mathematics and Computation, 142(1):99–111, 2003. https://doi.org/10.1016/S0096-3003(02)00286-2
G.I. Shishkin. Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems. Mathematical Modelling and Analysis, 10(4):393–412, 2005. https://doi.org/10.3846/13926292.2005.9637296
L. Sun, J. Niu and J. Hou. A high order convergence collocation method based on the reproducing kernel for general interface problems. Applied Mathematics Letters, 112:106718, 2021. https://doi.org/10.1016/j.aml.2020.106718
J. Wu. Theory and applications of partial functional differential equations, volume 119. Springer Science & Business Media, 1996.
M. Xu, R. Lin and Q. Zou. A C0 linear finite element method for a second order elliptic equation in non-divergence form with Cordes coefficients. Numerical Methods for Partial Differential Equations, 39(3):2244–2269, 2023.
M. Xu and C. Shi. A Hessian recovery-based finite difference method for biharmonic problems. Applied Mathematics Letters, 137:108503, 2023. https://doi.org/10.1016/j.aml.2022.108503
M. Xu, E. Tohidi, J. Niu and Y. Fang. A new reproducing kernelbased collocation method with optimal convergence rate for some classes of BVPs. Applied Mathematics and Computation, 432(1):127343, 2022. https://doi.org/10.1016/j.amc.2022.127343
M. Xu, L. Zhang and E. Tohidi. A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems. Applied Numerical Mathematics, 162:124–136, 2021. https://doi.org/10.1016/j.apnum.2020.12.015
M. Xu, L. Zhang and E. Tohidi. An efficient method based on least-squares technique for interface problems. Applied Mathematics Letters, 136:108475, 2022. https://doi.org/10.1016/j.aml.2022.108475
M.-Q. Xu and Y.-Z. Lin. Simplified reproducing kernel method for fractional differential equations with delay. Applied Mathematics Letters, 52:156–161, 2016. https://doi.org/10.1016/j.aml.2015.09.004
Y. Yu, J. Niu, J. Zhang and S. Ning. A reproducing kernel method for nonlinear C-q-fractional IVPS. Applied Mathematics Letters, 125:107751, 2022. https://doi.org/10.1016/j.aml.2021.107751
B.-G. Zhang and X. Deng. Oscillation of delay differential equations on time scales. Mathematical and Computer Modelling, 36(11-13):1307–1318, 2002. https://doi.org/10.1016/S0895-7177(02)00278-9
J. Zhang and J. Niu. Lobatto-reproducing kernel method for solving a linear system of second order boundary value problems. Journal of Applied Mathematics and Computing, 63:3631–3653, 2021. https://doi.org/10.1007/s12190-021-01685-9