Abstract
In this article, a closed-form iterative analytic approximation to a class of nonlinear singularly perturbed parabolic partial differential equation is developed and analysed for convergence. We have considered both parabolic reaction diffusion and parabolic convection diffusion type of problems in this paper. The solution of this class of problem is polluted by a small dissipative parameter, due to which solution often shows boundary and interior layers. A sequence of approximate analytic solution for the above class of problems is constructed using Lagrange multiplier approach. Within a general frame work, the Lagrange multiplier is optimally obtained using variational theory. The sequence of approximate analytical solutions so obtained is proved to converge the exact solution of the problem. To demonstrate the proposed method’s efficiency and accuracy, linear and nonlinear test problems have been taken into account. From numerical experiments, it is observed that the proposed method is highly accurate, straightforward.
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References
Ali I, Malik N (2014) Hilfer fractional advection diffusion equations with power law initial condition: a numerical study using variational iteration method. Comput Math Appl 68:1161–1179. https://doi.org/10.1016/j.camwa.2014.08.021
Al-Sawoor A, Al-Amr M (2014) A new modification of variational iteration method for solving reaction diffusion system with fast reversible reaction. J Egypt Math Soc 22:396–401. https://doi.org/10.1016/j.joems.2013.12.011
Alshabanat A, Jleli M, Kumar S, Samet B (2020) Generalization of caputo-fabrizio fractional derivative and applications to electrical circuits. Front Phys 8:64. https://doi.org/10.3389/fphy.2020.00064
Amrein M, Wihler TP (2017) An adaptive space-time newton-galerkin approach for semilinear singularly perturbed parabolic evolution equations. IMA J Numer Anal 37(4):2004–2019. https://doi.org/10.1093/imanum/drw049
Boglaev I (2004) Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem. J Comput Appl Math 172(2):313–335. https://doi.org/10.1016/j.cam.2004.02.010
Boglaev I (2012) An inexact monotone method for solving semilinear parabolic problems. Appl Math Comp 219(6):3253–3263. https://doi.org/10.1016/j.amc.2012.09.067
Chandru M, Das P, Ramos H (2018) Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math Meth Appl Sci 14:5359–5387. https://doi.org/10.1002/mma.5067
Cheng Y, Song C, Mei Y (2021) Local discontinuous galerkin method for time-dependent singularly perturbed semilinear reaction-diffusion problems. Comput Methods Appl Math 21(1):31–52. https://doi.org/10.1515/cmam-2019-0185
Das P (2019) An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer Algor 81:465–487. https://doi.org/10.1007/s11075-018-0557-4
Das P, Natesan S (2012) Higher order parameter uniform convergent schemes for Robin type reaction diffusion problems using adaptively generated grid. Int J Comput Methods 9:125–152
Das P, Natesan S (2013) A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion robin type boundary-value problems. J Appl Math Comput 41:447–471. https://doi.org/10.1007/s12190-012-0611-7
Das P, Natesan S (2014) Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction diffusion boundary value problems. Appl Math Comput 249:265–277
Das P, Rana S (2021) Theoretical prospects of fractional order weakly singular volterra integro differential equations and their approximations with convergence analysis. Math Meth Appl Sci 44(3):9419–9440. https://doi.org/10.1002/mma.7369
Das P, Rana S, Ramos H (2019) Homotopy perturbation method for solving caputo-type fractional-order volterra-fredholm integro-differential equations. Comp Math Methods 1:2577–7408. https://doi.org/10.1002/cmm4.1047
Das P, Rana S, Ramos H (2022) On the approximate solutions of a class of fractional order nonlinear volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. J Comput Appl Math 404:0377–0427. https://doi.org/10.1016/j.cam.2020.113116
Doeva O, Masjedi PK, Weaver PM (2021) A semi-analytical approach based on the variational iteration method for static analysis of composite beams. Compos Struct 257:113110. https://doi.org/10.1016/j.compstruct.2020.113110
El-Sayed T, El-Mongy H (2018) Application of variational iteration method to free vibration analysis of a tapered beam mounted on two-degree of freedom subsystems. Appl Math Model 58:349–364. https://doi.org/10.1016/j.apm.2018.02.005
Ghaneai H, Hosseini M (2015) Variational iteration method with an auxiliary parameter for solving wave-like and heat-like equations in large domains. Comput Math Appl 69:363–373. https://doi.org/10.1016/j.camwa.2014.11.007
He J (1997) Variational iteration method for delay differential equations. Commun Nonlinear Sci Numer Simul 2:235–236. https://doi.org/10.1016/S1007-5704(97)90008-3
He J (1999) Variational iteration method - a kind of non-linear analytical technique: some examples. Int J Non Linear Mech 34:699–708. https://doi.org/10.1016/S0020-7462(98)00048-1
He J-H, Kong H-Y, Chen R-X, Hu M-s, Chen Q-l (2014) Variational iteration method for bratu-like equation arising in electrospinning. Carbohydr Polym 105:229–230. https://doi.org/10.1016/j.carbpol.2014.01.044
Inokuti M, Sekine H, Mura T (1978) General use of the Lagrange multiplier in non-linear mathematical physics. Pergamon Press, Oxford
Kabeto MJ, Duressa GF (2021) Robust numerical method for singularly perturbed semilinear parabolic differential difference equations. Math Comput Simul 188:537–547. https://doi.org/10.1016/j.matcom.2021.05.005
Kanth AR, Aruna K (2010) He’s variational iterational method for treating nonlinear singular boundary value problems. Comput Math Appl 60:821–829. https://doi.org/10.1016/j.camwa.2010.05.029
Khari K, Kumar V (2022) An efficient numerical technique for solving nonlinear singularly perturbed reaction diffusion problem. J Math Chem 60:1356–1382. https://doi.org/10.1007/s10910-022-01365-4
Khari K, Kumar V (2022) Finite element analysis of the singularly perturbed parabolic reaction-diffusion problems with retarded argument. Numer Methods Partial Differ Eq 38:997–1014. https://doi.org/10.1002/num.22785
Kopteva N, Linss T (2012) Maximum norm a posteriori error estimation for a time-dependent reaction-diffusion problem. Comput Methods Appl Math 12:189–205. https://doi.org/10.2478/cmam-2012-0013
Kopteva N, Linss T (2013) Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions. SIAM J Numer Anal 51(3):1494–1524. https://doi.org/10.1137/110830563
Kumar S, Gupta V (2021) An application of variational iteration method for solving fuzzy time-fractional diffusion equations. Neural Comput Applic 33:17659–17668. https://doi.org/10.1007/s00521-021-06354-3
Kumar V, Leugeringb G (2021) Singularly perturbed reaction-diffusion problems on a k-star graph. Math Meth Appl Sci 44:14874–14891. https://doi.org/10.1002/mma.7749
Kumar V, Leugeringb G (2023) Convection dominated singularly perturbed problems on a metric graph. J Comput Appl Math. https://doi.org/10.1016/j.cam.2023.115062
Kumar S, Nisar KS, Kumar R, Cattani C, Samet B (2020) A new rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force. Math Meth Appl Sci 43(7):4460–4471. https://doi.org/10.1002/mma.6208
Kumar K, Podila P, Das P, Ramos H (2021) A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems. Math Meth Appl Sci 44:12332–12350. https://doi.org/10.1002/mma.7358
Mohammadi H, Kumar S, Rezapour S, Etemad S (2021) A theoretical study of the caputo-fabrizio fractional modeling for hearing loss due to mumps virus with optimal control, Chaos. Solitons Fractals 144:110668. https://doi.org/10.1016/j.chaos.2021.110668
Odibat ZM (2010) A study on the convergence of variational iteration method. Math Comput Model 51(9):1181–1192. https://doi.org/10.1016/j.mcm.2009.12.034
Ramos J (2008) On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl Math Comp 199:39–69. https://doi.org/10.1016/j.amc.2007.09.024
Sakar M, Ergören H (2015) Alternative variational iteration method for solving the time fractional Fornberg-Whitham equation. Appl Math Model 14:3972–3979. https://doi.org/10.1016/j.apm.2014.11.048
Shakti D et al (2022) A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction-diffusion problems with arbitrary small diffusion terms. J Comput Appl Math 404(3):0377–0427. https://doi.org/10.1016/j.cam.2020.113167
Veeresha P, Prakasha DG, Kumar S (2020) A fractional model for propagation of classical optical solitons by using nonsingular derivative. Math Meth Appl Sci. https://doi.org/10.1002/mma.6335
Wazwaz A-M (2017) Solving the non-isothermal reaction-diffusion model equations in a spherical catalyst by the variational iteration method. Chem Phys Lett 679:132–136. https://doi.org/10.1016/j.cplett.2017.04.077
Wazwaz A-M (2020) Optical bright and dark soliton solutions for coupled nonlinear schrödinger (cnls) equations by the variational iteration method. Optik 207:164457. https://doi.org/10.1016/j.ijleo.2020.164457
Wu G-C, Baleanu D (2013) Variational iteration method for the Burger’s flow with fractional derivatives: New Lagrange multipliers. Appl Math Model 37:6183–6190. https://doi.org/10.1016/j.apm.2012.12.018
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VK would like to acknowledge the National board of higher mathematics (NBHM) for research Grant No.–Ref. No. 2/48(6)/2016/NBHM(R.P.)/R \(, \) D II /15455.
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KK helped in conceptualization, methodology, writing—original draft, supervision
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Khari, K., Kumar, V. An iterative analytic approximation for a class of nonlinear singularly perturbed parabolic partial differential equations. Soft Comput 27, 16279–16291 (2023). https://doi.org/10.1007/s00500-023-08057-4
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DOI: https://doi.org/10.1007/s00500-023-08057-4