Implicit equations play a crucial role in Engineering.
Based on this importance, several techniques have been applied to
solve this particular class of equations. When it comes to practical
applications, in general, iterative procedures are taken into account.
On the other hand, with the improvement of computers, other
numerical methods have been developed to provide a more
straightforward methodology of solution. Analytical exact approaches
seem to have been continuously neglected due to the difficulty
inherent in their application; notwithstanding, they are indispensable
to validate numerical routines. Lagrange-s Inversion Theorem is a
simple mathematical tool which has proved to be widely applicable to
engineering problems. In short, it provides the solution to implicit
equations by means of an infinite series. To show the validity of this
method, the tree-parameter infiltration equation is, for the first time,
analytically and exactly solved. After manipulating these series,
closed-form solutions are presented as H-functions.
[1] E. T. Whittaker , G. N. Watson, A Course of Modern Analysis. 4th
Edition, Cambridge Univ. Press, Cambridge, U.K.,1990; pg. 133.
[2] E. Artin, The Gamma Function. Rinehart and Winston, NY: Holt, New
York, 1964.
[3] P. K. Swamee, P. N. Rathie, L. C. de S. M. Ozelim, "Explict equations
for infiltration," Journal of Hydrology, 426-427, 2012, pp. 151-153.
[4] R. E. Smith, J. Y. Parlange, "A parameter-efficient hydrologic
infiltration model," Water Resour. Res., 14 (3), 1977, pp. 533-538.
[5] W. H. Green, G. A. Ampt, 1911. "Studies in soil physics I: the flow of
air and water through soils," J. Agric. Sci., 4 (1), 1911, pp. 1-24.
[6] T. Talsma, J. Y. Parlange, "One-dimensional vertical infiltration," Aust.
J. Soil Res., 10 (2), 1972, pp. 143-150.
[7] J. Y. Parlange, I. Lisle, R. D. Braddock, R. E. Smith, "The three
parameter infiltration equation," Soil Sci., 133 (6), 1982, pp. 337-341.
[8] J. Y. Parlange, D. A. Barry, R. Haverkamp, "Explicit infiltration
equations and the Lambert W-function," Adv. Water Resour., 25 (8-12),
2002, pp. 1119-1124.
[9] L. C. de S. M. Ozelim, A. L. B. Cavalcante, P. K. Swamee, P. N. Rathie,
"Métodos Numéricos Iterativos Aplicados a Equa├º├Áes de Infiltra├º├úo,"
In: VII Simpósio Brasileiro de Solos Não Saturados, 2011, Pirenópolis.
Anais do VII Simpósio Brasileiro de Solos Não Saturados. Goiânia,
Editora Kelps, v. 1, 2011, p. 209-213.
[10] A. M. Mathai, R. K. Saxena, H. J. Haubold, The H-Function: Theory
and Applications. Springer, New York, 2010, pp. 2-23.
[1] E. T. Whittaker , G. N. Watson, A Course of Modern Analysis. 4th
Edition, Cambridge Univ. Press, Cambridge, U.K.,1990; pg. 133.
[2] E. Artin, The Gamma Function. Rinehart and Winston, NY: Holt, New
York, 1964.
[3] P. K. Swamee, P. N. Rathie, L. C. de S. M. Ozelim, "Explict equations
for infiltration," Journal of Hydrology, 426-427, 2012, pp. 151-153.
[4] R. E. Smith, J. Y. Parlange, "A parameter-efficient hydrologic
infiltration model," Water Resour. Res., 14 (3), 1977, pp. 533-538.
[5] W. H. Green, G. A. Ampt, 1911. "Studies in soil physics I: the flow of
air and water through soils," J. Agric. Sci., 4 (1), 1911, pp. 1-24.
[6] T. Talsma, J. Y. Parlange, "One-dimensional vertical infiltration," Aust.
J. Soil Res., 10 (2), 1972, pp. 143-150.
[7] J. Y. Parlange, I. Lisle, R. D. Braddock, R. E. Smith, "The three
parameter infiltration equation," Soil Sci., 133 (6), 1982, pp. 337-341.
[8] J. Y. Parlange, D. A. Barry, R. Haverkamp, "Explicit infiltration
equations and the Lambert W-function," Adv. Water Resour., 25 (8-12),
2002, pp. 1119-1124.
[9] L. C. de S. M. Ozelim, A. L. B. Cavalcante, P. K. Swamee, P. N. Rathie,
"Métodos Numéricos Iterativos Aplicados a Equa├º├Áes de Infiltra├º├úo,"
In: VII Simpósio Brasileiro de Solos Não Saturados, 2011, Pirenópolis.
Anais do VII Simpósio Brasileiro de Solos Não Saturados. Goiânia,
Editora Kelps, v. 1, 2011, p. 209-213.
[10] A. M. Mathai, R. K. Saxena, H. J. Haubold, The H-Function: Theory
and Applications. Springer, New York, 2010, pp. 2-23.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58475", author = "Pushpa N. Rathie and Prabhata K. Swamee and André L. B. Cavalcante and Luan Carlos de S. M. Ozelim", title = "Lagrange-s Inversion Theorem and Infiltration", abstract = "Implicit equations play a crucial role in Engineering.
Based on this importance, several techniques have been applied to
solve this particular class of equations. When it comes to practical
applications, in general, iterative procedures are taken into account.
On the other hand, with the improvement of computers, other
numerical methods have been developed to provide a more
straightforward methodology of solution. Analytical exact approaches
seem to have been continuously neglected due to the difficulty
inherent in their application; notwithstanding, they are indispensable
to validate numerical routines. Lagrange-s Inversion Theorem is a
simple mathematical tool which has proved to be widely applicable to
engineering problems. In short, it provides the solution to implicit
equations by means of an infinite series. To show the validity of this
method, the tree-parameter infiltration equation is, for the first time,
analytically and exactly solved. After manipulating these series,
closed-form solutions are presented as H-functions.", keywords = "Green-Ampt Equation, Lagrange's Inversion
Theorem, Talsma-Parlange Equation, Three-Parameter Infiltration
Equation", volume = "6", number = "7", pages = "761-6", }