B-Spline Collocation Solution for Burgers’ equation arising in Longitudinal Dispersion Phenomena in Fluid Flow through Porous Media

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2022 by IJMTT Journal
Volume-68 Issue-7
Year of Publication : 2022
Authors : Nilesh Sonara, Dr. D C Joshi, Dr. N B Desai
 10.14445/22315373/IJMTT-V68I7P503

How to Cite?

Nilesh Sonara, Dr. D C Joshi, Dr. N B Desai, "B-Spline Collocation Solution for Burgers’ equation arising in Longitudinal Dispersion Phenomena in Fluid Flow through Porous Media," International Journal of Mathematics Trends and Technology, vol. 68, no. 7, pp. 13-20, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I7P503

Abstract
This paper investigates B-Spline Collocation Solution for Burgers’ equation arising in longitudinal dispersion phenomenon in the fluid flow through porous media. In the porous medium clean water, saltwater or tainted water disperse longitudinal way offers to increase to a non-linear partial differential condition as Burgers’ equation. The equation is solved by utilizing the B-Spline Collocation method with suitable initial and boundary conditions. The issue of miscible displacement can be found in the seaside territories, where new water beds are step by step uprooted via ocean water. An unequivocally steady B-spline Collocation method has been utilized to discover the concentration C(X,T) of salty or polluted water dispersion in uni-direction. It is completed, that the concentration C(X,T) reduce as distance X just as time T increments. The tables and figures are created by utilizing MATLAB coding.

Keywords : longitudinal dispersion, Burgers’ Equation, B-Spline Collocation Method.

Reference

[1] A. Al-Niami and K. Rushton, “Analysis of flow against dispersion in porous media,” Journal of Hydrology, vol. 33, no. 1-2, pp. 87–97, 1977.
[2] J. Bear, “Hydraulics of groundwater mcgraw-hill intl,” Book Co., New York, 1979.
[3] P. M. Blair, “Calculation of Oil Displacement by Countercurrent Water Imbibition,” Society of Petroleum Engineers Journal, vol. 4, no. 03, pp. 195–202, 09 1964. [Online]. Available: https://doi.org/10.2118/873-PA
[4] J. Caldwell, “Application of cubic splines to the nonlinear burgers’ equation,” Numerical Methods for Nonlinear Problems, vol. 3, pp. 253–261, 1987.
[5] Z. Chen, G. Huan, and Y. Ma, Computational methods for multiphase flows in porous media. SIAM, 2006.
[6] I. Dag, D. Irk, and B. Saka, “A numerical solution of the burgers’ equation using cubic b-splines,” ˘ Applied Mathematics and Computation, vol. 163, no. 1, pp. 199–211, 2005.
[7] A. Daga, K. Desai, and V. Pradhan, “Variational homotopy perturbation method for longitudinal dispersion arising in fluid flow through porous media,” International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS), 2013.
[8] B. Jacob, Dynamics of fluids in porous media. Courier Corporation, 2013.
[9] M. S. Joshi, N. B. Desai, and M. N. Mehta, “Solution of the burger’s equation for longitudinal dispersion phenomena occurring in miscible phase flow through porous media,” ITB Journal of Engineering Sciences, vol. 44, no. 1, pp. 61–76, 2012.
[10] M. A. Marino, “Flow against dispersion in nonadsorbing porous media,” Journal of Hydrology, vol. 37, no. 1-2, pp. 149–158, 1978.
[11] R. Meher and M. N. Mehta, “A new approach to backlund transformations of burger equation arising in longitudinal dispersion of miscible fluid flow through porous media,” International Journal of Applied Mathematics and Computation, vol. 2, no. 3, pp. 17–24, 2010.
[12] M. N. Mehta and T. Patel, “A solution of burger’s equation type one dimensional ground water recharge by spreading in porous media,” Journal of Indian Acad. Math, vol. 28, no. 1, pp. 25–32, 2006.
[13] M. Mehta and T. Patel, “A solution of burger’s equation arising in the longitudinal dispersion of concentration of fluid flow through porous media,” African Journal of Mathematics and Science research, vol. 28, no. 5, pp. 208–213, 2011.
[14] R. C. Mittal and G. Arora, “Efficient numerical solution of fisher’s equation by using b-spline method,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 3039–3051, 2010.
[15] R. C. Mittal and R. K. Jain, “Numerical solutions of nonlinear burgers’ equation with modified cubic b-splines collocation method,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7839–7855, 2012.
[16] R. C. Mittal and A. Tripathi, “Numerical solutions of generalized burgers–fisher and generalized burgers–huxley equations using collocation of cubic b-splines,” International Journal of Computer Mathematics, vol. 92, no. 5, pp. 1053–1077, 2015.
[17] R. C. Mittal and R. K. Jain, “Numerical solutions of nonlinear fisher’s reaction–diffusion equation with modified cubic b-spline collocation method,” Mathematical Sciences, vol. 7, no. 1, pp. 1–10, 2013.
[18] M. A. Patel and N. Desai, “An approximate analytical solution of the burger’s equation for longitudinal dispersion phenomenon arising in fluid flow through porous medium,” International Journal on Recent and Innovation Trends in Computing and Communication, vol. 5, no. 5, pp. 1103–1107, 2017.
[19] P. I. Polubarinova-Koch, “Theory of ground water movement,” in Theory of Ground Water Movement. Princeton university press, 2015.
[20] P. M. Prenter et al., Splines and variational methods. Courier Corporation, 2008.
[21] A. E. Scheidegger, “General theory of dispersion in porous media,” Journal of Geophysical Research, vol. 66, no. 10, pp. 3273–3278, 1961.
[22] A. E. Scheidegger and E. F. Johnson, “The statistical behavior of instabilities in displacement processes in porous media,” Canadian Journal of physics, vol. 39, no. 2, pp. 326–334, 1961.
[23] G. D. Smith, G. D. Smith, and G. D. S. Smith, Numerical solution of partial differential equations: finite difference methods. Oxford university press, 1985.
[24] A. P. Verma, “Fingero imbibition in artificial replenishment of ground water through cracked porous medium,” Water resources research, vol. 6, no. 3, pp. 906–911, 1970.
[25] D. U. Von Rosenberg, Methods for the numerical solution of partial differential equations. North-Holland, 1969.