A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

Gürbüz, Burcu

doi:10.2478/fcds-2021-0015

Abstract

In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.

References

[1] Aizenshtadt V. S., Vladimir I. K., Metel’skii A. S., (2014). Tables of Laguerre Polynomials and Functions: Mathematical Tables Series, 39, Elsevier, Pergamon Press, London.
Search in Google Scholar Back to article
[2] Ansari A. R., Bakr S. A., Shishkin G. I., (2007). A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. Journal of Computational and Applied Mathematics, 205, 1, 552-566. DOI: 10.1016/j.cam.2006.05.03210.1016/j.cam.2006.05.032
Search in Google Scholar Back to article
[3] Avudai Selvi P., Ramanujam N., (2017). A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition. Applied Mathematics and Computation, 296, 101-115. DOI: 10.1016/j.amc.2016.10.02710.1016/j.amc.2016.10.027
Search in Google Scholar Back to article
[4] Bashier E. B. M., Patidar K. C., (2011). A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Applied Mathematics and Computation, 217, 4728-4739. DOI: 10.1016/j.amc.2010.11.02810.1016/j.amc.2010.11.028
Search in Google Scholar Back to article
[5] Bhrawy A. H., AlZahrani A., Baleanu D., Alhamed Y., (2014). A modified generalized Laguerre-Gauss collocation method for fractional neutral functional-differential equations on the half-line, In Abstract and Applied Analysis, 2014.10.1155/2014/692193
Search in Google Scholar Back to article
[6] Brunner H., Liang H., (2010). Stability of collocation methods for delay differential equations with vanishing delays. BIT Numerical Mathematics, 50, 4, 693-711. DOI: 10.1007/s10543-010-0285-110.1007/s10543-010-0285-1
Search in Google Scholar Back to article
[7] Bulut H., Sulaiman T. A., Baskonus H. M., Rezazadeh H., Eslami M., Mirzazadeh M., (2018). Optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. Optik, 172, 20-27.10.1016/j.ijleo.2018.06.108
Search in Google Scholar Back to article
[8] Bülbül B., Sezer M., (2011). A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation, Applied Mathematics Letters, 24, 10, 1716-1720.10.1016/j.aml.2011.04.026
Search in Google Scholar Back to article
[9] Bülbül B., Sezer M., (2013). A new approach to numerical solution of nonlinear Klein-Gordon equation, Mathematical Problems in Engineering, 2013.10.1155/2013/869749
Search in Google Scholar Back to article
[10] Cerutti J. H., Parter S. V., (1976). Collocation methods for parabolic partial differential equations in one space dimension. Numerische Mathematik, 26, 3, 227-254.10.1007/BF01395944
Search in Google Scholar Back to article
[11] Chen X., Collocation Methods for Nonlinear Parabolic Partial Differential Equations, (2017). (Doctoral dissertation, Concordia University).
Search in Google Scholar Back to article
[12] Das A., Natesan S., (2015). Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh, Applied Mathematics and Computation, 271, 168-186.10.1016/j.amc.2015.08.137
Search in Google Scholar Back to article
[13] Gao W., Veeresha P., Prakasha D. G., Baskonus H. M., Yel G., (2020). New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function. Chaos, Solitons & Fractals, 134, 109696.10.1016/j.chaos.2020.109696
Search in Google Scholar Back to article
[14] Gürbüz B., Sezer M., (2017). Laguerre polynomial solutions of a class of delay partial functional differential equations, Acta Physica Polonica, A, 132, 3, 558-560. DOI: 10.12693/APhysPolA.132.55810.12693/APhysPolA.132.558
Search in Google Scholar Back to article
[15] Gürbüz B., Sezer M., (2018). Modified Laguerre collocation method for solving 1-dimensional parabolic convection-diffusion problems, Mathematical Methods in the Applied Sciences, 41, 18, 8481-8487. DOI: 10.1002/mma.472110.1002/mma.4721
Search in Google Scholar Back to article
[16] Gürbüz B., Husein, I., Weber, G. W., (2021). Rumour propagation: an operational research approach by computational and information theory, Central European Journal of Operations Research, 1-21.10.1007/s10100-020-00727-0
Search in Google Scholar Back to article
[17] Gürbüz B., Sezer M., (2017). A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics, 7, 1, 49-58.10.17706/ijapm.2017.7.1.49-58
Search in Google Scholar Back to article
[18] Gürbüz B., Sezer M., (2020). A Modified Laguerre Matrix Approach for Burgers–Fisher Type Nonlinear Equations, Numerical Solutions of Realistic Nonlinear Phenomena, Springer, Cham. 107-123.
Search in Google Scholar Back to article
[19] Hemker P. W., Shishkin G. I., Shishkin L. P., (2003). Novel defect-correction high-order, in space and time, accurate schemes for parabolic singularly perturbed convectiondiffusion problems. Journal of Computational and Applied Mathematics, 3, 3, 387-404. DOI: 10.2478/cmam-2003-002510.2478/cmam-2003-0025
Search in Google Scholar Back to article
[20] Parthiban S., Valarmathi S., Franklin V., (2015). A numerical method to solve singularly perturbed linear parabolic second order delay differential equation of reaction-diffusion type. Malaya Journal of Matematik, 2, 412-420.
Search in Google Scholar Back to article
[21] Polyanin A. D., Zhurov A. I., (2014). Exact separable solutions of delay reaction-diffusion equations and other nonlinear partial functional-differential equations. Communications in Nonlinear Science and Numerical Simulation, 19, 409-416. DOI: 10.1016/j.cnsns.2013.07.01910.1016/j.cnsns.2013.07.019
Search in Google Scholar Back to article
[22] Jiang J., Guirao J. L. G., Chen H., Cao D., (2019). The boundary control strategy for a fractional wave equation with external disturbances, Chaos, Solitons & Fractals, 121, 92-97.10.1016/j.chaos.2019.01.031
Search in Google Scholar Back to article
[23] Kadalbajoo M. K., Sharma K. K., (2004). Numerical analysis of singularly perturbed delay differential equations with layer behavior, Applied Mathematics and Computation, 157, 1, 11-28.10.1016/j.amc.2003.06.012
Search in Google Scholar Back to article
[24] Kumar S., Kumar B. V. R., (2017). A domain decomposition Taylor Galerkin finite element approximation of a parabolic singularly perturbed differential equation. Applied Mathematics and Computation, 293, 508-522. DOI: 10.1016/j.amc.2016.08.03110.1016/j.amc.2016.08.031
Search in Google Scholar Back to article
[25] Lin J., Reutskiy S., (2020). A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems, Applied Mathematics and Computation, 371, 124944.10.1016/j.amc.2019.124944
Search in Google Scholar Back to article
[26] Mahzoun M. R., Kim J., Sawazaki S., Okazaki K., Tamura S., (1999). A scaled multigrid optical flow algorithm based on the least RMS error between real and estimated second images, Pattern Recognition, 32, 4, 657-670.10.1016/S0031-3203(98)00105-8
Search in Google Scholar Back to article
[27] Mirzaee F., Bimesl S., (2013). A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising from physics and engineering, Results in Physics, 3, 241-247.10.1016/j.rinp.2013.10.002
Search in Google Scholar Back to article
[28] Mirzaee F., Bimesl S., Tohidi E., Kilicman A., On the numerical solution of a class of singularly perturbed parabolic convection-diffusion equations.
Search in Google Scholar Back to article
[29] Müller S., Sverák V., (1998). Unexpected solutions of first and second order partial differential equations.
Search in Google Scholar Back to article
[30] Rai P., Sharma K. K., (2015). Singularly perturbed parabolic differential equations with turning point and retarded arguments. International Journal of Applied Mathematics and Computer Science, 45, 4.
Search in Google Scholar Back to article
[31] Russell R. D., Shampine L. F., (1972). A collocation method for boundary value problems. Numerische Mathematik, 19, 1, 1-28. DOI: 10.1007/BF0139592610.1007/BF01395926
Search in Google Scholar Back to article
[32] Russell R. D., (1977). A comparison of collocation and finite differences for two-point boundary value problems. SIAM Journal on Numerical Analysis, 14, 1, 19-39. DOI: 10.1137/071400310.1137/0714003
Search in Google Scholar Back to article
[33] Salama A. A., Al-Amery D. G., (2017). A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations. International Journal of Computer Mathematics, 94, 12, 2520-2546. DOI: 10.1080/00207160.2017.128431710.1080/00207160.2017.1284317
Search in Google Scholar Back to article
[34] Sun W., Wu J., Zhang X., (2007). Nonconforming spline collocation methods in irregular domains. Numerical Methods for Partial Differential Equations: An International Journal, 23, 6, 1509-1529. DOI: 10.1137/071400310.1137/0714003
Search in Google Scholar Back to article
[35] Wang Y., Tian D., Li Z., (2017). Numerical method for singularly perturbed delay parabolic partial differential equations. Thermal Science, 21, 4, 1595-1599. DOI: 10.2298/TSCI160615040W10.2298/TSCI160615040W
Search in Google Scholar Back to article
[36] Yamaç ÇalıŞkan S.,Özbay H., (2009). Stability analysis of the heat equation with time-delayed feedback. IFAC Proceedings Volumes (IFAC-PapersOnline), 6, 1, 220-224. DOI: 10.3182/20090616-3-IL-2002.0056
Search in Google Scholar Back to article
[37] Yavuz M.,Özdemir N., Baskonus H. M., (2018). Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel. The European Physical Journal Plus, 133, 6, 1-11.10.1140/epjp/i2018-12051-9
Search in Google Scholar Back to article
[38] Yüzbaşı Ş., Şahin N., (2013). Numerical solutions of singularly perturbed one-dimensional parabolic convection–diffusion problems by the Bessel collocation method. Applied Mathematics and Computation, 220, 305-315.10.1016/j.amc.2013.06.027
Search in Google Scholar Back to article
[39] Yüzbaşı Ş., Karaçayır M., (2020). An approximation technique for solutions of singularly perturbed one-dimensional convection-diffusion problem. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 33, 1, 2686.
Search in Google Scholar Back to article
[40] Zhao Y., Wu Y., Chai Z., Shi B., (2020). A block triple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations, Computers & Mathematics with Applications, 79, 9, 2550-2573.10.1016/j.camwa.2019.11.018
Search in Google Scholar Back to article

A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

Abstract

Paradigm

My account