ON FRACTIONAL HELMHOLTZ EQUATIONS. Abstract

Transcription

1 ON FRACTIONAL HELMHOLTZ EQUATIONS M. S. Samuel and Anitha Thomas Abstract In this paper we derive an analtic solution for the fractional Helmholtz equation in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented b means of the Mittag-Leffler function. In all three cases the solutions are represented also in terms of Fox s H-function. MSC 21: 26A33, 33E12, 33C6, 35R11 Ke Words and Phrases: fractional Helmholtz equation, Caputo fractional derivative, Wel fractional derivative, Mittag-Leffler function, Fox s H-function 1. Introduction The Helmholtz equation 2 Ψ(x,, z) + k 2 Ψ(x,, z) = is named after Hermann Von Helmholtz ( ). It represents the time-independent form of the wave equation or diffusion equation obtained while appling the technique of separation of variables to reduce the complexities of the solution procedure of the original equations. The two-dimensional Helmholtz equation appears in phsical phenomena and engineering applications such as heat conduction, acoustic radiation, water wave propagation and even in biolog. It plas essential role for estimating the geodesic sea floor properties, the proper prediction of acoustic propagation in shallow water as well as at low frequencies, for solutions provided to such problems, we refer to c 21, FCAA Diogenes Co. (Bulgaria). All rights reserved.

2 296 M.S. Samuel, A. Thomas Liu et al. 9. The Helmholtz equation solves also problems in pattern formation of animal coating, see Murra and Merscough 12; in electromagnetics, where its two-dimensional form appears as the governing equation for waveguide problems, etc. Different numerical methods like the Ritz-Galerkin method (Itoh 7), the surface integral equation method and the finite element method have been emploed to solve this equation. In the Ritz-Galerkin method and in the integral equation method, the application of the method of moments to the solution of the integral equation leads to homogeneous sstems of linear equations. The matrix coefficients of these sstems of equations are given b infinite summation in the case of the Ritz-Galerkin method, while in the surface integral method, integrals containing Hankel functions have to be numericall computed, and this consumes large central processing unit time. In the finite element method (we refer e.g. to Pregla 16), the basis functions representing the field singularities lead to matrices in the generalized eigenvalue problem, whose coefficients have to be computed b means of numerical integration. If we ignore these field singularities when appling finite element method, inaccurate results ma be obtained. To reduce the memor requirements and inaccurac, it is advantageous to reformulate the boundar value problem for the Helmholtz equation as an initial value problem of fractional order α, where 1 < R(α) 2. In this paper, we introduce a model fractional Helmholtz equation in two-dimensions, in which both the space variables x and are allowed to take fractional order changes. This model is suitable for the stud of electromagnetic waves propagating in the upper halfspace of the cartesian plane and is defined as D α x N(x, ) + D α N(x, ) + k 2 N(x, ) = Φ(x, ); (1) k >, x R, R +, 1 < R(α) 2, where k is the wave number given b k = λ, λ is the wavelength, N(x, ) is the field variable of interest, which could be acoustic pressure, wave elevation or electromagnetic potential, among man other possibilities and Φ(x, ) is a non-linear function in the field. Here Dx α denotes the Wel fractional derivative of order α, and D α is the Caputo fractional derivative of order α, which are two alternative forms of the Riemann-Liouville fractional derivative a Dx α of order α: ad α t N(x, t) := R a D α t N(x, t) = that are defined as follows: 1 Γ(m α) dt m d m t a N(x, u) du; (2) (t u) α m+1

3 ON FRACTIONAL HELMHOLTZ EQUATIONS 297 Dt α N(x, t) := C Dt α 1 N(x, t) = Γ(m α) Dt α 1 N(x, t) = Γ(m α) dt m d m t t N (m) (x, u) du, (3) (t u) α m+1 N(x, u) du, (4) (t u) α m+1 where t >, m 1 < α m, m N, and dm dt m N(x, t) is the m th derivative of order m of the function N(x, t) with respect to t. The Mittag-Leffler function is a special function having an essential role in the solutions of fractional order integral and differential equations. This function is frequentl used recentl in modeling phenomena of fractional order appearing in phsics, biolog, engineering and applied other sciences. After being introduced and studied b Mittag-Leffler ( ), Wiman (195) and Agarwal (1953), the Mittag-Leffler function, in its two forms: E α (z) = E α,β (z) = n= n= z n, α C, R(α) >, (5) Γ(nα + 1) z n, α, β C, R(α) >, R(β) >, (6) Γ(nα + β) has been studied in details b Dzherbashan 1. Both functions (5)-(6) are entire functions of order ρ = 1 α and tpe σ = 1. Firstl in 192, Hille and Tamarkin 6 have presented a solution of the Abel-Volterra tpe integral equation φ(x) λ x φ(t) dt = f(x), < x < 1, Γ(α) (x t) 1 α in terms of Mittag-Leffler function. We are motivated b the recent works of Saxena-Mathai-Haubold (18, 19, 2) on fractional kinetic and fractional diffusion equations to obtain solutions in terms of the Mittag-Leffler function. For an extensive bibliograph on these kind of studies, see e.g. in Povstenko 14. The objective of this paper is to represent a solution of the fractional Helmholtz equation (1) in terms of the Mittag-Leffler function as well as in Fox s H-function, using the Laplace and Fourier transforms and their inverse transforms. Mathematicall, the Poisson and the Laplace equations are two special cases of the Helmholtz equation. We appl this fact also to the fractional case. In Section 3, we derive the solution of the fractional Laplace equation in Mittag-Leffler function and Fox s H-function. Section 4

4 298 M.S. Samuel, A. Thomas is devoted to the fractional Poisson equation. The solution of the fractional Helmholtz equation is given in Section 5. Its proof and the convergence and the series representation of the H-function are given in the Appendix. 2. Mathematical preliminaries The main results on Mittag-Leffler functions (5), (6) are available in the handbook of Erdéli et al. 4, the monographs b Dzherbashan (1, 2), some recent books as b Kirakova (1994), Podlubn 15, Kilbas, Srivastava and Trujillo (26), Mainardi (21), among them see e.g. the book of Mathai-Saxena-Haubold 1. The H-function is defined b means of a Mellin-Barnes tpe integral in the following manner (we refer to Mathai- Saxena-Haubold 1), Hp,q m,n (a p,a p) (z) z = 1 i L (b q,b q) { m } { j=1 Γ(b n j + B j s) j= Γ(1 b j B j s)} { q } { j=m+1 Γ(1 b p z j B j s) j=n+1 Γ(a j + A j s)} s ds, where: an empt product is alwas interpreted as unit; m, n, p, q N with n p, 1 m q, A j, B j R +, a i, b j R or C, i = 1,..., p; j = 1,..., q such that A i (b j + k) B j (a i l 1), k, l N ; i = 1,..., n; j = 1,..., m, and we emplo the usual notations: N = (, 1,..., ); R = (, ), R + = (, ) and C being the complex number field. For the details about the contour L and the existence conditions see, Mathai-Saxena-Haubold 1, Prudnikov-Brchkov-Marichev 17 and Kilbas-Saigo 8. The Wright s generalized hpergeometric function (Wright (21, 22) is defined b the series representation p pψ q (z) = p ψ q z (a p,a p ) j=1 Γ(a j + A j s) z s (b q,b q) = q, r= j=1 Γ(b j + B j s) s! where z C, a j, b j C, A j, B j R + ; i = 1,..., p; j = 1,..., q; q j=1 b j p j=1 A j > 1; C is the complex number field. The Mittag-Leffler function (7) is a special case of this function, E α,β (z) = 1 ψ 1 z (1,1) (β,α) = H 1,1 1,2 z (,1) (,1)(1 β,α). (8)

5 ON FRACTIONAL HELMHOLTZ EQUATIONS 299 The Laplace transform of the function N(x, t) with respect to t is L N(x, t) = e st N(x, t)dt = Ñ(x, s), x R, R(s) >, (9) and its inverse transform with respect to s is given b L 1 Ñ(x, s) = 1 γ+i e st Ñ(x, s)ds = N(x, t), (1) i γ i γ being a fixed real number. The Fourier transform of a function N(x, t) with respect to x is defined as F N(x, t) = e ipx N(x, t)dx = N (p, t), x R, (11) the inverse Fourier transform with respect to p: F 1 N (p, t) = 1 e ipx N (p, t) dp = N(x, t). (12) The space of functions for the above two transforms is LF = L(R + ) F (R), where L(R + ) is the space of summable functions on R + with norm f = f(t) dt < and F (R) is the space of summable functions on R with norm f = f(t) dt <. From Saxena-Mathai-Haubold 1 and Prudnikov-Brchkov-Marichev 17, the cosine transform of the H- function is given b t ρ 1 cos(kt)hp,q m,n dt (13) (b q,b q ) = π p µ (1 b q,b q )( 1+ρ 2 k ρ Hn+1,m, µ 2 ) q+1,p+2 a, (ρ,µ), (1 a p,a p ),( 1+ρ 2, µ 2 ) where R ρ + µ min ( b j ) >, R ρ + µ min 1 j m B (a j 1 ) <, arg a < j 1 j n A j 1 2 πθ; θ = n j=1 A j p j=n+1 A j + m j=1 B j q j=m+1 B j >. The Laplace transform of the Caputo fractional derivative is (see e.g. Podlubn 15) L D α t N(x, t) = s α N(x, s) at µ (a p,a p ) n r=1 s r 1 D α r t N(x, t) t=, n 1 < R(α) n. (14) The Fourier transform of the Wel fractional derivative, as given b Metzler- Klafter 11 is F D α t N(x, t) = (ik) α Ñ(p, t), R(α) >, (15)

6 3 M.S. Samuel, A. Thomas where Ñ(p, t) is the Fourier transform of N(x, t) with respect to the variable x of N(x, t). B adopting Saxena-Mathai-Haubold 2 and also from Gorenflo-Luchko-Umarov 5, F D α t N(x, t) = k α Ñ(p, t), R(α) >. (16) We also need the following propert of H-function, well known from the recent books on special functions (see e.g. in Mathai-Saxena-Haubold 1): (a p,a p ) Hp,q m,n z δ = 1 (a p, A p δ δ Hm,n p,q z ), (17) (b q,b q) (b q, B q δ ) and the following result (see e.g. Podlubn 15) ( ) s L 1 β 1 s α + a ; t = t α β E α,α β+1 ( at α ), R(s) >, R(α β) > 1, a < 1. s α (18) 3. Fractional Laplace equation Let us start with the fractional Laplace equation. integer order of the standard Laplace equation 2 B replacing the 2 N(x, ) + N(x, ) =, (19) x2 2 b a fractional order α, where 1 < R(α) 2, we get the fractional Laplace equation Dx α N(x, ) + D α N(x, ) =. In this section we derive a solution of the fractional Laplace equation in terms of the Mittag-Leffler function and derive also its Fox s H-function form. The case we consider is related to a Cauch problem Cauch problem. The solution to the fractional Laplace equation Dx α N(x, ) + D α N(x, ) = ;, x R, 1 < R(α) 2, (2) with the initial conditions D α 1 N(x, ) = f(x), D α 2 N(x, ) = g(x), x R, N(x, ) = is: lim x ± N(x, ) = α 2 + α 1 g(p)e α,α 1 ( p α α )e ipx dp f(p)e α,α ( p α α )e ipx dp, (21) where indicates the Fourier transform with respect to space variable x.

7 ON FRACTIONAL HELMHOLTZ EQUATIONS 31 Solution. If we appl the Laplace transform with respect to the space variable and use (14), then (2) becomes D α x N (x, s) + s α N (x, s) f(x) sg(x) =, (22) where depicts the Laplace transform with respect to the space variable. Now appling the Fourier transform with respect to the space variable x and using initial conditions and (16), the above equation takes the form Ñ (p, s) = Using (18), it is seen that f(p) s α + p α + s g(p) s α + p α. Ñ(p, ) = f(p) α 1 E α,α ( p α α ) + g(p) α 2 E α,α 1 ( p α α ). Taking the inverse Fourier transform on both sides, the solution is N(x, ) = α 1 + α 2 f(p)e α,α ( p α α )e ipx dp g(p)e α,α 1 ( p α α )e ipx dp. In the initial conditions of Cauch problem 3.1, put g(x) =, and with the help of the convolution theorem of the Fourier transforms, see the changes in the solution of the above problem. Corollar 3.1. The solution of the fractional Laplace equation x R, where = α 1 πα D α x N(x, ) + D α N(x, ) =,, (23) D α 1 N(x, = ) = f(x), D α 2 N(x, = ) =, lim x ± = α 1 α x H2,1 3,3 N(x, ) =, 1 < R(α) 2, is given b N(x, ) = G(x τ, )f(τ)dτ, (24) G(x, ) = α 1 e ipx E α,α ( p α α )dp cos(px)h 1,1 (, 1 α 1,2 p ) dp, using propert (17), (, 1 α ),(1 α,1) x (1, 1 α ),(α,1),(1, 1 2 ), using eq. (13), where 1 < R(α) 2. (1,1),(1, 1 α ),(1, 1 2 )

8 32 M.S. Samuel, A. Thomas Again, b changing the initial conditions of the Cauch problem 1, as f(x) = δ(x), where δ(x) is the Dirac-delta { function defined b 1, x =, δ(x) =, elsewhere, and g(x) =, we can express the above solution in terms of a Fox s H- function. We shall consider this in the following result. Corollar 3.2. The solution to the fractional Laplace equation D α x N(x, ) + D α N(x, ) = ;, x R, 1 < R(α) 2, (25) with the initial conditions D α 1 N(x, ) = δ(x), D α 2 N(x, ) =, x R, lim x ± N(x, ) =, is: N(x, ) = α 1 = α 1 α x H2,1 3,3 e ipx E α,α ( p α α )dp, x (1, 1 α ),(α,1)(1, 1 2 ) (1,1),(1, 1 α ),(1, 1 2 ), 1 < R(α) 2. (26) 4. Fractional Poisson equation To generalize Cauch problem 3.1, we consider the Mittag-Leffler solution of the fractional Poisson equation D α x N(x, ) + D α N(x, ) = Φ(x, ). (27) Namel, we consider the following 4.1. Cauch problem. The solution to the fractional Poisson equation D α x N(x, ) + D α N(x, ) = Φ(x, );, x R, 1 < R(α) 2, (28) where Φ(x, ) is a non-linear function, with the initial conditions D α 1 N(x, ) = f(x), D α 2 N(x, ) = g(x), x R, lim x ± N(x, ) =, N(x, ) = α 1 + α f(p)e α,α ( p α α )e ipx dp, g(p)e α,α 1 ( p α α )e ipx dp, (29) ξ α 1 Φ(k, ξ)e α,α ( p α ξ α )e ipx dpdξ.

9 ON FRACTIONAL HELMHOLTZ EQUATIONS 33 Solution. Appling the Laplace transform with respect to the space variable and Fourier transform with respect to the space variable x and using the initial conditions and (14), we have Using the result (16), s α + p α Ñ (p, s) = f(p) s g(p) + φ (p, s). Ñ(p, ) = f(p) α 1 E α,α ( p α α ) + g(p) α 2 E α,α 1 ( p α α ) + Taking the inverse Fourier transform, the solution is N(x, ) = α 1 + α f(p)e α,α p α α e ipx dp φ(p, ξ)ξ α 1 E α,α ( p α )ξ α dξ. g(p)e α,α 1 p α α e ipx dp ξ α 1 Now we shall tr to get an H-function solution. φ(k, ξ)e α,α ( p α )ξ α e ipx dpdξ. Corollar 4.1. The solution of the fractional Poisson equation D x α N(x, ) + D α N(x, ) = Φ(x, ),, (3) D α 1 N(x, =) = f(x), D α 2 N(x, =) = for x R, lim x ± N(x, ) =, 1 < R(α) 2 and Φ(x, ) a non-linear function of x and, is given b N(x, ) = where Similarl, G 1 (x τ, )f(τ)dτ+ G 1 (x, ) = tα 1 α x H2,1 3,3 G 2 (x, ) = 1 α x H2,1 3,3 x x x ( ξ) α 1 G 2 (x τ, ξ)φ(τ, ξ)dτdξ, (1, 1 α ),(α,1),(1, 1 2 ) (1,1),(1, 1 α ),(1, 1 2 ) (1, 1 α ),(α,1),(1, 1 2 ) (1,1),(1, 1 α ),(1, 1 2 ), 1 < R(α) 2,, 1 < R(α) 2. (31)

10 34 M.S. Samuel, A. Thomas 5. Fractional Helmhotz equation In this section, we solve the fractional Helmholtz equation (1) to generate a solution in terms of the Mittag-Leffler function, using the Laplace and Fourier transforms and their inverses. Theorem 5.1. Consider the fractional Helmholtz equation Dx α N(x, ) + D α N(x, ) + k 2 N(x, ) = Φ(x, ); k >, x R,, (32) 1 < R(α) 2, with initial conditions D α 1 N(x, ) = f(x), D α 2 N(x, ) = g(x), x R, lim N(x, ) means the Riemann-Liouville frac- N(x, ) =, where D α 1 x ± tional derivative of order α 1 with respect to and D α 2 N(x, ) means the Riemann-Liouville fractional derivative of order α 2 with respect to, when =. The quantit k is a constant and Φ(x, ) is a nonlinear function. Then the solution of (32) subject to the initial conditions, is: N(x, ) = α 1 + α f(p)e α,α ( p α + k 2 ) α e ipx dp (33) g(p)e α,α 1 ( p α + k 2 ) α e ipx dp ξ α 1 Φ(k, ξ)e α,α ( p α + k 2 )ξ α e ipx dpdξ, where indicates the Fourier transform with respect to the space variable x. The proof of this theorem and the H-function solution of the fractional Helmholtz equation are discussed in the Appendix. 6. Conclusion The closed form solutions in terms of the Mittag-Leffler function and Fox s H-function are obtained for the fractional Laplace and the fractional Poisson equations. It is seen that, the solutions in terms of the Mittag- Leffler function as well as in the H-function to the fractional Helmholtz equation are the master solutions to the solutions of the fractional Laplace equation and the fractional Poisson equation.

11 ON FRACTIONAL HELMHOLTZ EQUATIONS 35 Acknowledgements The authors acknowledge gratefull the encouragement of Professor A.M. Mathai, Department of Mathematics and Statistics, McGill Universit, Montreal, Canada and Dr. Vincent Mathew, Department of Phsics, St. Thomas College, Pala, Kerala, India in this research work. The authors would like to thank the Centre for Mathematical Sciences, Pala Campus for providing facilities for carring out this work. Appendix A.1. The Mittag-Leffler solution of the fractional Helmholtz equation If we appl the Laplace transform with respect to the space variable and use (14), the equation (32) becomes D α x N (x, s) + s α N (x, s) f(x) sg(x) + k 2 N (x, s) = Φ (x, s). (34) Now appling the Fourier transform with respect to the space variable x and use initial conditions and (16) to obtain Ñ (p, s) = Using the result (18), it is seen that f(p) s α + p α + k 2 + s g(p) s α + p α + k 2 + Φ (p, s) s α + p α + k 2. Ñ(p, ) = f(p) α 1 E α,α ( p α + k 2 ) α + g(p) α 2 E α,α 1 ( p α + k 2 ) α + φ(p, ξ)ξ α 1 E α,α ( p α + k 2 )ξ α dξ. Taking the inverse Fourier transform on both sides, the solution is N(x, ) = α 1 + α f(p)e α,α ( p α + k 2 ) α e ipx dp g(p)e α,α 1 ( p α + k 2 ) α e ipx dp ξ α 1 φ(k, ξ)e α,α ( p α + k 2 )ξ α e ipx dpdξ. Hence Theorem 5.1 follows. Using the equation (8), the solution (33) of the fractional Helmholtz equation can be given in terms of the Fox s H-function.

12 36 M.S. Samuel, A. Thomas We shall consider the case as follows. A.2. The Fox s H-function solution of the fractional Helmholtz equation The solution of the fractional Helmholtz equation D α x N(x, ) + D α N(x, ) + k 2 N(x, ) = Φ(x, ), (35) >, k >, x R, 1 < R(α) 2 with the initial conditions D α 1 N(x, ) = f(x), Dx α 2 N(x, ) =, for x R, lim N(x.) =, and Φ(x, ) is a x ± non-linear function, given b N(x, )= where G 1 (x τ, )f(τ)dτ + G 1 (x, ) = α 1 e ipx H 1,1 1,2 G 2 (x, ) = 1 e ipx H 1,1 1,2 x ( ξ) α 1 G 2 (x τ, ξ)φ(τ, ξ)dτdξ, ( p α + k 2 ) α,1 ( p α + k 2 ) α (,1) (,1),(1 α,α) (,1),(1 α,α) (36) dp, (37) dp. (38) Convergence and the series representation of the solution (26) B Mathai-Saxena-Haubold 1, the H 2,1 3,3 x (1, 1 α ),(α,1),(1, 1 2 ) (1,1),(1, 1 α ),(1, 1 2 ) converges for < x < 1. Again, b Mathai-Saxena-Haubold 1, the series expansion of the H-function is given as follows: We have: H 2,1 x (1, 1 α ),(α,1),(1, 1 2 ) 3,3 = 1 Γ(1 + s)γ(1 + s s α )Γ( α ) (1,1),(1, 1 α ),(1, 1 2 ) i L Γ( s 2 )Γ(α + s)γ(1 + s 2 ) x s ds. (39) Let us assume that the poles of the integrand are simple. The region of convergence is L = L iγ is a contour starting at the point γ i and terminating at the point γ + i, where γ R = (, + ) such that all the poles of Γ(b j + B j s), j = 1,, m are separated from those of Γ(1 a j A j s), j = 1,, n. B calculating the residues at the poles of Γ(1+s) and Γ(1+ s α ) where the poles are given b 1+s = ν, ν =, 1, 2, and 1 + s α = ν, ν =, 1, 2, we will get the series representation of (26).

13 ON FRACTIONAL HELMHOLTZ EQUATIONS 37 N(x, ) = α 1 α + x α 1 where R(1 1+ν α ( 1) ν Γ(1 1+ν 1+ν α )Γ( α ) x ν (ν)!γ( 1+ν 1+ν 2 )Γ(α 1 ν)γ(1 2 ) ( 1) ν Γ(1 α(1 + ν))γ((1 + ν)) (ν)!γ((1 + ν) α ν= 2 )Γ(α α(1 + ν))γ(1 (1 + ν) α 2 ) ) >, < x < 1. ν= x αν, References 1 M. M.Dzherbashan, Integral Transforms and Representation of Functions in Complex Domain (in Russian). Nauka, Moscow (1966). 2 M. M. Dzherbashan, Harmonic Analsis and Boundar Value Problems in the Complex Domain. Birkhauser Verlag, Basel - London (1993). 3 A. Erdéli, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. I. McGraw-Hill, New York (1953). 4 A. Erdéli, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. III. McGraw-Hill, New York (1955). 5 R. Gorenflo, Yu.F. Luchko, S.R. Umarov, Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, No 1 (2), E. Hille and J.D. Tamarkin, On the theor of linear integral equations. Annals of Mathematics 31, (193), T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. John Wile & Sons, New York (1989). 8 A.A. Kilbas and M. Saigo, H Transforms: Theor and applications. Analtic Methods and Special Functions. Chapman & Hall, CRC Press, Boca Raton (24). 9 Jin-Yuan Liu, Sheng-Hsuing Tsai, Chau-Chang Wang and Chung-Ra Chu, Acoustic wave reflection from a rough seabed with a continuousl varing sediment laer overling an elastic basement. Sound and Vibration 275, (24), A.M. Mathai, R.K. Saxena and H.J. Haubold, The H - Function: Theor and Applications. Springer, New York (21). 11 R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: A fractional dnamics approach. Phs. Rep. 339 (2), J.D. Murra and M.R. Merscough, Pigmentation pattern formation on snakes. Journal of Theoretical Biolog 149 (1991),

14 38 M.S. Samuel, A. Thomas 13 K.B. Oldham and J. Spanier, The Fractional Calculus: Theor and Applications of Differentiation and Integration to Arbitrar Order. Academic Press, New York, and Dover Publications, New York (1974). 14 Y.Z. Povstenko, Two-dimensional axismmetric stress exerted b instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Solids and Structures 44 (27), I. Podlubn, Fractional Differential Equations. Academic Press, San Diego - Boston - N. York etc. (1999). 16 R. Pregla, Analsis of Electromagnetic Fields and Waves. Research Studies Press Limited, John Wile & Sons, Chichester (28). 17 A.P. Prudnikov, Yu.A. Brchkov, O.I. Marichev, Integrals and Series, More Special Functions, Vol. 3. Gordon and Breach, New York (1989). 18 R.K. Saxena, A.M. Mathai and H.J. Haubold, On fractional kinetic equations. Astrophsics and Space Science 282 (22), R.K. Saxena, A.M. Mathai and H.J. Haubold, Unified fractional kinetic equation and a fractional diffusion equation. Astrophsics & Space Science 29 (24), R.K. Saxena, A.M. Mathai and H.J. Haubold, Solution of generalized fractional diffusion equations. Astrophsics & Space Science 35 (27), E.M. Wright, The asmptotic expansion of the generalized hpergeometric functions. J. London Math. Soc. 1 (1935), E.M. Wright, The asmptotic expansion of the generalized hpergeometric functions. Proc. London Math. Soc. 46 (2) (194), Department of Computer Applications Mar Athanasios College for Advanced Studies Thiruvalla - 1, Pathanamthitta, Kerala, INDIA mssamuel@macfast.org Department of Mathematics Bishop Abraham Memorial College Thuruthicadu P.O., Pathanamthitta Kerala , INDIA Received: April 14, 21 anithadaniel.bam@gmail.com