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ON FRACTIONAL HELMHOLTZ EQUATIONS M. S. Samuel

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ON FRACTIONAL HELMHOLTZ EQUATIONS M. S. Samuel View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bulgarian Digital Mathematics Library at IMI-BAS ON FRACTIONAL HELMHOLTZ EQUATIONS M. S. Samuel ¤ and Anitha Thomas ¤¤ Abstract In this paper we derive an analytic solution for the fractional Helmholtz equation in terms of the Mittag-Le²er function. The solutions to the frac- tional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Le²er function. In all three cases the solutions are represented also in terms of Fox's H-function. MSC 2010: 26A33, 33E12, 33C60, 35R11 Key Words and Phrases: fractional Helmholtz equation, Caputo frac- tional derivative, Weyl fractional derivative, Mittag-Le²er function, Fox's H-function 1. Introduction The Helmholtz equation r2ª(x; y; z) + k2ª(x; y; z) = 0 is named after Hermann Von Helmholtz (1821-1894). It represents the time-independent form of the wave equation or di®usion equation obtained while applying the technique of separation of variables to reduce the complexities of the solu- tion procedure of the original equations. The two-dimensional Helmholtz equation appears in physical phenomena and engineering applications such as heat conduction, acoustic radiation, water wave propagation and even in biology. It plays essential role for estimating the geodesic sea floor proper- ties, 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 2010, FCAA { Diogenes Co. (Bulgaria). All rights reserved. 296 M.S. Samuel, A. Thomas Liu et al. [9]. The Helmholtz equation solves also problems in pattern for- mation of animal coating, see Murray and Myerscough [12]; in electromag- netics, where its two-dimensional form appears as the governing equation for waveguide problems, etc. Di®erent numerical methods like the Ritz-Galerkin method (Itoh [7]), the surface integral equation method and the ¯nite element method have been employed 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 systems of linear equations. The matrix coe±cients of these systems of equations are given by in¯nite summation in the case of the Ritz-Galerkin method, while in the surface integral method, integrals containing Hankel functions have to be numerically computed, and this consumes large central processing unit time. In the ¯nite element method (we refer e.g. to Pregla [16]), the basis functions representing the ¯eld singularities lead to matrices in the generalized eigenvalue problem, whose coe±cients have to be computed by means of numerical integration. If we ignore these ¯eld singularities when applying ¯nite element method, inaccurate results may be obtained. To reduce the memory requirements and inaccuracy, it is advantageous to reformulate the boundary value problem for the Helmholtz equation as an initial value problem of fractional order ®, where 1 < <(®) · 2. In this paper, we introduce a model fractional Helmholtz equation in two-dimensions, in which both the space variables x and y are allowed to take fractional order changes. This model is suitable for the study of elec- tromagnetic waves propagating in the upper halfspace of the cartesian plane and is de¯ned as ® ® 2 ¡1Dx N(x; y) + 0Dy N(x; y) + k N(x; y) = ©(x; y); (1) k > 0; x 2 <; y 2 <+; 1 < <(®) · 2; where k is the wave number given 2¼ by k = ¸ , ¸ is the wavelength, N(x; y) is the ¯eld variable of interest, which could be acoustic pressure, wave elevation or electromagnetic poten- tial, among many other possibilities and ©(x; y) is a non-linear function in ® the ¯eld. Here ¡1Dx denotes the Weyl fractional derivative of order ®, and ® 0Dy is the Caputo fractional derivative of order ®, which are two alternative ® forms of the Riemann-Liouville fractional derivative aDx of order ®: m Z t ® R ® 1 d N(x; u) aDt N(x; t) := a Dt N(x; t) = m ®¡m+1 du; (2) ¡(m ¡ ®) dt a (t ¡ u) that are de¯ned as follows: ON FRACTIONAL HELMHOLTZ EQUATIONS 297 Z t (m) ® C ® 1 N (x; u) 0Dt N(x; t) := 0 Dt N(x; t) = ®¡m+1 du; (3) ¡(m ¡ ®) 0 (t ¡ u) m Z t ® 1 d N(x; u) ¡1Dt N(x; t) = m ®¡m+1 du; (4) ¡(m ¡ ®) dt ¡1 (t ¡ u) dm th where t > 0; m ¡ 1 < ® · m; m 2 N, and dtm N(x; t) is the m derivative of order m of the function N(x; t) with respect to t. The Mittag-Le²er function is a special function having an essential role in the solutions of fractional order integral and di®erential equations. This function is frequently used recently in modeling phenomena of fractional order appearing in physics, biology, engineering and applied other sciences. After being introduced and studied by Mittag-Le²er (1903-1905), Wiman (1905) and Agarwal (1953), the Mittag-Le²er function, in its two forms: X1 zn E (z) = ; ® 2 C; <(®) > 0; (5) ® ¡(n® + 1) n=0 X1 zn E (z) = ; ®; ¯ 2 C; <(®) > 0; <(¯) > 0; (6) ®;¯ ¡(n® + ¯) n=0 has been studied in details by Dzherbashyan [1]. Both functions (5)-(6) are 1 entire functions of order ½ = ® and type σ = 1: Firstly in 1920, Hille and Tamarkin [6] have presented a solution of the Abel-Volterra type integral equation Z ¸ x Á(t) Á(x) ¡ 1¡® dt = f(x); 0 < x < 1; ¡(®) 0 (x ¡ t) in terms of Mittag-Le²er function. We are motivated by the recent works of Saxena-Mathai-Haubold ([18], [19], [20]) on fractional kinetic and frac- tional di®usion equations to obtain solutions in terms of the Mittag-Le²er function. For an extensive bibliography 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-Le²er function as well as in Fox's H-function, using the Laplace and Fourier transforms and their inverse transforms. Mathematically, the Poisson and the Laplace equations are two special cases of the Helmholtz equation. We apply this fact also to the fractional case. In Section 3, we derive the solution of the fractional Laplace equation in Mittag-Le²er function and Fox's H-function. Section 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-Le²er functions (5), (6) are available in the handbook of Erd¶elyiet al. [4], the monographs by Dzherbashyan ([1], [2]), some recent books as by Kiryakova (1994), Podlubny [15], Kilbas, Srivastava and Trujillo (2006), Mainardi (2010), among them see e.g. the book of Mathai-Saxena-Haubold [10]. The H-function is de¯ned by means of a Mellin-Barnes type integral in the following manner (we refer to Mathai- Saxena-Haubold [10]), " ¯ # ¯(ap;Ap) m;n ¯ Hp;q (z) z ¯ (bq;Bq) n o n o (7) Z Qm Qn ¡(bj + Bjs) ¡(1 ¡ bj ¡ Bjs) 1 j=1 j=0 ¡s = nQ o nQ o z ds; 2¼i L q p j=m+1 ¡(1 ¡ bj ¡ Bjs) j=n+1 ¡(aj + Ajs) where: an empty product is always interpreted as unity; m; n; p; q 2 N0 with 0 · n · p; 1 · m · q; Aj;Bj 2 <+; ai; bj 2 < or C; i = 1; : : : ; p; j = 1; : : : ; q such that Ai(bj + k) 6= Bj(ai ¡ l ¡ 1); k; l 2 N0; i = 1; : : : ; n; j = 1; : : : ; m; and we employ the usual notations: N0 = (0; 1;:::; ); < = (¡1; 1); <+ = (0; 1) and C being the complex number ¯eld. For the details about the contour L and the existence conditions see, Mathai-Saxena-Haubold [10], Prudnikov-Brychkov-Marichev [17] and Kilbas-Saigo [8]. The Wright's generalized hypergeometric function (Wright ([21], [22]) is de¯ned by the series representation h i Qp s h ¯ i X1 ¡(aj + Ajs) z (ap;Ap) j=1 pÃq(z) =p Ãq z¯ = h i ; (bq;Bq) Qq r=0 j=1 ¡(bj + Bjs) s! Pq where z 2 C; aj; bj 2 C;Aj;Bj 2 <+; i = 1; :::; p; j = 1; :::; q; j=1 bj ¡ Pp j=1 Aj > ¡1; C is the complex number ¯eld. The Mittag-Le²er function is a special case of this function, " ¯ # " ¯ # ¯(1;1) ¯(0;1) ¯ 1;1 ¯ E®;¯(z) = 1Ã1 z¯ = H1;2 ¡z¯ : (8) (¯;®) (0;1)(1¡¯;®) ON FRACTIONAL HELMHOLTZ EQUATIONS 299 The Laplace transform of the function N(x; t) with respect to t is Z 1 L [N(x; t)] = e¡stN(x; t)dt = N~(x; s); x 2 <; <(s) > 0; (9) 0 and its inverse transform with respectZ to s is given by h i 1 γ+i1 L¡1 N~(x; s) = estN~(x; s)ds = N(x; t); (10) 2¼i γ¡i1 γ being a ¯xed real number. The Fourier transform of a function N(x; t) with respect to x is de¯ned as Z 1 F [N(x; t)] = eipxN(x; t)dx = N ¤(p; t); x 2 <; (11) ¡1 the inverse Fourier transform with respect to p: Z 1 1 F ¡1 [N ¤(p; t)] = e¡ipxN ¤(p; t) dp = N(x; t): (12) 2¼ ¡1 The space of functions for the above two transforms is LF = L(<+) £ F (<), where L(<+) is the space of summable functions on <+ with norm R 1 jjfjj = 0 jf(t)jdt < 1 and F (<) is the space of summable functions R 1 on < with norm jjfjj = ¡1 jf(t)jdt < 1: From Saxena-Mathai-Haubold [10] and Prudnikov-Brychkov-Marichev [17], the cosine transform of the H- function is given by Z " ¯ # 1 ¯(ap;Ap) ½¡1 m;n ¹¯ t cos(kt)Hp;q at ¯ dt (13) 0 (b ;B ) q q " # ¯(1¡b ;B )( 1+½ ; ¹ ) ¹ ¯ q q 2 2 ¼ n+1;m p ¯ = ½ Hq+1;p+2 ¯ ; k a (½;¹); (1¡a ;A );( 1+½ ; ¹ ) · ¸ · p¸ p 2 2 b a ¡ 1 where < ½ + ¹ min ( j ) > 0; < ½ + ¹ min ( j ) < 0; jarg aj < 1·j·m Bj 1·j·n Aj 1 Pn Pp Pm Pq 2 ¼θ; θ = j=1 Aj ¡ j=n+1 Aj + j=1 Bj ¡ j=m+1 Bj > 0: The Laplace transform of the Caputo fractional derivative is (see e.g.
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