Solution of One – Dimensional Ground Water Recharge Through Porous Media Via Reduced Differential Transform Method

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2021 by IJMTT Journal
Volume-67 Issue-9
Year of Publication : 2021
Authors : Priti Tandel, Hardik Patel
  10.14445/22315373/IJMTT-V67I9P509

MLA

MLA Style: Priti Tandel, Hardik Patel "Solution of One – Dimensional Ground Water Recharge Through Porous Media Via Reduced Differential Transform Method" International Journal of Mathematics Trends and Technology 67.9 (2021):81-86. 

APA Style: Priti Tandel, Hardik Patel(2021). Solution of One – Dimensional Ground Water Recharge Through Porous Media Via Reduced Differential Transform Method  International Journal of Mathematics Trends and Technology, 67(9), 81-86.

Abstract
This paper contains a problem of one-dimensional ground water recharge by spreading through porous media. The mathematical formulation is obtained with variable permeability and constant average coefficient of diffusivity over the entire range of moisture content. Reduced differential transform method (RDTM) is applied to obtain solutions of governing equation. Graphical and numerical representation of the variations in moisture content of soil with the depth and time have been discussed. For validation of the method, results obtained by RDTM are compared with numerical solutions. This research also emphasizes the applicability of RDTM for solving non-linear partial differential equation.

Reference

[1] A. Klute, A numerical method for solving the flow equation of water in unsaturated materials, Soil sciences, 73( 2) (1952) 105-116.
[2] A. Roozi, A. Alibeiki, S. S. Husseini, S. M. Shafiof, M. Ebrahimi, Homotopy perturbation method for special nonlinear partial differential equatios, Journal of King Saud University, 23(2011) 99-103.
[3] A. P.Verma, The Laplace transform solution of a one dimensional groundwater recharge by spreading, . Ricevuto il 21 Gennaio, (1969) 25-31.
[4] A. P Verma., S.K.,Mishra, A similarity solution of unidimensional vertical groundwater recharge through porous media, Revue Roumaine des Sciences Techniques Serie de Mechanique Appliquee, 18(2) (1973) 345-351.
[5] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, Inc., New York, (1972).
[6] K.Shah, T. Sing., A Solution of the Burger’s Equation Arising in the Longitudinal Dispersion Phenomenon in Fluid Flow through Porous Media by Mixture of New Integral Transform and Homotopy Perturbation Method, Journal of Geoscience and Environment Protection,3(4) (2015) 24-30, DOI: 10.4236/gep.2015.34004
[7] M.Rawashdeh,Improved, Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method, Journal of Applied Mathematics & Bioinformatics, 3(2) (2013) 1-14.
[8] N.S.Rabari, A.S Gor., P.H Bhathawala., Finite Element Solution of One Dimensional Flow in Unsaturated Porous Media, IOSR Journal of Mathematics,10(3),Ver. V,(2014) 49-57.
[9] M. N.Mehta ,S. Yadav,Solution of Problem arising during vertical groundwater recharge by spreading in slightly saturated Porous Media, Journal of Ultra Scientists of Physical Sciences, volume 19(3)(2007) 541-546.
[10] M. N. Mehta,T. Patel, A solution of Burger’s equation type one-dimensional Groundwater Recharge by spreading in Porous Media, The Journal of the Indian Academy of Mathematics, volume 28(1)(2006) 25-32
[11] M. O. Al-Amr, New applications of reduced differential transform method, Alexandria Engineering Journal, 53(1) (2014) 243–247.
[12] N. Taghizadeh and S. R. M. Noori, Reduced differential transform method for solving parabolic-like and hyperbolic like equations, SeMA Journal, Volume 74(4) (2017) 559–567.
[13] R.K.Meher, M.N. Mehta., S.K. Meher,:Adomian decomposition method for moisture content in one dimensional fluid flow through unsaturated porous media, International Journal of Applied Mathematics and mechanics 6(7) (2010) 13-23.
[14] R.K. Meher,M.N. Mehta, Adomian Decomposition Method for Dispersion Phenomenon Arising in Longitudinal Dispersion of Miscible Fluid Flow through Porous Media. Advances in Theoretical and Applied Mechanics, 3(2010) 211-220.
[15] R.N Borana., V. H Pradhan., M. N Mehta.,Numerical solution of Bergers’ equation in a one-dimensional groundwater recharge by spreading using finite difference method, International Journal of Advance Research in Science and Engineering, 2(11) (2013) 121-130.
[16] S.Pathak, T. Singh, The solution of non-linear problem arising in infiltration phenomenon in unsaturated soil by optimal homotopy analysis method, International Journal of Advances in Applied Mathematics and Mechanics, 4(2) (2016) 21 – 28.
[17] S.R.M. Noori , N. Taghizadeh, Study of Convergence of Reduced Differential Transform Method for Different Classes of Differential Equations, International Journal of Differential Equations Volume 2021, Article ID 6696414, 16 pages https://doi.org/10.1155/2021/6696414
[18] S.M.Mohmed, A.G.Khaled, Reduced differential transform method for nonlinear integral member of Kadomtsv-Petviashivili heirarchy differential equations, Journal of Egiptian Methematical Society, 25-(2017) 1-7.
[19] S.S.Patel, Unidmensional Flow through Unsaturated Porous Media: A Problem of Groundwater Recharge with Perturbation Technique, International Journal of Computational Science and Mathematics, 2(3) (2010) 123-128
[20] T. A. Marwan, Applying Differential Transform Method to Nonlinear Partial Differential Equations: A Modified Approach, Applications and Applied Mathematics, 7(1) (2012) 155 – 163.
[21] V.K.Shrivastava., N.Mishra, S.Kumar,Singh,B.Kumar, M.K Awasthi., Reduced differential transform method for (1+n)-dimensional Burgers’equation, Egyptian Journal of Applied and basic Sciences I (2014) 115-119.
[22] Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 10(6) (2009) 741-750.
[23] Y. Keskin, G. Oturanc, The reduced differential transform method: a new approach to factional partial differential equations. Nonlinear Science Letters- A, 1(2) (2010) 207-217.
[24] Y. Keskin, G. Oturanc., Reduced differential transform method for generalized KdV equations. Mathematical and Computational applications, 15(3) (2010) 382-393.

Keywords : Ground water,Moisture content, Porous media, Reduced Differential Transform Method.