Remarks on Green function of space-fractional biharmonic heat equation using Ramanujan’s master theorem


  • Alireza Ansari


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.


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.