Analytical solutions of linear fractional partial differential equations using fractional Fourier transform

doi:10.1016/j.cam.2020.113202

Journal of Computational and Applied Mathematics

Volume 385, 15 March 2021, 113202

https://doi.org/10.1016/j.cam.2020.113202 Get rights and content

Abstract

This paper discusses the analytical solutions of fractional partial differential equations using Integral Transform method. The fractional derivatives are considered with reference to modified Riemann–Liouville derivatives. Fractional Fourier transform (FrFT) is applied to solve fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. The method proposed here is effective enough to work on these equations efficiently.

Introduction

Integral transforms have been considered as one of the prominent mathematical tools to solve ordinary differential and partial differential equations and applied in almost every domain of science and engineering for a long time now [1], [2]. Applications of fractional integral transforms are a pioneering area of investigation and many integral transforms, such as fractional Fourier, fractional Hankel, fractional Laplace, fractional Sumudu, fractional Wavelet, and fractional Mellin transforms are being employed to solve fractional differential equations [3], [4], [5], [6], [7], [8].

The Fourier transform is one of the most extensively used integral transforms because it has multifarious applications in the diverse disciplines [9], [10], [11]. The Fourier transform of the function $ϕ (x)$ is denoted by $\hat{ϕ} (ϖ)$ and is defined as $\hat{ϕ} (ϖ) = \int_{- \infty}^{+ \infty} e^{- i ϖ x} ϕ (x) d x, ϖ > 0$ and its inversion formula is given by $ϕ (x) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{i ϖ x} \hat{ϕ} (ϖ) d ϖ, ϖ > 0$ provided the above integrals converge.

The Fourier transform of fractional order has been employed tremendously after 1980 when the definition was consolidated by Namias [3]. He applied it to solve certain differential equations in the field of quantum mechanics. Credit for its first application goes to Wiener [12] in 1929. As a generalization of the Fourier transform, FrFT has extra degree of freedom and has become more effectual tool than the classical Fourier transform because of its wider range of applications in the various sectors of applied mathematics, electrical engineering, optics, signal processing, signal analysis, optical communication, quantum mechanics, filter designing, estimation and signal recovery [13], [14], [15], [16], [17], [18], [19].

Multiple definitions of FrFT were experimented using different mathematical aspects [18]. In 2003 Bolangna and West presented a novel definition of FrFT using concepts of fractional calculus, but it could not grab much attention from investigators [20]. In the year 2008, Y.F. Luchko, H. Martinez, J.J. Trujillo [21] and Jumarie [22] also used concepts of fractional calculus and introduced novel definitions of FrFT of real order $α$ . In the present work, we have applied Jumarie’s FrFT to obtain analytical solutions of fractional partial differential equations.

The theory of derivatives and integrals of non-integer order, that is the fractional calculus is used to describe many phenomena in engineering and science [23], [24]. The fractional differential equations have been used enormously for last two decades because of their varied applications in many spheres of physical and biological sciences [25], [26], [27].

There are many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials where fractional derivatives are involved. Fractional differential equations are used to describe phenomena occurring in electro magnetics, acoustics, viscoelasticity, electro chemistry and material science. In fact fractional differential equations most efficiently describe certain physical and engineering systems and therefore we need a reliable and effective technique to solve fractional differential equations [26], [27], [28], [29]. In fact no analytical method was available before 1992 even for linear fractional differential equations. To begin with Liao [30] proposed Homotopy Analysis Method (HAM) to solve fractional differential equations.

Fractional Partial Differential Equations (FPDEs) are generalization of classical integer order partial differential equations. They significantly describe the problems in various field of science, engineering and other disciplines such as astrophysics, dynamical systems, control systems and mathematical physics [31], [32], [33], [34], [35].

In the literature, many analytical methods are available to solve FPDEs such as Adomian decomposition method [36], fractional variational iteration method [37], variable separable method [38], integral transform method [39], differential transform method [40] and optimal Homotopy Analysis Method [41]. The different types of solutions of FPDEs had been obtained by several researchers using the aforementioned mathematical methods [36], [37], [38], [39], [40], [41], [42], [43].

Recently Laplace-Adomian decomposition method [44] and natural transform decomposition method [45] have been used to obtain exact and approximate analytical solutions for FPDEs.

The fractional integral transform method is also very simple, reliable, accurate and useful method for solving FDEs and FPDEs. This method has been employed since 1992. In this context Laplace transform is being used by the most of researchers to solve these equations [46], [47], [48], [49], [50], [51]. Fourier [10], Mellin [52] and Sumudu transforms [53] have been also applied successfully to solve them.

In the recent years, Zhiaiang and Xuemei [49] have used Laplace transform method for a free boundary value problem of time FPDEs, Mohamed et al. [54] have used Integral transform method for linear and nonlinear partial differential equations. Also Kumar [53] has obtained exact solutions of FPDEs using Sumudu transform iterative method. However Prakash et al. [51] have applied homotopy technique via Laplace transform in the solution of fractional order multidimensional telegraph equation.

With this background, the fractional Fourier transform has been used to obtain analytical solutions of fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. In this study, the proposed method gives the exact solutions of fractional heat diffusion and fractional wave equations in terms of Mittag-Leffler function and fractional Dirac delta function respectively. The solution of fractional telegraph equation has been derived in integral form, and solution of fractional kinetic equation is obtained in a series form.

This paper is organized as follows: Section 2 contains some relevant definitions of fractional calculus. In Section 3, the derivations of some useful results are given, which will be used later. Section 4 contains main results i.e. analytical solutions of FPDEs. Finally, Section 5 concludes the article.

Section snippets

Preliminaries

This section presents some basic concepts of fractional calculus.

Definition 2.1

Let $ϕ : R \to R$ , $u \to ϕ (u)$ , be a continuous function, but not necessarily differentiable, and let $h > 0$ be a constant discretization span, then fractional derivative of order $α$ , $0 < α < 1$ , of the function $ϕ (u)$ is given by the following limit [55] $ϕ^{α} (u) = lim_{h ↓ 0} \frac{Δ^{α} ϕ (u)}{h^{α}} = lim_{h ↓ 0} h^{- α} \sum_{n = 0}^{\infty} {(- 1)}^{n} (\binom{α}{n}) ϕ [u + (α - n) h], 0 < α < 1$ where

$Δ^{α} ϕ (u) = {(F W - 1)}^{α} ϕ (u)$ is a fractional difference of order $α$ , $0 < α < 1$ and $F W (h) ϕ (u) = ϕ (x + h)$ is a forward operator.

Assume that function $ϕ (u)$ is

Dirac delta function of fractional order

Definition 3.1

Consider a function $δ_{α} (x, ε) = \{\begin{matrix} 0 & if x \notin [0, ε] \\ \frac{ε^{- α}}{2} & if 0 < x < ε \end{matrix}$ In the limiting case, when $ε \to 0$ we have the limit $lim_{ε \to 0} δ_{α} (x, ε) = δ_{α} (x)$

Lemma 3.2

If $δ_{α} (x - b)$ is a function of fractional order $α$ , for $0 < α \leq 1$ , then following formula holds $\int_{- \infty}^{+ \infty} ϕ (x) δ_{α} (x - b) {(d x)}^{α} = α ϕ (b)$

Proof

We have $\int_{b - ε}^{b + ε} ϕ (x) δ_{α} (x - b) {(d x)}^{α} = α \int_{b - ε}^{b + ε} {(b + ε - x)}^{(α - 1)} ϕ (x) δ_{α} (x - b) d x$ $= α \int_{b - ε}^{b + ε} ε^{α - 1} ϕ (b) δ_{α} (x, ε) d x$ using (9) we get the desired result.

(ii) We can easily obtain fractional Fourier transform (FrFT) of Dirac delta function of fractional order using (11) $F_{α} [δ_{α} (x - b)] = \int_{- \infty}^{+ \infty} E_{α} (i ϖ^{α}$

Applications of fractional Fourier transform to the fractional partial differential equations

In this section, we have derived the analytical solutions of some fractional partial differential equations using the method of fractional Fourier transform.

Conclusion

Analytical solutions of fractional diffusion, fractional wave, fractional telegraph and fractional kinetic equations have been reported by using fractional Fourier transform. This method gives exact solutions of fractional heat diffusion and fractional wave equations in terms of Mittag-Leffler function and Dirac delta function of fractional order respectively. However solution of fractional telegraph equation is obtained in integral form and series solution of fractional kinetic equation is

Acknowledgments

This work was supported by the University Grant Commission (UGC) New Delhi, India [Grant No.: RGNF-2015-17-SC-MAD-20534]. The authors express their special thanks to Professor Rajiv Saxena and R.K. Johar for their guidance and suggestions.

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Analytical solutions of linear fractional partial differential equations using fractional Fourier transform

Abstract

Introduction

Section snippets

Preliminaries

Dirac delta function of fractional order

Applications of fractional Fourier transform to the fractional partial differential equations

Conclusion

Acknowledgments

J. Math. Anal. Appl.

Math. Comput. Modelling

Entropy

Abstr. Appl. Anal.

C. R. Acad. Sci., Paris

Curr. Dev. Math.

The Use of Integral Transforms

Integral Transform and their Applications

The fractional order Fourier transform and its application to quantum mechanics

J. Inst. Math. Appl.

Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative

Appl. Math. Lett.

A note on fractional sumudu transform

J. Appl. Math.

A novel fractional wavelet transform and its applications

Sci. China Inf. Sci.

On fractional order Mellin transform and some of its properties

Tbilisi Math. J.

Fourier Transform

Fourier Transform with Applications in Physics and Engineering

The Fourier Transform and Its Applications

Hermitian polynomials and Fourier analysis

Stud. Appl. Math.

Namias’s fractional Fourier transform

IMA J. Appl. Math.

The fractional Fourier transform and applications

SIAM Rev.

Image rotation, wigner rotation, and the fractional Fourier transform

J. Opt. Soc. Amer. A

The fractional Fourier transform and time frequency representation

IEEE Trans. Signal Process.

Digital computation of the fractional Fourier transform

IEEE Trans. Signal Process.

The Fractional Fourier Transform with Applications in Optics and Signal Processing

Short-time fractional Fourier transform and its applications

IEEE Trans. Signal Process.

Physics of Fractal Operators

Fractional Fourier transform and some of its applications

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Fourier’s transform of fractional order via Mittag–Leffler function and modified Riemann–Liouville derivative

J. Appl. Inform.

Some basic problems in continuum and statistical mechanics