1. bookVolume 46 (2021): Issue 3 (September 2021)
Journal Details
License
Format
Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
Languages
English
access type Open Access

A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

Published Online: 17 Sep 2021
Page range: 221 - 233
Received: 16 Jun 2020
Accepted: 17 Feb 2021
Journal Details
License
Format
Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
Languages
English
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.

Keywords

[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

[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.2006.05.032 Search in Google Scholar

[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.2016.10.027 Search in Google Scholar

[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.2010.11.028 Search in Google Scholar

[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. Search in Google Scholar

[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/s10543-010-0285-1 Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[9] Bülbül B., Sezer M., (2013). A new approach to numerical solution of nonlinear Klein-Gordon equation, Mathematical Problems in Engineering, 2013. Search in Google Scholar

[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. Search in Google Scholar

[11] Chen X., Collocation Methods for Nonlinear Parabolic Partial Differential Equations, (2017). (Doctoral dissertation, Concordia University). Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[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.132.558 Search in Google Scholar

[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.4721 Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[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

[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-2003-0025 Search in Google Scholar

[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

[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.2013.07.019 Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[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.2016.08.031 Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[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

[29] Müller S., Sverák V., (1998). Unexpected solutions of first and second order partial differential equations. Search in Google Scholar

[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

[31] Russell R. D., Shampine L. F., (1972). A collocation method for boundary value problems. Numerische Mathematik, 19, 1, 1-28. DOI: 10.1007/BF01395926 Search in Google Scholar

[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: 10.1137/0714003 Search in Google Scholar

[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: 10.1080/00207160.2017.1284317 Search in Google Scholar

[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: 10.1137/0714003 Search in Google Scholar

[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/TSCI160615040W Search in Google Scholar

[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

[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. Search in Google Scholar

[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. Search in Google Scholar

[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

[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. Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo