Modified approach to solve nonlinear equation arising in infiltration phenomenon. (English) Zbl 1453.76189
Summary: In the present analysis, the modified homotopy analysis method has been employed to find an approximate analytical solution of Richards’ equation. This method is the slight modification of standard homotopy analysis method. Some standard cases of Richards’ equation have been discussed as an example to illustrate the high accuracy and reliability of modified homotopy analysis method. The result obtained from the proposed method is very close to the exact solution of the problem. It is concluded that modified homotopy analysis method is the better alternative to some standard existing methods to solve some realistic problems arising in science and technology.
MSC:
| 76M99 | Basic methods in fluid mechanics |
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
infiltration phenomenon; Richards’ equation; modified homotopy analysis method; \(h\)-curveReferences:
| [1] | Asgari, A., Bagheripourb, M.H., Mollazadehb, M.: A generalized analytical solution for a nonlinear infiltration equation using the exp-function method. Sci. Iran. A. 18(1), 28-35 (2011) · Zbl 1282.76149 · doi:10.1016/j.scient.2011.03.004 |
| [2] | Barari, A., Omidvar, M., Ghotbi, A.R., Ganji, D.D.: Numerical analysis of Richards’ problem for water penetration in unsaturated soils. Hydrol. Earth Syst. Sci. Discuss. 6, 6359-6385 (2009) · doi:10.5194/hessd-6-6359-2009 |
| [3] | Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media. Hydrol. Pap. Colo. State Univ. Fort Collins 3, 24 (1964) |
| [4] | Buckingham, E.: Studies on the movement of soil moisture. Bur. Soils Bull. 38, 61 (1907) |
| [5] | Corey, A.T.: Mechanics of Immiscible Fluids in Porous Media. Colorado Co., Water Resource Publication, Littleton (1986) |
| [6] | El-Tawil, M.A., Huseen, S.N.: The q-homotopy analysis method. Int. J. Appl. Math. Mech. 8(15), 5175 (2012) |
| [7] | Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. J. Soc. Soil Sci. Am. 44, 892-898 (1980) · doi:10.2136/sssaj1980.03615995004400050002x |
| [8] | Liao, S.J.: Homotopy analysis method a new analytical technique for nonlinear problems. Commun. Nonlinear Sci. Numer. Simul. 2, 95-100 (1997) · Zbl 0927.65069 · doi:10.1016/S1007-5704(97)90047-2 |
| [9] | Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC Press, Boca Raton (2003) · Zbl 1051.76001 · doi:10.1201/9780203491164 |
| [10] | Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499-513 (2004) · Zbl 1086.35005 |
| [11] | Mays, L.W.: Water Resources Engineering, 2nd edn. Wiley, New York (2011) |
| [12] | Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318-333 (1931) · Zbl 0003.28403 · doi:10.1063/1.1745010 |
| [13] | Shah, K., Singh, T.: The modified homotopy algorithm for dispersion phenomena. Int. J. Appl. Comput. Math. 3(1), 785-799 (2017) · doi:10.1007/s40819-017-0382-9 |
| [14] | Shah, K., Singh, T.: Solution of Burger’s equation in a one-dimensional groundwater recharge by spreading using q-homotopy analysis method. Eur. J. Pure Appl. Math. 9(1), 114-124 (2016) · Zbl 1359.35171 |
| [15] | Shah, K., Singh, T.: An approximate solution of \[\theta\] θ-based Richards’ equation by combination of new integral transform and homotopy perturbation method. J. Niger. Math. Soc. 36, 85-100 (2017) · Zbl 1474.65414 |
| [16] | Soliman, A.A.: The modified extended tanh-function method for solving Burgers type equations. Physics A 361, 394-404 (2006) · doi:10.1016/j.physa.2005.07.008 |
| [17] | Wazwaz, A.M.: Traveling wave solutions for generalized forms of Burgers, Burgers-KDV and Burgers-Huxley equations. Appl. Math. Comput. 169, 639-656 (2005) · Zbl 1078.35109 |
| [18] | Whitman, B.G.: Linear and nonlinear waves. Wiley, New York (1974) · Zbl 0373.76001 |
| [19] | Witelski, T.P.: Perturbation analysis for wetting front in Richards’ equation. Trans. Porous Med. 27, 121-134 (1997) · doi:10.1023/A:1006513009125 |
| [20] | Witelski, T.P.: Motion of wetting fronts moving into partially pre-wet soil. Adv. Water Resour. 28, 1131-1141 (2005) · doi:10.1016/j.advwatres.2004.06.006 |
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.
