×
Nadeem, Muhammad; Li, Fengquan; Ahmad, Hijaz

Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients. (English) Zbl 1442.65317

Comput. Math. Appl. 78, No. 6, 2052-2062 (2019).
Summary: In the present study, a new amendment in Laplace variational iteration method for the solution of fourth-order parabolic partial differential equations with variable coefficients is revealed i.e. modified Laplace variational iteration method (ML-VIM). The proposed modification is made by coupling of two methods: one of them is variational iteration method (VIM) and the other one is Laplace transformation (LT). Our modification has an important beauty that one has no need to compute Lagrange multiplier via integration nor by taking convolution theorem and has a simple way to use in the implementation of our proposed scheme. Moreover, homotopy perturbation method (HPM) with He’s polynomials is employed for the computation of nonlinear terms. The main improvement of our proposed scheme is the reduction of the problem to a simple one, which also embraces for solving a nonlinear term. Some illustrated examples are interpreted which unveil the robustness and accuracy of the newly developed scheme.

Cited in 38 Documents

MSC:

65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs

Keywords:

modified Laplace variational iteration method; homotopy perturbation method; Lagrange multipliers; fourth order parabolic PDEqs
Cite Review PDF
Full Text: DOI

References:

[1] Dehghan, M.; Manafian, J., The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method, Z. Naturf. a, 64, 7-8, 420-430 (2009)
[2] Noor, M. A.; Noor, K. I.; Mohyud-Din, S. T., Modified variational iteration technique for solving singular fourth-order parabolic partial differential equations, Nonlinear Anal., 71, 12, 630-640 (2009) · Zbl 1238.65103
[3] Khaliq, A.; Twizell, E., A family of second order methods for variable coefficient fourth order parabolic partial differential equations, Int. J. Comput. Math., 23, 1, 63-76 (1987) · Zbl 0661.65096
[4] Liao, W.; Zhu, J.; Khaliq, A. Q., An efficient high-order algorithm for solving systems of reaction-diffusion equations, Numer. Methods Partial Differential Equations, 18, 3, 340-354 (2002) · Zbl 0997.65105
[5] Evans, D., A stable explicit method for the finite-difference solution of a fourth-order parabolic partial differential equation, Comput. J., 8, 3, 280-287 (1965) · Zbl 0134.33006
[6] Jain, M.; Iyengar, S.; Lone, A., Higher order difference formulas for a fourth order parabolic partial differential equation, Internat. J. Numer. Methods Engrg., 10, 6, 1357-1367 (1976) · Zbl 0345.65047
[7] Aziz, T.; Khan, A.; Rashidinia, J., Spline methods for the solution of fourth-order parabolic partial differential equations, Appl. Math. Comput., 167, 1, 153-166 (2005) · Zbl 1082.65564
[8] Evans, D.; Yousif, W., A note on solving the fourth order parabolic equation by the age method, Int. J. Comput. Math., 40, 1-2, 93-97 (1991) · Zbl 0736.65061
[9] Rashidinia, J.; Mohammadi, R., Sextic spline solution of variable coefficient fourth order parabolic equations, Int. J. Comput. Math., 87, 15, 3443-3454 (2010) · Zbl 1280.65088
[10] Caglar, H.; Caglar, N., Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations, Appl. Math. Comput., 201, 1-2, 597-603 (2008) · Zbl 1148.65080
[11] Wazwaz, A.-M., Analytic treatment for variable coefficient fourth-order parabolic partial differential equations, Appl. Math. Comput., 123, 2, 219-227 (2001) · Zbl 1027.35006
[12] He, J., A variational iteration approach to nonlinear problems and its applications, Mech. Appl., 20, 1, 30-31 (1998)
[13] He, J.-H., Variational iteration method — a kind of non-linear analytical technique: some examples, Internat. J. Non-Linear Mech., 34, 4, 699-708 (1999) · Zbl 1342.34005
[14] He, J.-H., Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114, 2-3, 115-123 (2000) · Zbl 1027.34009
[15] Zhou, X.-W.; Yao, L., The variational iteration method for Cauchy problems, Comput. Math. Appl., 60, 3, 756-760 (2010) · Zbl 1201.65189
[16] Ghaneai, H.; Hosseini, M., Variational iteration method with an auxiliary parameter for solving wave-like and heat-like equations in large domains, Comput. Math. Appl., 69, 5, 363-373 (2015) · Zbl 1443.65270
[17] Nadeem, M.; Li, F.; Ahmad, H., He’s variational iteration method for solving non-homogeneous cauchy euler differential equations, Nonlinear Sci. Lett. A, 9, 3, 231-237 (2018)
[18] Biazar, J.; Ghazvini, H., He’s variational iteration method for fourth-order parabolic equations, Comput. Math. Appl., 54, 7-8, 1047-1054 (2007) · Zbl 1267.65147
[19] He, J.-H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178, 3-4, 257-262 (1999) · Zbl 0956.70017
[20] He, J.-H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135, 1, 73-79 (2003) · Zbl 1030.34013
[21] He, J.-H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26, 3, 695-700 (2005) · Zbl 1072.35502
[22] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Comput. Math. Appl., 61, 8, 1963-1967 (2011) · Zbl 1219.65119
[23] Khuri, S. A.; Sayfy, A., A Laplace variational iteration strategy for the solution of differential equations, Appl. Math. Lett., 25, 12, 2298-2305 (2012) · Zbl 1252.65128
[24] Wu, G.-C.; Baleanu, D., Variational iteration method for fractional calculus — a universal approach by Laplace transform, Adv. Differ. Equ., 2013, 1, 18 (2013) · Zbl 1365.34022
[25] Kumar Mishra, H.; Nagar, A. K., He-Laplace method for linear and nonlinear partial differential equations, J. Appl. Math., 2012, 16 (2012), Article ID 180315 · Zbl 1251.65146
[26] Biazar, J.; Ghazvini, H., Convergence of the homotopy perturbation method for partial differential equations, Nonlinear Anal. RWA, 10, 5, 2633-2640 (2009) · Zbl 1173.35395
[27] Turkyilmazoglu, M., Convergence of the homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul., 12, 1-8, 9-14 (2011) · Zbl 1401.35024
[28] Biazar, J.; Aminikhah, H., Study of convergence of homotopy perturbation method for systems of partial differential equations, Comput. Math. Appl., 58, 11-12, 2221-2230 (2009) · Zbl 1189.65246
[29] Eslami, M.; Mirzazadeh, M., Study of convergence of homotopy perturbation method for two-dimensional linear volterra integral equations of the first kind, Int. J. Comput. Sci. Math., 5, 1, 72-80 (2014) · Zbl 1312.65227
[30] He, J.-H., Non-perturbative Methods for Strongly Nonlinear Problems (2006), Dissertation.de Verlag im Internet GmbH
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.