On a Fractional Master Equation

Abstract

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)≤2 and 1<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 two-dimensions ∇²Ψ𝑥,𝑦+𝑘²Ψ𝑥,𝑦=0, (1.1) where 𝑘>0 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 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 equation or diffusion 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_𝜃𝐸𝑥,𝑦+₀𝐷^𝛽_𝑦𝐸𝑥,𝑦+𝑘²𝐸𝑥,𝑦=Φ𝑥,𝑦 (1.2)𝑘>0, 𝑥∈ℜ, 𝑦∈ℜ⁺, 1<ℜ(𝛼)≤2, 1<ℜ(𝛽)≤2, where 𝑘 is the wave number given by 𝑘=2𝜋/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. _𝑥𝐷^𝛼_𝜃 is the Riesz-Feller space fractional derivative of order 𝛼 and asymmetry parameter (skewness) 𝜃, and _𝑦𝐷^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)=^∞_𝑛=0𝑧^𝑛Γ()(𝛼)𝑛𝛼+1,𝛼∈ℂ,ℜ>0, (1.3)𝐸_𝛼,𝛽(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(𝛼)𝛽𝑛𝛼+𝛽,𝛼,𝛽∈ℂ,ℜ>0,ℜ>0, (1.4) has been studied in details by Dzherbashyan [8]. Both functions (1.2)-(1.3) are entire functions of order 𝜌=1/𝛼 and type 𝜎=1. In 1920, Hille and Tamarkin [9] have presented a solution of the Abel-Volterra type integral equation 𝜙(𝑥)−𝜆Γ(𝛼)^𝑥₀𝜙(𝑡)(𝑥−𝑡)^1−𝛼d𝑡=𝑓(𝑥),0<𝑥<1, (1.5) in terms of Mittag-Leffler functions. Fox [10] used the 𝐻-function to give the most generalized symmetrical Fourier kernel. This function is defined in terms of Mellin-Barnes 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 _𝑥𝐷^𝛼_𝜃 of order 𝛼 and skewness 𝜃 is defined as _𝑥𝐷^𝛼_𝜃𝑓(𝑥)=Γ(1+𝛼)𝜋)𝜋sin(𝛼+𝜃2^∞₀𝑓(𝑥+𝜉)−𝑓(𝑥)𝜉^1+𝛼+Γ()d𝜉1+𝛼𝜋)𝜋sin(𝛼−𝜃2^∞₀𝑓(𝑥−𝜉)−𝑓(𝑥)𝜉^1+𝛼d𝜉, (2.1) where 0<𝛼≤2,|𝜃|≤min(𝛼,2−𝛼) and the Caputo derivative of order 𝛼 with respect to 𝑡,𝑡>0,_𝑡𝐷^𝛼_∗𝑁()=1𝑥,𝑡Γ(𝑚−𝛼)^𝑡₀𝑁^(𝑚)(𝑥,𝑢)(𝑡−𝑢)^{𝛼−𝑚+1}(𝛼)dd𝑢,𝑚−1<ℜ<𝑚,_𝑚d𝑡^𝑚𝑁(𝑥,𝑡),𝛼=𝑚,𝑚∈𝑁, (2.2) 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]. The 𝐻-function is defined by means of a Mellin-Barnes type integral in the following manner (Mathai and Haubold [19]): 𝐻^𝑚,𝑛_𝑝,𝑞(𝑧)𝑧|^{(𝑎𝑝,𝐴𝑝)}_{(𝑏𝑞,𝐵𝑞)}=12𝜋𝑖_𝐿∏^𝑚_𝑗=1Γ𝑏_𝑗+𝐵_𝑗𝑠∏^𝑛_𝑗=1Γ1−𝑏_𝑗−𝐵_𝑗𝑠∏^𝑞_𝑗=𝑚+1Γ1−𝑏_𝑗−𝐵_𝑗𝑠∏^𝑝_𝑗=𝑛+1Γ𝑎_𝑗+𝐴_𝑗𝑠𝑧^−𝑠d𝑠, (2.3) and an empty product is always interpreted as unity; 𝑚,𝑛,𝑝,𝑞∈𝑁₀ with 0≤𝑛≤𝑝, 1≤𝑚≤𝑞, 𝐴_𝑗,𝐵_𝑗∈ℜ₊,𝑎_𝑖,𝑏_𝑗∈ℂ, 𝑖=1,…,𝑝;  𝑗=1,…,𝑞 such that 𝐴_𝑖(𝑏_𝑗+𝑘)≠𝐵_𝑗(𝑎_𝑖−𝑙−1), 𝑘,𝑙∈𝑁₀; 𝑖=1,…,𝑛; 𝑗=1,…,𝑚, where we employ the usual notations: 𝑁₀=(0,1,…,); ℜ=(−∞,∞),ℜ₊=(0,∞); 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 [20].

The Wright's generalized hypergeometric function, Wright ([21, 22]); is defined by the series representation _𝑝𝜓_𝑞(𝑧)=_𝑝𝜓_𝑞𝑧|^{(𝑎𝑝,𝐴𝑝)}_{(𝑏𝑞,𝐵𝑞)}=^∞_𝑟=0∏^𝑝_𝑗=1Γ𝑎_𝑗+𝐴_𝑗𝑠𝑧^𝑟∏^𝑞_𝑗=1Γ𝑏_𝑗+𝐵_𝑗𝑠𝑟!, (2.4) where 𝑧∈ℂ,𝑎_𝑗,𝑏_𝑗∈ℂ,𝐴_𝑗,𝐵_𝑗∈ℜ₊;𝑖=1,…,𝑝;𝑗=1,…,𝑞;∑^𝑞_𝑗=1𝑏_𝑗−∑^𝑝_𝑗=1𝐴_𝑗>−1;ℂ is the complex number field. The Mittag-Leffler function is a special case of this function, 𝐸_𝛼,𝛽(𝑧)=₁𝜓₁𝑧|^(1,1)_(𝛽,𝛼)=𝐻^1,1_1,2−𝑧|₍^(0,1)_{0,1)1−𝛽,𝛼}. (2.5) The Laplace transform of the function 𝑁(𝑥,𝑡) with respect to 𝑡 is ℒ[𝑁(𝑥,𝑡)]=^∞₀𝑒^−𝑠𝑡𝑁(𝑥,𝑡)d𝑡=𝑁^∗(𝑥,𝑠),𝑥∈ℜ,ℜ(𝑠)>0, (2.6) and its inverse transform with respect to 𝑠 is given by ℒ⁻¹𝑁^∗(=1𝑥,𝑠)2𝜋𝑖^𝛾+𝑖∞_{𝛾−𝑖∞}𝑒^𝑠𝑡𝑁^∗(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^𝑖𝑝𝑥𝑁(𝑥,𝑡)d𝑥=𝑁𝑝,𝑡,𝑥∈ℜ, (2.8) the inverse Fourier transform with respect to 𝑝: ℱ⁻¹𝑁=1𝑝,𝑡2𝜋^∞_−∞𝑒^{−𝑖𝑝𝑥}𝑁𝑝,𝑡d𝑝=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‖=∫^∞₀|𝑓(𝑡)|d𝑡<∞ and ℱ(ℜ) is the space of summable functions on ℜ with norm ‖𝑓‖=∫^∞_−∞|𝑓(𝑡)|d𝑡<∞. From Mathai et al. [11], and Prudnikov et al. [23] the cosine transform of the 𝐻-function is defined as ^∞₀𝑡^𝜌−1cos(𝑘𝑡)𝐻^𝑚,𝑛_𝑝,𝑞𝑎𝑡^𝜇|^{(𝑎𝑝,𝐴𝑝)}_{(𝑏𝑞,𝐵𝑞)}=2d𝑡_𝜌−1𝜋𝑘^𝜌𝐻^𝑚,𝑛+1_𝑝+2,𝑞𝑎2𝑘^𝜇|^{(2−𝜌/2,𝜇/2),(𝑎𝑝,𝐴𝑝),((1−𝜌)/2,𝜇/2)}_{(𝑏𝑞,𝐵𝑞)}, (2.10) where ℜ𝜌+𝜇min_{1≤𝑗≤𝑚}ℜ𝑏_𝑗𝐵_𝑗>0,ℜ𝜌+𝜇max_{1≤𝑗≤𝑛}𝑎_𝑗−1𝐴_𝑗<1<1,arg𝑎2𝜋𝜃,𝜃=^𝑛_𝑗=1𝐴_𝑗−^𝑝_𝑗=𝑛+1𝐴_𝑗+^𝑚_𝑗=1𝐵_𝑗−^𝑞_𝑗=𝑚+1𝐵_𝑗>0. (2.11) The Laplace transform of the Caputo fractional derivative (see, e.g., Podlubny [16]) ℒ₀𝐷^𝛼_𝑡𝑁(𝑥,𝑡)=𝑠^𝛼𝑁^∗(𝑥,𝑠)−^𝑛_𝑟=1𝑠₀^𝑟−1𝐷_𝑡^𝛼−𝑟𝑁(𝑥,𝑡)∣_𝑡=0,𝑛−1<ℜ(𝛼)≤𝑛,𝑛∈𝑁. (2.12) From Mainardi et al. [4], the Fourier transform of the Riesz-Feller derivative is given by ℱ_𝑥𝐷^𝛼_𝜃𝑓(𝑥);𝑝=−𝜓^𝜃_𝛼𝑓𝑝, (2.13) where 𝜓^𝜃_𝛼𝑓(𝑝)=|𝑝|^𝛼𝑒^{𝑖(sign𝑝)𝜃𝜋/2},𝑝∈ℜ, and 𝑓=ℱ[𝑓(𝑥);𝑝]=∫^+∞_−∞𝑒^𝑖𝑝𝑥𝑓(𝑥)d𝑥.

We also need the property of 𝐻-function (Mathai et al. [11]) 𝐻^𝑚,𝑛_𝑝,𝑞𝑧|^{(𝑎1,𝐴1),…(𝑎𝑝,𝐴𝑝)}_{𝑏1,𝐵1𝑏,…𝑞−1,𝐵𝑞−1,(𝑎1,𝐴1)}=𝐻^{𝑚,𝑛−1}_{𝑝−1,𝑞−1}𝑧|^{(𝑎2,𝐴2),…(𝑎𝑝,𝐴𝑝)}_{𝑏1,𝐵1…(𝑏𝑞−1,𝐵𝑞−1)},𝑚≥1,𝑞>𝑚, (2.14) and the following result (Podlubny, [16]): ℒ⁻¹𝑠^𝛽−1𝑠^𝛼−𝑎;𝑡=𝑡^𝛼−𝛽𝐸_{𝛼,𝛼−𝛽+1}𝑎𝑡^𝛼(𝑠)𝑎,ℜ>0,ℜ𝛼−𝛽>−1,𝑠^𝛼<1. (2.15)

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.

3.1. The Exact Solution in Terms of Mittag-Leffler Function

Theorem 3.1. Consider the fractional Laplace equation _𝑥𝐷^𝛼_𝜃𝐸𝑥,𝑦+_𝑦𝐷^𝛽_∗𝐸𝑥,𝑦=0,𝑥∈ℜ,𝑦⩾0, (3.1)1<ℜ(𝛼)≤2, |𝜃|≤min(𝛼,2−𝛼), 1<ℜ(𝛽)≤2 with initial conditions 𝐸(𝑥,0)=𝑓(𝑥), 𝜕/𝜕𝑦𝐸(𝑥,𝑦)∣_𝑦=0=𝑔(𝑥), 𝑥∈ℜ, lim_𝑥→±∞𝐸(𝑥,𝑦)=0, 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 (3.1) subject to the initial conditions is 𝐸=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}𝑦d𝑝+2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−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_𝜃𝐸^∗(𝑥,𝑠)+𝑠^𝛽𝐸^∗(𝑥,𝑠)−𝑠^𝛽−1𝑓(𝑥)−𝑠^𝛽−2𝑔(𝑥)=0. (3.3) Now applying the Fourier transform with respect to the space variable 𝑥 to obtain 𝑠^𝛽−𝜓^𝜃_𝛼𝑝𝐸^∗𝑝,𝑠=𝑠^𝛽−1𝑓𝑝+𝑠^𝛽−2𝑔𝑝,𝐸^∗=𝑠𝑝,𝑠^𝛽−1𝑓𝑝𝑠^𝛽−𝜓^𝜃_𝛼𝑝+𝑠^𝛽−2𝑔𝑝𝑠^𝛽−𝜓^𝜃_𝛼𝑝. (3.4) Using the result (2.15), we get 𝐸𝑝,𝑦=𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽+𝑔𝑝𝑦𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝑦^𝛽. (3.5) Taking the inverse Fourier transform on both sides, we obtain the solution 𝐸=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}𝑦d𝑝+2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}d𝑝. (3.6) 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)=0, the exact solution or the Green function in Mittag-Leffler function is derived with the help of the Fourier convolution: 𝐸𝑥,𝑦=^∞_−∞𝐺^𝜃_𝛼,𝛽𝑥−𝜏,𝑦𝑓(𝜏)d𝜏, (3.7) where the exact solution in terms of the Mittag-leffler function 𝐺(𝑥,𝑦) is 𝐺^𝜃_𝛼,𝛽=1𝑥,𝑦2𝜋^∞_−∞𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}d𝑝. (3.8)

3.2. 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_𝜃𝐸𝑥,𝑦+₀𝐷^𝛽_𝑦𝐸𝑥,𝑦=0,𝑥∈ℜ,𝑦⩾0, (3.9)

1<ℜ(𝛼)≤2,  |𝜃|≤min(𝛼,2−𝛼),  1<ℜ(𝛽)≤2 with initial conditions 𝐸(𝑥,0)=𝑓(𝑥)=𝛿(𝑥),𝜕/𝜕𝑦𝐸(𝑥,𝑦)∣_𝑦=0=0, 𝑥∈ℜ, lim_𝑥→±∞𝐸(𝑥,𝑦)=0, is given by 𝐸𝑥,𝑦=^∞_−∞𝐺^𝜃_𝛼,𝛽𝑥−𝜏,𝑦𝑓(𝜏)𝑑𝜏, (3.10) where 𝐺^𝜃_𝛼,𝛽=1𝑥,𝑦2𝜋^∞_−∞𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}=1d𝑝,𝜋^∞₀cos𝑝𝑥𝐻^1,1_1,2−𝜓^𝜃_𝛼𝑝𝑦^𝛽|₍₎^(0,1)_0,1,(0,𝛽)=1d𝑝,𝜋^∞₀cos𝑝𝑥𝐻^1,1_1,2−𝑝^𝛼𝑦^𝛽𝑒^{𝑖𝜃(𝜋/2)}|₍₎^(0,1)_0,1,(0,𝛽)=1d𝑝,√𝜋|𝑥|𝐻^1,2_3,2−2^𝛼𝑒^{𝑖𝜃(𝜋/2)}|𝑥|^−𝛼𝑦^𝛽|^{(1/2,𝛼/2),(0,1),(0,𝛼/2)}_{(0,1),(0,𝛽)}𝛽=1,ℜ<2,√𝜋|𝑥|𝐻^1,1_2,1−2^𝛼𝑒^{𝑖𝜃(𝜋/2)}|𝑥|^−𝛼𝑦^𝛽|^{(1/2,𝛼/2),(0,𝛼/2)}_(0,𝛽),using(2.14). (3.11)

4. Fractional Poisson Equation

As a generalization of the Cauchy problem 3.1, the Mittag-Leffler solution of the fractional Poisson equation _𝑥𝐷^𝛼_𝜃𝐸𝑥,𝑦+₀𝐷^𝛽_𝑦𝐸𝑥,𝑦=Φ𝑥,𝑦 (4.1) is given as.

4.1. Cauchy Problem

The solution to the fractional Poisson equation _𝑥𝐷^𝛼_𝜃𝐸𝑥,𝑦+₀𝐷^𝛽_𝑦𝐸𝑥,𝑦=Φ𝑥,𝑦,𝑦⩾0,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)≤2 with initial conditions 𝐸(𝑥,0)=𝑓(𝑥), 𝜕/𝜕𝑦𝐸(𝑥,𝑦)∣_𝑦=0=𝑔(𝑥), 𝑥∈ℜ, lim_𝑥→±∞𝐸(𝑥,𝑦)=0 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=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}+𝑦d𝑝2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}+1d𝑝2𝜋^𝑦₀𝜉^𝛽−1∞_−∞Φ𝑝,𝑦−𝜉𝐸_𝛽,𝛽𝜓^𝜃_𝛼𝑝𝜉^𝛽𝑒^{−𝑖𝑝𝑥}d𝑝d𝜉, (4.3) where ̂‌ indicates the Fourier transform with respect to the space variable 𝑥.

Solution 4.1. 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_𝛼𝑝𝐸^∗𝑝,𝑠=𝑠^𝛽−1𝑓𝑝+𝑠^𝛽−2𝑔𝑝+Φ^∗,𝐸𝑝,𝑠_∗=𝑠𝑝,𝑠^𝛽−1𝑓𝑝𝑠^𝛽−𝜓^𝜃_𝛼𝑝+𝑠^𝛽−2𝑔𝑝𝑠^𝛽−𝜓^𝜃_𝛼𝑝+Φ^∗𝑝,𝑠𝑠^𝛽−𝜓^𝜃_𝛼𝑝. (4.4) Taking the inverse Laplace transform and using Laplace convolution, 𝐸𝑝,𝑦=𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽+𝑔𝑝𝑦𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝜉^𝛽+^∞₀Φ𝑝,𝑦−𝜉𝐸_𝛽,𝛽𝜓^𝜃_𝛼𝑝𝑦^𝛽d𝜉. (4.5) Taking the inverse Fourier transform on both sides and using Fourier convolution, we obtain the solution 𝐸=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}+𝑦d𝑝2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2𝜓^𝜃_𝛼𝑝𝑦^𝛽𝑒^{−𝑖𝑝𝑥}+1d𝑝2𝜋^𝑦₀𝜉^𝛽−1∞_−∞Φ𝑝,𝑦−𝜉𝐸_𝛽,𝛽𝜓^𝜃_𝛼𝑝𝜉^𝛽𝑒^{−𝑖𝑝𝑥}d𝑝d𝜉. (4.6)

5. The Fractional Master Equation

In this section, we solve the fractional master equation, namely, the fractional time-independent 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 𝑘²>𝜓^𝜃_𝛼(𝑝)=|𝑝|^𝛼𝑒^{𝑖(sign𝑝)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_𝜃𝐸𝑥,𝑦+_𝑦𝐷^𝛽_∗𝐸𝑥,𝑦+𝑘²𝐸𝑥,𝑦=Φ𝑥,𝑦, (5.1)𝑘>0, 𝑥∈ℜ, 𝑦∈ℜ⁺, with the initial conditions 𝐸(𝑥,0)=𝑓(𝑥), 𝜕/𝜕𝑦𝐸(𝑥,𝑦)∣_𝑦=0=𝑔(𝑥), lim_𝑥→±∞𝐸(𝑥,𝑦)=0, is 𝐸=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+𝑦exp−𝑖𝑝𝑥d𝑝2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+1exp−𝑖𝑝𝑥d𝑝2𝜋^∞₀𝜉^𝛽−1∞_−∞Φ𝑝,𝑦−𝜉𝐸_𝛽,𝛽−𝑘²−𝜓^𝜃_𝛼𝑝𝜉^𝛽exp−𝑖𝑝𝑥d𝑝d𝜉, (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 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_𝜃𝐸^∗(𝑥,𝑠)+𝑠^𝛼𝐸^∗(𝑥,𝑠)−𝑓(𝑥)−𝑠𝑔(𝑥)+𝑘²𝐸^∗(𝑥,𝑠)=Φ^∗(𝑥,𝑠). (A.1) Applying Fourier transform with respect to 𝑥,−𝜓^𝜃_𝛼𝑝𝐸^∗𝑝,𝑠+𝑠^𝛽𝐸^∗𝑝,𝑠−𝑠^𝛽−1𝑓𝑝−𝑠^𝛽−2𝑔𝑝+𝑘²𝐸^∗𝑝,𝑠=Φ^∗𝑝,𝑠. (A.2) Thus, 𝐸^∗=𝑠𝑝,𝑠^𝛽−1𝑠^𝛽+𝑘²−𝜓^𝜃_𝛼𝑝𝑓𝑝+𝑠^𝛽−2𝑠^𝛽+𝑘²−𝜓^𝜃_𝛼𝑝𝑔𝑝+Φ^∗𝑝,𝑠𝑠^𝛽+𝑘²−𝜓^𝜃_𝛼𝑝. (A.3) Taking Laplace inverse transform with respect to 𝑠,𝐸𝑝,𝑦=𝑓𝑝𝐸_𝛽,1−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+𝑔𝑝𝑦𝐸_𝛽,2−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+Φ𝑝,𝑦𝑦^𝛽−1𝐸_𝛽,𝛽−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽. (A.4) Using inverse Fourier transform with respect to 𝑝, the solution is 𝐸=1𝑥,𝑦2𝜋^∞_−∞𝑓𝑝𝐸_𝛽,1−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+𝑦exp−𝑖𝑝𝑥d𝑝2𝜋^∞_−∞𝑔𝑝𝐸_𝛽,2−𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽+1exp−𝑖𝑝𝑥d𝑝2𝜋^𝑦₀𝜉^𝛽−1∞_−∞Φ𝑝,𝑦−𝜉𝐸_𝛽,𝛽−𝑘²−𝜓^𝜃_𝛼𝑝𝜉^𝛽exp−𝑖𝑝𝑥d𝑝d𝜉. (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).

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_𝜃𝐸𝑥,𝑦+_𝑦𝐷^𝛽_∗𝐸𝑥,𝑦+𝑘²𝐸𝑥,𝑦=Φ𝑥,𝑦, (B.1)𝑘>0, 𝑥∈ℜ, 𝑦∈ℜ⁺, with the initial conditions 𝐸(𝑥,0)=𝑓(𝑥)=𝛿(𝑥), 𝜕/𝜕𝑦𝐸(𝑥,𝑦)∣_𝑦=0=0, lim_𝑥→±∞𝐸(𝑥,𝑦)=0 is 𝐸𝑥,𝑦=^∞_−∞𝐺^𝜃_𝛼,𝛽𝑥−𝜏,𝑦𝑓(𝜏)𝑑𝜏+^𝑦₀𝑦−𝜉^𝛼−1𝑥₀𝑔^𝜃_𝛼,𝛽𝑥−𝜏,𝑦−𝜉Φ(𝜏,𝜉)d𝜏d𝜉, (B.2) where the exact solution in terms of the 𝐻-function 𝐺^𝜃_𝛼,𝛽(𝑥,𝑦) is 𝐺^𝜃_𝛼,𝛽=1𝑥,𝑦2𝜋^∞_−∞𝐻^1,1_1,2𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽|₍₎^(0,1)_0,1,(0,𝛽)𝑒^{−𝑖𝑝𝑥}d𝑝, (B.3)𝑔^𝜃_𝛼,𝛽=1𝑥,𝑦2𝜋^∞_−∞𝐻^1,1_1,2𝑘²−𝜓^𝜃_𝛼𝑝𝑦^𝛽|₍₎^(0,1)_{0,1,(1−𝛽,𝛽)}𝑒^{−𝑖𝑝𝑥}d𝑝. (B.4)

C. Mellin-Barnes Representation and Region of Convergence of the Mittag-Leffler 𝐸_𝛽,1[(𝜓^𝜃_𝛼(𝑝))𝑦^𝛽]

For 0<ℜ(𝛽)≤2, 𝐸_𝛽,1[(𝜓^𝜃_𝛼(𝑝))𝑦^𝛽 is represented by the Mellin-Barnes integral as 𝐸_𝛽,1𝜓^𝜃_𝛼𝑝𝑦^𝛽=12𝜋𝑖_𝐿Γ(𝑠)Γ(1−𝑠)Γ𝑦1−𝛽𝑠^𝛽𝜓^𝜃_𝛼𝑝^−𝑠d𝑠,=𝐻^1,1_1,2𝑦^𝛽𝜓^𝜃_𝛼𝑝|^(0,1)_{(0,1),(0,𝛽)}, (C.1) where |arg(𝑦^𝛽𝜓^𝜃_𝛼(𝑝))|<𝜋; the contour of integration is at 𝑐−𝑖∞ and ending at 𝑐+𝑖∞,0<𝑐<1, separates all poles at 𝑠=−𝑘,𝑘=0,1,… to the left and all the poles at 𝑠=𝑛+1,𝑛=0,1,… to the right. From Mathai et al. [11], the integral converges for all 𝑧≠0, where 𝑧=𝑦^𝛽𝜓^𝜃_𝛼(𝑝),𝑝∈ℜ.

D. Series Representation of 𝐻^1,1_2,1[−𝑧|^{(1/2,𝛼/2),(0,𝛼/2)}_(0,𝛽)]

By Mathai et al. [11], the series expansion of the function 𝐻^1,1_2,1[−𝑧|^{(1/2,𝛼/2),(0,𝛼/2)}_(0,𝛽)] where 𝑧=−2^𝛼𝑒^{𝑖𝜃(𝜋/2)}|𝑥|^−𝛼𝑦^𝛽 is given as follows. We have 𝐻^1,1_2,1−𝑧|^{(1/2,𝛼/2),(0,𝛼/2)}_0,𝛽=12𝜋𝑖_𝐿Γ𝛽𝑠Γ((1/2)−(𝛼/2)𝑠)Γ()[]𝛼/2𝑠−𝑧^−𝑠d𝑠, (D.1) where 𝑧=2^𝛼𝑒^{𝑖𝜃(𝜋/2)}|𝑥|^−𝛼𝑦^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=0,1,2,…, we will get the series representation as 𝐻^1,1_2,1−𝑧|^{(1/2,𝛼/2),(0,𝛼/2)}_0,𝛽=2𝛼^∞_𝜈=0(−1)^𝜈Γ𝛽/𝛼[2𝜈+1]()()𝜈!Γ𝜈+1/2−𝑧^{−(2𝜈+1)/𝛼}. (D.2) The 𝐻-function exists for all 𝑧,𝑧≠0 where 𝑧=−2^𝛼𝑒^{𝑖𝜃(𝜋/2)}|𝑥|^−𝛼𝑦^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.