Research Article On a Fractional Master Equation

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1 Hindawi Publishing Corporation International Journal of Differential Equations Volume 211, Article ID , 13 pages doi:1.1155/211/ Research Article On a Fractional Master Equation Anitha Thomas Department of Mathematics, B.A.M. College, Thuruthicadu P.O., Mallapally, Kerala, Pathanamthitta , India Correspondence should be addressed to Anitha Thomas, anithadaniel.bam@gmail.com Received 9 February 211; Accepted 25 August 211 Academic Editor: Peiguang Wang Copyright q 211 Anitha Thomas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A fractional order time-independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time-independent form of the wave equation or diffusion equation in two-dimensions by replacing the integer order partial derivatives by fractional Riesz-Feller derivative and Caputo derivative of order α, β, 1 < R α 2and1< R β 2 respectively. In this paper, we derive an analytic solution for the fractional time-independent form of the wave equation or diffusion equation in two dimensions 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 by means of the Mittag-Leffler function. In all three cases, the solutions are represented also in terms of Fox s H-function. 1. Introduction The standard time-independent form of the wave equation or diffusion equation in twodimensions 2 Ψ ( x, y ) k 2 Ψ ( x, y ), 1.1 where k > is the wave number, is mathematically considered as the master equations to different classes of partial differential equations, namely, Poisson equation and Laplace equation. It represents the time-independent form of the wave equation or diffusion equation obtained while applying the technique of separation of variables to reduce the complexities of the solution procedure of the original equations. This equation appears in physical phenomena and engineering applications such as heat conduction, acoustic radiation, water wave propagation, and even in biology. For estimating the geodesic sea floor properties, the proper prediction of acoustic propagation in shallow water as well as at low frequencies is

2 2 International Journal of Differential Equations very essential. It also provides the solution to such problems, refer Liu et al. 1. It also solves the problems in pattern formation in animal coating, see Murray and Myerscough 2. In electromagnetics, the two-dimensional time-independent form of the wave equation or diffusion equation appears as the governing equation for waveguide problems. There is huge mathematical and engineering interest in electromagnetic wave scattering problems driven by many applications such as modeling radar, sonar, acoustic noise barriers, atmospheric particle scattering, and ultrasound since both the incident and scattered electric field satisfy the two-dimensional time-independent form of the wave equation or diffusion equation which is also known as the scalar Helmholtz equation see Budiarto and Takada 3. This paper introduces a new fractional-model time-independent form of the wave equationordiffusion equation in two-dimensions, in which both the space variables x and y are allowed to take fractional order changes. Such models are described in Mainardi et al. 4 and are defined as xd α θ E( x, y ) D β ye ( x, y ) k 2 E ( x, y ) Φ ( x, y ) 1.2 k >, x R, y R,1< R α 2, 1 < R β 2, where k is the wave number given by k /λ where λ is the wavelength, E x, y is the field variable of interest, which could be acoustic pressure, wave elevation, or electromagnetic potential, among many other possibilities, and Φ x, y is a nonlinear function in the field. x D α is the Riesz-Feller space θ fractional derivative of order α and asymmetry parameter skewness θ, and y D β is the Caputo fractional derivative of order β. These fractional derivatives are integrodifferential operators, and are defined in the section on mathematical preliminaries. The Mittag-Leffler function is a special function having an essential role in the solutions of fractional order integral and differential equations. Recently, this function is frequently used in modeling phenomena of fractional order appearing in physics, biology, engineering and applied sciences. After being introduced and studied by Mittag-Leffler 5, Wiman 6 and Agarwal 7, the Mittag-Leffler function, in its two forms: E α z z n, α C, R α >, 1.3 Γ nα 1 n E α,β z n z n Γ ( nα β ),α,β C, R α >, R( β ) >, 1.4 has been studied in details by Dzherbashyan 8. Both functions are entire functions of order ρ 1/α and type σ 1. In 192, Hille and Tamarkin 9 have presented a solution of the Abel-Volterra type integral equation φ x λ x Γ α φ t dt f x, 1 α x t <x<1, 1.5 in terms of Mittag-Leffler functions. Fox 1 used the H-function to give the most generalized symmetrical Fourier kernel. This function is defined in terms of Mellin-Barnes

3 International Journal of Differential Equations 3 integrals and is a generalization of the Meijer G-function. Most of the special functions are available as the special cases of this function Mathai et al. 11. The objective of this paper is to develop a solution of the fractional time-independent form of the wave equation or diffusion equation 1.1 in terms of Mittag-Leffler function and then in Fox s H-function, using the Laplace and Fourier transforms and their inverse transforms. Mathematically, the Poisson and the Laplace equations are the two special cases of the two-dimensional time-independent form of the wave equation or diffusion equation. We apply this fact to fractional case also. This paper is divided as follows. Section 2 is devoted to mathematical preliminaries used to solve the Cauchy problems. In Section 3, we derive the solution of the fractional Laplace equation in Mittag-Leffler function and then in Fox s H-function. Section 4 is devoted to the fractional Poisson equation. The solution of the fractional master equation is given in Section 5. Its proof and the convergence and the series representation of the H-function are given in the appendix. The Mellin-Barnes representation and the series representation of the special functions in the fundamental solutions are given in the appendix. 2. Mathematical Preliminaries The space fractional Riesz-Feller derivative x D α of order α and skewness θ is defined as θ xd α θ Γ 1 α f x sin α θ π π 2 Γ 1 α π sin α θ π 2 f x ξ f x dξ ξ 1 α f x ξ f x dξ, ξ 1 α 2.1 where <α 2, θ min α, 2 α and the Caputo derivative of order α with respect to t, t>, 1 t N m x, u du, td α Γ m α α m 1 N x, t t u d m N x, t, α m, m N, dtm m 1 < R α <m, where is the integer part. The main results on Mittag-Leffler functions of 1.3, 1.4 are available in the handbook of Erdélyi et al. 12, 13, the monographs by Dzherbashyan 8, 14, some recent books by Mathai et al. 11, Kiryakova 15, Podlubny 16, Kilbas et al. 17,and Mainardi 18. TheH-function is defined by means of a Mellin-Barnes type integral in the following manner Mathai and Haubold 19 : 2.2 Hp,q m,n z z a p,a p b q,b q 1 i L { m j 1 Γ( b j B j s )}{ n j 1 Γ( 1 b j B j s )} { q j m 1 Γ( 1 b j B j s )}{ p j n 1 Γ( a j A j s )}z s ds, 2.3 and an empty product is always interpreted as unity; m, n, p, q N with n p, 1 m q, A j,b j R,a i,b j C, i 1,..., p; j 1,..., qsuch that A i b j k / B j a i l 1,

4 4 International Journal of Differential Equations k, l N ; i 1,...,n; j 1,...,m, where we employ the usual notations: N, 1,..., ; R,, R, ; C being the complex number field. For the details about the contour L and existence conditions see; Mathai et al. 11 and Kilbas and Saigo 2. The Wright s generalized hypergeometric function, Wright 21, 22 ; is defined by the series representation p pψ q z p ψ q z a p,a p j 1 Γ( a j A j s ) z r b q,b q q r j 1 Γ( b j B j s ), 2.4 r! 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 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. 2.5 The Laplace transform of the function N x, t with respect to t is L N x, t e st N x, t dt N x, s, x R, R s >, 2.6 and its inverse transform with respect to s is given by L 1 N x, s 1 i γ i γ i e st N x, s ds N x, t, 2.7 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, 2.8 the inverse Fourier transform with respect to p: F 1 N ( p, t ) 1 e ipx N ( p, t ) dp N x, t. 2.9 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

5 International Journal of Differential Equations 5 the space of summable functions on R with norm f f t dt <. FromMathaietal. 11, and Prudnikov et al. 23 the cosine transform of the H-function is defined as t ρ 1 cos kt Hp,q m,n at μ a p,a p b q,b q dt 2ρ 1 π k ρ H m,n 1 p 2,q a ( ) 2 μ 2 ρ/2,μ/2, a p,a p, 1 ρ /2,μ/2 k b q,b q, 2.1 where R ( ρ ) μ min 1 j m R bj B j >, R ( ρ ) ( aj 1 ) μ max < 1, 1 j n θ n A j j 1 p j n 1 A j m B j j 1 A j q j m 1 B j >. arg a < 1 2 πθ, 2.11 The Laplace transform of the Caputo fractional derivative see, e.g., Podlubny 16 L n D α t N x, t s α N x, s s r 1 Dt α r N x, t t, n 1 < R α n, n N. r From Mainardi et al. 4, the Fourier transform of the Riesz-Feller derivative is given by F { xd α θ f x ; p} ψ θ α f ( p ), 2.13 where ψ θ α f p p α e i sign p θπ/2, p R,and f F f x ; p eipx f x dx. We also need the property of H-function Mathai et al. 11 Hp,q m,n z a 1,A 1,... a p,a p b 1,B 1,... b q 1,B q 1, a 1,A 1 H m,n 1 p 1,q 1 z a 2,A 2,... a p,a p b 1,B 1... b q 1,B q 1, m 1, q > m, 2.14 and the following result Podlubny, 16 : ( ) 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 s α <

6 6 International Journal of Differential Equations 3. The Exact Solution of the Fractional Laplace Equation The exact solution of the fractional Laplace equation 1.2 in terms of three special functions namely, Mittag-Leffler function, Fox s H-function, and Mainardi function, respectively, are derived in the following subsections The Exact Solution in Terms of Mittag-Leffler Function Theorem 3.1. Consider the fractional Laplace equation xd α θ E( x, y ) y D β E ( x, y ), x R, y, < R α 2, θ min α, 2 α, 1 < R β 2 with initial conditions E x, f x, / ye x, y y g x, x R, lim x ± E x, y, where x D α is the Riesz-Feller fractional θ derivative of order α and symmetry θ and y D β is the Caputo fractional derivative of order β. Then the solution of 3.1 subject to the initial conditions is E ( x, y ) 1 f ( p ) ( E β,1 ψ θ ( ) ) α p y β e ipx dp y ĝ ( p ) E β,2 (ψ θ ) α( ) p y β e ipx dp, 3.2 where indicates the Fourier transform with respect to the space variable x. Proof. If we apply Laplace transform with respect to the space variable y and use 2.12, 3.1 becomes xd α θ E x, s s β E x, s s β 1 f x s β 2 g x. 3.3 Now applying the Fourier transform with respect to the space variable x to obtain s β ψ θ α( p ) Ê ( p, s ) s β 1 f( p ) s β 2ĝ ( p ), Ê ( p, s ) ) sβ 1 f( p s β ψα θ ( ) sβ 2 ĝ ( p ) p s β ψα θ ( ). p 3.4 Using the result 2.15, weget Ê ( p, y ) f ( p ) E β,1 ψ α θ ( ) p y β ĝ ( p ) ye β,2 ψα( θ ) p y β. 3.5 Taking the inverse Fourier transform on both sides, we obtain the solution E ( x, y ) 1 f ( p ) E β,1 ( ψ θ α ( p ) ) y β e ipx dp y ĝ ( p ) E β,2 (ψ θ α( p ) ) y β e ipx dp. 3.6

7 International Journal of Differential Equations 7 Hence Theorem 3.1 follows. From the above Cauchy problem, by introducing the initial conditions f x δ x where δ x is the Dirac delta function and g x, the exact solution or the Green function in Mittag-Leffler function is derived with the help of the Fourier convolution: E ( x, y ) G θ α,β( x τ, y ) f τ dτ, 3.7 where the exact solution in terms of the Mittag-leffler function G x, y is G θ ( ) 1 α,β x, y E β,1 (ψ θ ) α( ) p y β e ipx dp Exact Solution in Terms of Fox s H-Function From the above theorem, exact solution in Fox s H-function for the fractional Laplace equation xd α θ E( x, y ) D β ye ( x, y ), x R, y, < R α 2, θ min α, 2 α, 1< R β 2 with initial conditions E x, f x δ x, / ye x, y y, x R, lim E x, y, is given by x ± E ( x, y ) G θ α,β( x τ, y ) f τ dτ, 3.1 where G θ ( ) 1 α,β x, y 1 π 1 π E β,1 (ψ θ α( p ) ) y β e ipx dp, cos ( px ) H 1,1 1,2 ( ψα θ ( ) ) p y β,1 dp,,1,,β cos ( px ) ( H 1,1 1,2 p α y β e iθ π/2 ),1 dp,,1,,β 1 H 1,2 π x 3,2 2 α e iθ π/2 x α y β 1/2,α/2,,1,,α/2, R ( β ) < 2,,1,,β 1 H 1,1 π x 2,1 2 α e iθ π/2 x α y β 1/2,α/2,,α/2, using 2.14.,β 3.11

8 8 International Journal of Differential Equations 4. Fractional Poisson Equation As a generalization of the Cauchy problem 3.1, the Mittag-Leffler solution of the fractional Poisson equation xd α θ E( x, y ) D β ye ( x, y ) Φ ( x, y ) 4.1 is given as Cauchy Problem The solution to the fractional Poisson equation xd α θ E( x, y ) D β ye ( x, y ) Φ ( x, y ), y, x R, 1 < R α 2, 4.2 where Φ x, y is a nonlinear function, 1 < R α 2, θ min α, 2 α, 1< R β 2with initial conditions E x, f x, / ye x, y y g x, x R, lim x ± E x, y where xd α θ is the Riesz-Feller fractional derivative of order α and symmetry θ and yd β is the Caputo fractional derivative of order β. Then, the solution of 4.2 subject to the initial conditions is E ( x, y ) 1 f ( p ) ( E β,1 ψ θ ) α( ) p y β e ipx dp y 1 y ĝ ( p ) E β,2 (ψ θ α( p ) ) y β e ipx dp ξ β 1 Φ ( p, y ξ ) E β,β ( ψ θ α( p ) ) ξ β e ipx dpdξ, 4.3 where indicates the Fourier transform with respect to the space variable x. Solution. Applying the Laplace transform with respect to the space variable y and Fourier transform with respect to the space variable x and using initial conditions, we have s β ψ θ α( p ) Ê ( p, s ) s β 1 f( p ) s β 2ĝ ( p ) Φ ( p, s ), Ê ( p, s ) ) sβ 1 f( p s β ψα θ ( ) sβ 2 ĝ ( p ) p s β ψα θ ( ) Φ ( p, s ) p s β ψα θ ( ). p 4.4 Taking the inverse Laplace transform and using Laplace convolution, Ê ( p, y ) f ( p ) E β,1 (ψ α θ ( ) ) p y β ĝ ( p ) ye β,2 ψα θ ( ) p ξ β Φ ( p, y ξ ) ( E β,β ψ θ ) α( ) p y β dξ. 4.5

9 International Journal of Differential Equations 9 Taking the inverse Fourier transform on both sides and using Fourier convolution, we obtain the solution E ( x, y ) 1 f ( p ) ( E β,1 ψ θ ) α( ) p y β e ipx dp y 1 y ĝ ( p ) E β,2 (ψ θ α( p ) ) y β e ipx dp ξ β 1 Φ ( p, y ξ ) E β,β ( ψ θ α( p ) ) ξ β e ipx dpdξ The Fractional Master Equation In this section, we solve the fractional master equation, namely, the fractional timeindependent form of the wave or diffusion equation fractional Helmholtz equation 1.2 to generate a solution in terms of the Mittag-Leffler function, using the Laplace and Fourier transforms and their inverses. The wave number k is selected such a way that k 2 >ψ θ α p p α e i sign p θπ/2,p R. Thus, the model is made suitable to handle the scattering problems of electromagnetic waves of large wave number or short wavelength. Theorem 5.1. The analytical solution of the fractional time-independent form of the wave equation or diffusion equation xd α θ E( x, y ) y D β E ( x, y ) k 2 E ( x, y ) Φ ( x, y ), 5.1 k >, x R, y R, with the initial conditions E x, f x, / ye x, y y E x, y, is lim x ± E ( x, y ) 1 f ( p ) ( E β,1 k 2 ψ θ ) α( ) p y β exp ( ipx ) dp y 1 ĝ ( p ) ( E β,2 k 2 ψα( θ ) ) p y β exp ( ipx ) dp ξ β 1 Φ ( p, y ξ ) ( E β,β k 2 ψα( θ ) ) p ξ β exp ( ipx ) dpdξ, g x, 5.2 Where is the Fourier transform with respect to x. The proof of this theorem and the H-function solution of the fractional time-independent form of the wave equation or diffusion 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

10 1 International Journal of Differential Equations the fractional time-independent form of the wave equation or diffusion equation or the fractional Helmholtz equation are the master solutions to the solutions of the fractional Laplace equation and the fractional Poisson equation. Appendices A. The Mittag-Leffler Solution of the Fractional Time-Independent form of the Wave Equation or Diffusion Equation If we apply the Laplace transform with respect to the space variable y and use 2.12, 5.1 becomes xd α θ E x, s s α E x, s f x sg x k 2 E x, s Φ x, s. A.1 Applying Fourier transform with respect to x, ψ θ α( p )Ê ( p, s ) s β Ê ( p, s ) s β 1 f( p ) s β 2ĝ ( p ) k 2 Ê ( p, s ) Φ ( p, s ). A.2 Thus, Ê ( p, s ) s β 1 s β k 2 ψα θ ( ) f ( p ) p s β 2 s β k 2 ψα θ ( ) ĝ( p ) Φ ( p, s ) p s β k 2 ψα θ ( ). A.3 p Taking Laplace inverse transform with respect to s, Ê ( p, y ) f ( p ) ( E β,1 k 2 ψα θ ( ) ) p y β ĝ ( p ) ( ye β,2 k 2 ψα θ ( ) ) p y β Φ ( p, y ) ( y β 1 E β,β k 2 ψα( θ ) ) p y β. A.4 Using inverse Fourier transform with respect to p, the solution is E ( x, y ) 1 f ( p ) ( E β,1 k 2 ψ θ ) α( ) p y β exp ( ipx ) dp y 1 y ĝ ( p ) ( E β,2 k 2 ψα( θ ) ) p y β exp ( ipx ) dp ξ β 1 Φ ( p, y ξ ) ( E β,β k 2 ψα( θ ) ) p ξ β exp ( ipx ) dpdξ. A.5 It is seen that the above solution is a master solution of the solutions to the fractional Laplace equation 3.6 and the fractional Poisson equation 4.6.

11 International Journal of Differential Equations 11 B. The Fox s H-Function Solution of the Fractional Time-Independent form of the Wave Equation or Diffusion Equation. The Fox s H-function solution of the fractional time-independent form of the wave equation or diffusion equation is xd α θ E( x, y ) y D β E ( x, y ) k 2 E ( x, y ) Φ ( x, y ), B.1 k>, x R, y R, with the initial conditions E x, f x δ x, / ye x, y y, lim x ± E x, y is E ( x, y ) G θ α,β ( ) y ( ) x α 1 x τ, y f τ dτ y ξ gα,β( θ ) x τ, y ξ Φ τ, ξ dτdξ, B.2 where the exact solution in terms of the H-function G θ x, y is α,β G θ ( ) 1 α,β x, y ( H 1,1 1,2 k 2 ψα θ ( ) ) p y β,1 e ipx dp,,1,,β B.3 g θ ( ) 1 α,β x, y ( H 1,1 1,2 k 2 ψα θ ( ) ) p y β,1 e ipx dp.,1, 1 β,β B.4 C. Mellin-Barnes Representation and Region of Convergence of the Mittag-Leffler E β,1 ψ θ α p y β For < R β 2, E β,1 ψ θ α p y β is represented by the Mellin-Barnes integral as E β,1 (ψ α θ ( ) ) p y β 1 i H 1,1 1,2 Γ s Γ 1 s ( L Γ ( 1 βs ) y β ψα( θ ) ) s p ds, y β ψα θ ( ),1 p,,1,,β C.1 where arg y β ψ θ α p <π; the contour of integration is at c i and ending at c i, < c < 1, separates all poles at s k, k, 1,... to the left and all the poles at s n 1, n, 1,...to the right. From Mathai et al. 11, the integral converges for all z /, where z y β ψ θ α p, p R.

12 12 International Journal of Differential Equations D. Series Representation of H 1,1 2,1 α/2,, α/2 z 1/2,,β By Mathai et al. 11, the series expansion of the function H 1,1 2,1 2 α e iθ π/2 x α y β is given as follows. We have α/2,, α/2 z 1/2,,β where z H 1,1 2,1 z 1/2, α/2,, α/2,β 1 i L Γ ( βs ) Γ 1/2 α/2 s z s ds, Γ α/2s D.1 where z 2 α e iθ π/2 x α y β. Let us assume that the poles of the integrand are simple. The region of convergence is L L is a loop beginning and ending at and encircling all the poles of Γ 1/2 α/2 s once in the negative direction but none of the poles of Γ βs. By calculating the residues at the poles of Γ 1/2 α/2 s where the poles are given by 1/2 α/2 s ν, ν, 1, 2,..., we will get the series representation as H 1,1 2,1 z 1/2, α/2,, α/2,β 2 1 ν Γ (( β/α ) 2ν 1 ) z 2ν 1 /α. α ν!γ ν 1/2 ν D.2 The H-function exists for all z, z / where z 2 α e iθ π/2 x α y β. Acknowledgments The author acknowledges gratefully the encouragements given by Professor A. M. Mathai, Department of Mathematics and Statistics, McGill University, Montreal, Canada, and Dr. M. S. Samuel, Director, Department of Computer Science, MACFAST, Thiruvalla, Kerala, India, in this research work. The author would like to thank the Centre for Mathematical Sciences, Pala Campus, for providing facilities for carrying out this work. References 1 J. Y. Liu, S. H. Tsai, C. C. Wang, and C. R. Chu, Acoustic wave reflection from a rough seabed with a continuously varying sediment layer overlying an elastic basement, Journal of Sound and Vibration, vol. 275, no. 3-5, pp , J. D. Murray and M. R. Myerscough, Pigmentation pattern formation on snakes, Journal of Theoretical Biology, vol. 149, no. 3, pp , H. Budiarto and J. Takada, Numerical estimatiom of electromagnetic wave scattering frombuilding surfaces for the mobile propagation modeling, ITE Technical Report, vol. 25, no. 41, pp. 7 11, F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp , G. M. Mittag-Leffler, Sur la nouvelle fonction E(X), Comptes Rendus de l Académie des Sciences, vol. 137, no. 2, pp , A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen E(X), Acta Mathematica, vol. 29, no. 1, pp , R. P. Agarwal, A propos d une note de M. Pierre Humbert, Comptes Rendus de l Académie des Sciences, vol. 236, pp , M. M. Dzherbashyan, Integral Transforms and Representation of Functions in Complex Domain(in Russian), Nauka, Moscow, Russia, E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Annals of Mathematics, vol. 31, no. 3, pp , 193.

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