Solution of the porous media equation by Adomian's decomposition method

Pamuk, Serdal

doi:10.1016/j.physleta.2005.06.068

Cited by 55 publications

(26 citation statements)

References 8 publications

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“…A number of researchers investigated analytical solutions to the 1D Richards' equation by Variational Iteration Method (VIM) [24][25][26] and Adomian Decomposition Method (ADM) [27][28][29][30][31]. They used ADM and VIM in an initial value problem for the equation; however, the series solution obtained by ADM and VIM often did not satisfy the PDE.…”

Section: Introductionmentioning

confidence: 99%

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

2017

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Abstract. Fluid in ltration into unsaturated soil is of vital signi cance from many perspectives. Mathematically, such transient in ltrations are described by Richards' equation: a nonlinear parabolic Partial Di erential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1D horizontal and vertical water in ltration are derived and compared to numerical nite-di erence method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water in ltration are derived for constant, no-ow, and sinusoidal boundary and initial conditions. The presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use di erent computational techniques.

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Section: Introductionmentioning

confidence: 99%

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

2017

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“…A number of researchers investigated analytical solutions to the 1D Richards' equation by variational iteration method (VIM) (He 1998;Moghimi and Hejazi 2007;Wazwaz 2007), and Adomian decomposition method (ADM) (Nasseri et al 2008;Serrano and Adomian 1996;Serrano 1998Serrano , 2004Pamuk 2005). They used ADM and VIM in an initial value problem for the equation; however, the series solution obtained by ADM and VIM often did not satisfy the PDE.…”

Section: Introductionmentioning

confidence: 99%

New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

Sanayei

Rakhshandehroo

Talebbeydokhti

2016

Iran J Sci Technol Trans Civ Eng

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Fluid infiltration and imbibition into unsaturated soil are of vital significance from many perspectives. Mathematically, such transient flows are described by Richards' equation, a nonlinear parabolic partial differential equation with limited analytical solutions in the literature. However, the choice of exponential model for water content and hydraulic conductivity linearizes the nonlinear Richards' equation, making it possible to obtain an analytical solution via classical approaches. In this study, separation of variables and Fourier series expansion techniques are used to derive new analytical solutions to 2-D vertical and horizontal infiltration and imbibition into unsaturated soils for nonsymmetrical boundary and nonuniform initial conditions. A total of 11 cases are considered, where high water content is imposed on the top, side, or bottom edges of the sample and water is infiltrated (from the top and/or side boundaries) and imbibed (from the bottom boundary) into the sample. Residual water content and/or noflow boundary condition are assumed on other edges of the sample. Initial conditions include 7 cases of constant residual water content, 2 cases of sinusoidal, and 2 cases of exponential water content functions over the sample. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed-form solution. Two-dimensional and 3-D plots of water content are presented for the transient as well as steady-state conditions. To illustrate the use of the derived equations, water content values from numerical solutions are compared to those from analytical solutions for four cases, showing a maximum error of\2 %. The presented analytical solutions may be used as a benchmark for verification and accuracy assessment of numerical approaches where nonsymmetrical boundary and/or nonuniform initial conditions exist.

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“…Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Lane, Emden, Clairaut, Euler and Lagrange. The various powerful numerical and analytic methods such as perturbation methods, Nayfeh (2011), Rand et al (1987), He (1999), Ganji (2006), , Yıldırım et al (2009), Shakeri et al (2008), Aminikhah (2010) and Aminikhah et al (2009), Adomian decomposition method, Bildik et al (2006), Biazar et al (2004), Babolian et al (2004) and Pamuk (2005), homotopy analysis method, Liao (2004), and Jafari et al (2012) and differential transform method, Odibat et al (2008), Ertürk et al (2008) and Ayaz (2004) have been developed for solving them.…”

Section: Introductionmentioning

confidence: 99%

A New Approximate Analytical Method for ODEs

Aminikhah

2014

Journal of Applied Mathematics, Statistics and Informatics

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In this paper, we propose a new algorithm for solving ordinary differential equations. We show the superiority of this algorithm by applying the new method for some famous ODEs. Theoretical considerations are discussed. The first He's polynomials have used to reach the exact solution of these problems. This method which has good stability and accuracy properties is useful in deal with linear and nonlinear system of ordinary differential equations.

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Solution of the porous media equation by Adomian's decomposition method

Cited by 55 publications

References 8 publications

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

A New Approximate Analytical Method for ODEs

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