Geodesic
Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator
Agarwal, Ritu1 ; Jain, Sonal1 ; Agarwal, R. P.2
1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India
2 Department of Mathematics, Texas A & M University, Kingsville 700 University Blvd. Kingsville, TX 78363-8202
Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 6, p. 3545-3554.

Voir la notice de l'article provenant de la source International Scientific Research Publications

The aim of this paper is to investigate the solutions of Time-space fractional advection-dispersion equation with Hilfer composite fractional derivative and the space fractional Laplacian operator. The solution of the equation is obtained by applying the Laplace and Fourier transforms, in terms of Mittag-leffler function. The work by R. K. Saxena (2010) and Haung and Liu (2005) follows as particular case of our results.
DOI : 10.22436/jnsa.009.06.09
Classification : 26A33, 49K20, 44A10
Keywords: Time-space fractional advection-dispersion equation, Fourier transform, Laplace transform, composite fractional derivative, H-function, Mittag-Leffler function, fractional Laplace operator.

Agarwal, Ritu 1 ; Jain, Sonal 1 ; Agarwal, R. P. 2

1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India
2 Department of Mathematics, Texas A & M University, Kingsville 700 University Blvd. Kingsville, TX 78363-8202 
@article{JNSA_2016_9_6_a8,
     author = {Agarwal, Ritu and Jain, Sonal and Agarwal, R. P.},
     title = {Analytic solution of generalized space time advection-dispersion equation with fractional {Laplace} operator},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {3545-3554},
     publisher = {mathdoc},
     volume = {9},
     number = {6},
     year = {2016},
     doi = {10.22436/jnsa.009.06.09},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.09/}
}
TY  - JOUR
AU  - Agarwal, Ritu
AU  - Jain, Sonal
AU  - Agarwal, R. P.
TI  - Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator
JO  - Journal of nonlinear sciences and its applications
PY  - 2016
SP  - 3545
EP  - 3554
VL  - 9
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.09/
DO  - 10.22436/jnsa.009.06.09
LA  - en
ID  - JNSA_2016_9_6_a8
ER  - 
%0 Journal Article
%A Agarwal, Ritu
%A Jain, Sonal
%A Agarwal, R. P.
%T Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator
%J Journal of nonlinear sciences and its applications
%D 2016
%P 3545-3554
%V 9
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.09/
%R 10.22436/jnsa.009.06.09
%G en
%F JNSA_2016_9_6_a8
Agarwal, Ritu; Jain, Sonal; Agarwal, R. P. Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator. Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 6, p. 3545-3554. doi : 10.22436/jnsa.009.06.09. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.09/

[1] Basu, M.; Acharya, D. P. On quadratic fractional generalized solid bi-criterion transportation problem, J. Appl. Math. Comput., Volume 10 (2002), pp. 131-144

[2] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M. Application of a fractional advection-dispersion equation, Water Resources Research, Volume 36 (2000), pp. 1403-1412

[3] Brockmann, D.; Sokolov, I. M. Levy flights in external force fields: from model to equations, Chem. Phys., Volume 284 (2002), pp. 409-421

[4] Caputo, M. Elasticita e dissipazione, Zani-Chelli, Bologana, 1969

[5] Debnath, L.; Bhatta, D. Integral Transforms and their Applications, CRC Press, Boca Raton, 1995

[6] El-Sayed, M. A. A.; Aly, M. A. E. Continuation theorem of fractional order evolutionary integral equations, Korean J. Comput. Appl. Math., Volume 9 (2002), pp. 525-533

[7] Haubold, H. J.; Mathai, A. M.; Saxena, R. K. Solution of reaction-diffusion equations in terms of the H-function, Bull. Astro. Soc. India., Volume 35 (2007), pp. 681-689

[8] Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000

[9] Huang, F.; F. Liu The Fundamental Solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput., Volume 18 (2005), pp. 339-350

[10] Javandel, L.; Doughly, C.; Tsang, F. C. Groundwater Transport: Handbook of Mathematical Models, American Geophysical Union, Michigan, 1984

[11] Liu, F.; Anh, V.; Turner, I. Numerical solution of the space fractional Fokker-Plank Equation, J. Comp. Appl. Math., Volume 166 (2004), pp. 209-319

[12] Liu, F.; Turner, I.; Anh, V. An unstructured mesh finite volume method for modelling saltwater intrusion into coatal aquifer, Korean J. Comput. Appl. Math., Volume 9 (2002), pp. 391-407

[13] Liu, F.; Turner, I. W.; Anh, V.; Su, N. A two-dimensional finite volume method for transient simulation of time scale and density-dependent transport in heterogeneous aquifer systems, Korean J. Comput. Appl. Math., Volume 11 (2003), pp. 215-241

[14] Mainardi, F. Fractals and Fractional Calculus in Continuum Mechanics Fraction(A. Carpinteri, F. Mainardi, Eds.) , chapter: fractional calculus, 291-348, Springer, Wien, 1997

[15] Mathai, A. M.; Saxsena, R. K.; Haubold, H. J. The H-function: Theory and Applications, Springer, New York, 2010

[16] Meerschaert, M. M.; Benson, D. A.; Baumer, B. Multidimensional advection and fractional dispersion, Phys. Rev. E., Volume 59 (1999), pp. 5026-5028

[17] Pandey, R. K.; Singh, O. P.; Baranwal, V. K.; M. P. Tripathi An analytic solution for the space-time fractional advection-dispersion equation using the optimal homotopy asymptotic method, Comput. Phys. Commun., Volume 183 (2012), pp. 2098-2106

[18] Samko, S. G.; Kilbas, A. A.; O. I. Marichev Fractional Integrals and derivatives-Theory and applications, CRC Press, Linghorne, 1993

[19] Saxena, R. K.; Saxena, R.; S. L. Kalla Solution of the space-time fractional Schrödinger equation equation occuring in quantum mechanics, Fract. Calc. Appl. Anal., Volume 13 (2010), pp. 177-190

[20] Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Wheatcraft, S. W. Eulerian derivation of the factional adverction- dispersion equation, J. Contam. Hydrol., Volume 48 (2001), pp. 69-88

[21] Sneddon, I. N. Fourier Transform, MacGraw-Hill, New York, 1951

[22] Wiman, A. ber den Fundamentalsatz in der Teorie der Funktionen Ea(x), (German), Acta Math., Volume 29 (1905), pp. 191-201

Cité par Sources :