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Erschienen in: Neural Computing and Applications 2/2013

01.08.2013 | Original Article

A method for solving nonlinear time-dependent drainage model

verfasst von: Yasir Khan

Erschienen in: Neural Computing and Applications | Ausgabe 2/2013

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Abstract

The purpose of this paper is to present a method for solving nonlinear time-dependent drainage model. This method is based on the perturbation theory and Laplace transformation. The proposed technique allows us to obtain an approximate solution in a series form. The computed results are in good agreement with the results of Adomian decomposition method. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present approach provides a reliable technique, which avoids the tedious work needed by classical techniques and existing numerical methods. The nonlinear time-dependent drainage model is solved without linearizing or discretizing the nonlinear terms of the equation. The method does not require physically unrealistic assumptions, linearization or discretization in order to find the solutions of the given problems.

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Literatur
1.
Zurück zum Zitat Weaire D, Hutzler S (2000) The physic of foams. Oxford University Press, Oxford Weaire D, Hutzler S (2000) The physic of foams. Oxford University Press, Oxford
2.
Zurück zum Zitat Weaire D, Hutzler S, Cox S, Alonso MD, Drenckhan D (2003) The fluid dynamics of foams. J Phys Condens Matter 15:65–72CrossRef Weaire D, Hutzler S, Cox S, Alonso MD, Drenckhan D (2003) The fluid dynamics of foams. J Phys Condens Matter 15:65–72CrossRef
3.
Zurück zum Zitat Helal MA, Mehanna MS (2007) The tanh method and Adomian decomposition method for solving the foam drainage equation. Appl Math Comput 190:599–609MathSciNetMATHCrossRef Helal MA, Mehanna MS (2007) The tanh method and Adomian decomposition method for solving the foam drainage equation. Appl Math Comput 190:599–609MathSciNetMATHCrossRef
4.
Zurück zum Zitat Hilgenfeldt S, Koehler SA, Stone HA (2001) Dynamics of coarsening foams: accelerated and self-limiting drainage. Phys Rev Lett 86:4704–4707CrossRef Hilgenfeldt S, Koehler SA, Stone HA (2001) Dynamics of coarsening foams: accelerated and self-limiting drainage. Phys Rev Lett 86:4704–4707CrossRef
5.
Zurück zum Zitat Verbist G, Weaire D, Kraynik AM (1996) The foam drainage equation. J Phys Condens Matter 83:715–3731 Verbist G, Weaire D, Kraynik AM (1996) The foam drainage equation. J Phys Condens Matter 83:715–3731
6.
Zurück zum Zitat Koehler SA, Stone HA, Brenner MP, Eggers J (1998) Dynamics of foam drainage. Phys Rev E 58:2097–2106CrossRef Koehler SA, Stone HA, Brenner MP, Eggers J (1998) Dynamics of foam drainage. Phys Rev E 58:2097–2106CrossRef
7.
Zurück zum Zitat Dahmani Z, Mesmoudi MM, Bebbouchi R (2008) The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method. Electron J Qual Theory Differ Equ 30:1–10MathSciNet Dahmani Z, Mesmoudi MM, Bebbouchi R (2008) The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method. Electron J Qual Theory Differ Equ 30:1–10MathSciNet
8.
Zurück zum Zitat Mirmoradi SH, Hosseinpour I, Barari A, Ghotbi AR (2009) Analysis of foam drainage problem using variational iteration method. Adv Stud Theor Phys 3:283–292 Mirmoradi SH, Hosseinpour I, Barari A, Ghotbi AR (2009) Analysis of foam drainage problem using variational iteration method. Adv Stud Theor Phys 3:283–292
9.
Zurück zum Zitat Dahmani Z, Anber A (2010) The variational iteration method for solving the fractional foam drainage equation. Int J Nonlinear Sci 10:39–45MathSciNetMATH Dahmani Z, Anber A (2010) The variational iteration method for solving the fractional foam drainage equation. Int J Nonlinear Sci 10:39–45MathSciNetMATH
10.
Zurück zum Zitat Fadravi HH, Nik HS, Buzhabadi R (2011) Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives. Int J Differ Equ 2011:12, Art ID 237045 Fadravi HH, Nik HS, Buzhabadi R (2011) Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives. Int J Differ Equ 2011:12, Art ID 237045
11.
Zurück zum Zitat Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61:1963–1967MathSciNetMATHCrossRef Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61:1963–1967MathSciNetMATHCrossRef
12.
Zurück zum Zitat Khan Y, Mohyud-Din ST (2010) Coupling of He’s polynomials and Laplace transformation for MHD viscous flow over a stretching sheet. Int J Nonlinear Sci Num Simul 11:1103–1107CrossRef Khan Y, Mohyud-Din ST (2010) Coupling of He’s polynomials and Laplace transformation for MHD viscous flow over a stretching sheet. Int J Nonlinear Sci Num Simul 11:1103–1107CrossRef
13.
Zurück zum Zitat He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Nonlinear Mech 35:37–43MATHCrossRef He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Nonlinear Mech 35:37–43MATHCrossRef
14.
Zurück zum Zitat Abbasbandy S (2006) Application of He’s homotopy perturbation method for Laplace transform. Chaos Solitons Fractals 30:1206–1212MATHCrossRef Abbasbandy S (2006) Application of He’s homotopy perturbation method for Laplace transform. Chaos Solitons Fractals 30:1206–1212MATHCrossRef
15.
Zurück zum Zitat Mohyud-Din ST, Noor MA, Noor KI, Hosseini MM (2010) On the coupling of He’s polynomials and Laplace transformation. Int J Nonlinear Sci Num Simul 11:93–96 Mohyud-Din ST, Noor MA, Noor KI, Hosseini MM (2010) On the coupling of He’s polynomials and Laplace transformation. Int J Nonlinear Sci Num Simul 11:93–96
16.
Zurück zum Zitat Madani M, Fathizadeh M, Khan Y, Yildirim A (2011) On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 53:1937–1945MathSciNetMATHCrossRef Madani M, Fathizadeh M, Khan Y, Yildirim A (2011) On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 53:1937–1945MathSciNetMATHCrossRef
17.
Zurück zum Zitat Khan Y, Wu Q, Faraz N, Yildirim A (2011) The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Comput Math Appl 61:3391–3399MathSciNetMATHCrossRef Khan Y, Wu Q, Faraz N, Yildirim A (2011) The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Comput Math Appl 61:3391–3399MathSciNetMATHCrossRef
18.
Zurück zum Zitat Khan Y, Faraz N, Yildirim A, Wu Q (2011) A series solution of the long porous slider. Tribol Trans 54:187–191CrossRef Khan Y, Faraz N, Yildirim A, Wu Q (2011) A series solution of the long porous slider. Tribol Trans 54:187–191CrossRef
19.
Zurück zum Zitat He JH (2011) Analytical methods for thermal science-an elementary introduction. Therm Sci 15:S1–S3CrossRef He JH (2011) Analytical methods for thermal science-an elementary introduction. Therm Sci 15:S1–S3CrossRef
20.
Zurück zum Zitat Hesameddini E, Latifzadeh H (2009) Reconstruction of variational iteration algorithm using the Laplace transform. Int J Nonlinear Sci Num Simul 10:1377–1382 Hesameddini E, Latifzadeh H (2009) Reconstruction of variational iteration algorithm using the Laplace transform. Int J Nonlinear Sci Num Simul 10:1377–1382
22.
Zurück zum Zitat Mohyud-Din ST, Yildirim A (2010) Variation of parameters method using Laplace transformation method. World Appl Sci J 9:300–302 Mohyud-Din ST, Yildirim A (2010) Variation of parameters method using Laplace transformation method. World Appl Sci J 9:300–302
23.
Zurück zum Zitat Khan Y (2009) An effective modification of the Laplace decomposition method for nonlinear equations. Int J Nonlinear Sci Num Simul 10:1373–1376 Khan Y (2009) An effective modification of the Laplace decomposition method for nonlinear equations. Int J Nonlinear Sci Num Simul 10:1373–1376
24.
Zurück zum Zitat Khan M, Gondal MA (2010) A new analytical solution of foam drainage equation by Laplace decomposition method. J Adv Res Differ Equ 2:53–64 Khan M, Gondal MA (2010) A new analytical solution of foam drainage equation by Laplace decomposition method. J Adv Res Differ Equ 2:53–64
25.
Zurück zum Zitat Khan Y, Austin F (2010) Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations. Z Naturforsch 65a:849–853 Khan Y, Austin F (2010) Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations. Z Naturforsch 65a:849–853
26.
Zurück zum Zitat Šmarda Z, Archalousova O (2010) Adomian decomposition method for certain singular initial value problems II. J Appl Math 3:91–98 Šmarda Z, Archalousova O (2010) Adomian decomposition method for certain singular initial value problems II. J Appl Math 3:91–98
27.
Zurück zum Zitat Turkyilmazoglu M (2011) An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. J Appl Mech Trans ASME 78:021005CrossRef Turkyilmazoglu M (2011) An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. J Appl Mech Trans ASME 78:021005CrossRef
28.
29.
Zurück zum Zitat Diblík J, Šmarda Z, Berezansky L (2010) Positive solutions of a second-order delay differential equations with a damping term. Comput Math Appl 62:1332–1352 Diblík J, Šmarda Z, Berezansky L (2010) Positive solutions of a second-order delay differential equations with a damping term. Comput Math Appl 62:1332–1352
30.
Zurück zum Zitat Bastinec J, Diblík J, Šmarda Z (2011) An explicit criterion for the existence of positive solutions of the linear delayed equation \( \dot{x}(t) = - c(t)x(t - \tau (t)). \) Abstract Appl Anal 2011:12, Art ID 561902 Bastinec J, Diblík J, Šmarda Z (2011) An explicit criterion for the existence of positive solutions of the linear delayed equation \( \dot{x}(t) = - c(t)x(t - \tau (t)). \) Abstract Appl Anal 2011:12, Art ID 561902
31.
Zurück zum Zitat Bastinec J, Berezansky L, Diblík J, Šmarda Z (2011) A final result on the oscillation of solutions of the linear discrete delayed equation ∆x(n) = −p(n)x(n – k) with a positive coefficient. Abstract Appl Anal 2011:28, Art ID 586328 Bastinec J, Berezansky L, Diblík J, Šmarda Z (2011) A final result on the oscillation of solutions of the linear discrete delayed equation ∆x(n) = −p(n)x(n  k) with a positive coefficient. Abstract Appl Anal 2011:28, Art ID 586328
33.
Zurück zum Zitat Zayed EME, Abdel Rahman HM (2012) On using the He’s polynomials for solving the nonlinear coupled evolution equations in mathematical physics. WSEAS Trans Math 11:294–302 Zayed EME, Abdel Rahman HM (2012) On using the He’s polynomials for solving the nonlinear coupled evolution equations in mathematical physics. WSEAS Trans Math 11:294–302
Metadaten
Titel
A method for solving nonlinear time-dependent drainage model
verfasst von
Yasir Khan
Publikationsdatum
01.08.2013
Verlag
Springer London
Erschienen in
Neural Computing and Applications / Ausgabe 2/2013
Print ISSN: 0941-0643
Elektronische ISSN: 1433-3058
DOI
https://doi.org/10.1007/s00521-012-0933-2

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