Remarks on Green function of space-fractional biharmonic heat equation using Ramanujan’s master theorem
Abstract
In this paper, we show an addition formula for a class of Mellin transformable functions using Ramanujan’s master theorem. Also, we get the associated addition formula of Gaussian functions in terms of the Bessel functions and Green function of the space-fractional biharmonic heat equation is obtained with respect to the quartic Levy stable functions.
References
Amdeberhan, T., Espinosa, O., Gonzalezm, I., Harrison,
M., Moll, V. H. & Straub, A. (2012). Ramanujan’s master
theorem, The Ramanujan Journal, 29:103 -120.
Abramowitz, M. & Stegun, I. A. (1972). Handbook
of mathematical functions with formulas, graphs, and
mathematical tables, Dover, New York.
Apelblat, A. (2008). Volterra Functions, Nova Science
Publishers, New York.
Ansari, A. (2015). Some inverse fractional Legendre
transforms of gamma function form, Kodai Mathematical
Journal, 38:658 -671.
Ansari, A., Ahamadi Darani, M. & Moradi, M. (2012a). On
fractional Mittag-Leffler operators, Reports on Mathematical
Physics, 70(1):119- 131.
Ansari, A., Refahi Sheikhani, A. & Saberi Najafi, H.
(2012b). Solution to system of partial fractional differential
equation using the fractional exponential operators,
Mathematical Methods in the Applied Sciences, 35:119 -123.
Ansari, A. (2012). Fractional exponential operators and timefractional
telegraph equation, Boundary Value Problems, 125.
Ansari, A., Refahi Sheikhani, A. & Kordrostami, S. (2013).
On the generating function and its fractional calculus,
Central European Journal of Physics, 11(10):1457- 1462.
Bertram, W. (1997). Ramanujan’s master theorem and
duality of symmetric spaces, Journal of Functional
Analysis, 148(1):117 -151.
Borosa, G. & Moll, V. H. (2001). The double square root,
Jacobi polynomials and Ramanujan’s master theorem,
Journal of Computational and Applied Mathematics,
(12):337 -344.
Ding, H. (1997). Ramanujan’s master theorem for
Hermitian symmetric spaces, The Ramanujan Journal,
(1):35- 52.
Ferrero, A., Gazzola, F. & Grunau, H. C. (2008). Decay
and eventual local positivity for biharmonic parabolic
equations, Discrete and Continuous Dynamical Systems,
(4):1129- 1157.
Furati, K. M., Iyiola, O. S. & Kirane, M. (2014). An
inverse problem for a generalized fractional diffusion,
Applied Mathematics and Computation, 249:24 -31.
Garg, M. & Manohar, P. (2013). Analytical solution of
the reaction-diffusion equation with space-time fractional
derivatives by means of the generalized differential
transform method, Kuwait Journal of Science, 40(1):23- 34.
Gazzola, F. (2013). On the moments of solutions to
linear parabolic equations involving the biharmonic
operator, Discrete and Continuous Dynamical Systems,
(8):3583 -3597.
Gazzola, F. & Grunau, H. C. (2008). Eventual local
positivity for a biharmonic heat equation in Rn, Discrete
and Continuous Dynamical Systems Series S, 1(1):83 -87.
Ghany, H. A. & Hyder, A. (2014). Exact solutions for
the wick-type stochastic time-fractional KdV equations,
Kuwait Journal of Science, 41(1):75 -84.
Gonzalez, I., Moll, V. H. & Schmidt, I. (2015). Ramanujan’s
master theorem applied to the evaluation of Feynman
diagrams, Advances in Applied Mathematics, 63:214- 230.
Gorska, K., Babusci, D., Dattoli, G., Duchampe, G.H.E.
& Penson, K.A. (2012).
The Ramanujan master theorem and its implications for
special functions, Applied Mathematics and Computation,
(23):11466- 11471.
Gorska, K., Horzela, A., Penson, K. A. & Dattoli, G.
(2013). The higher-order heat-type equations via signed
Levy stable and generalized Airy functions, Journal of
Physics A: Mathematical and Theoretical, 46:16 pages.
Gradshteyn, I.S. & Ryzhik, I. M. (2007). Table of Integrals,
Series and Products. Seventh Edition, Academic Press,
New York.
Hardy, G.H. (1940). Ramanujan: Twelve lectures on
subjects suggested by his life and work, Cambridge
University Press, Cambridge.
Kilbas, A. A., Srivastava, H. M. & Trujillo, J. J. (2006).
Theory and applications of fractional differential
equations, Elsevier, Amsterdam.
Mainardi, F. (2010). Fractional Calculus and waves in
Linear Viscoelasticity, Imperial College Press, London.
Olafssona, G. & Pasquale, A. (2012). Ramanujan’s master
theorem for Riemannian symmetric spaces, Journal of
Functional Analysis, 262(11):4851- 4890.
Olafssona, G. & Pasquale, A. (2013). Ramanujans master
theorem for the hyperge-ometric Fourier transform
associated with root systems, Journal of Fourier Analysis
and Applications, 19(6):1150 -1183.
Proskurin, N.V. (1997). Integral addition formulae for
Airy function, Integral Transforms and Special Functions,
(1):143 -146.
Vallee, O. & Soares, M. (2004). Airy functions and
applications to Physics, Imperial College Press, London.
