Fractional order solutions to fractional order partial differential equations

Abstract

In this paper, we formulate a class of fractional order partial differential equations by concentrating on fractional order partial derivatives. We consider an initiative towards the formulation of fractional order Schrödinger equations, heat equation, and wave equation in the realm of fractional calculus. For a given fractional number, we obtain their respective fractional order solutions. In the light of fractional order quantum mechanics, we extend our analysis further for various potential functions for the case of the time-independent and time-dependent Schrödinger equations. Hereby, our proposal gives a refined solution to discrete dynamical systems. Namely, instead of taking the integral-valued steps in the evolution of a chosen dynamical system, it takes fractional-valued steps while performing an integral or solving a system of differential equations. Finally, concerning the extended analysis of our fractional order solutions, we anticipate interesting applications of the above formulation in other domains of physical sciences and natural systems.

Appendix: physical motivations

In this Appendix, we offer a brief review of the physical motivations in the realm of ordinary calculus whose generalizations we have sought in the light of fractional differentiation. These are the three well-known equations, namely, the Schrödinger, heat, and wave equations that play a major role in understanding the laws of physics.

1.1 Schrödinger equation

First of all, the Schrödinger equation is based on De Broglie’s hypothesis, laws of the conservation of energy and momentum, and the classical plane wave equation. For instance, the Schrödinger equation explicates phenomena of the barrier penetration, whereby the emission of alpha particles [25] and proton fusion process, see [14] in the light of leading order effective theories.

Physically, pertaining to the atomic description of the world, it is known that the Schrödinger equation, when applied to hydrogen atom [21] yields a wave function that depends upon a triple of the quantum numbers, viz. the principal, azimuthal and magnetic quantum numbers. When the given potential energy $V$ is independent of the time, the time-independent Schrödinger equation [7] takes the following form

$$ - \frac{{\hbar ^{2} }}{{2{\text{m}}}}\frac{{\partial ^{2} \psi }}{{\partial {\text{x}}^{2} }} + {\text{V}}\left( {\text{x}} \right)\psi = {\text{E}}\psi , $$

where $\psi$ is termed as the wave function. More precisely, $\psi$ is an element of the Hilbert space, for instance, see [4] towards the logical understanding of quantum mechanics.

1.2 Wave equation

Traditionally, in 1746, De Alembert introduced the one-dimensional wave equation, whereby within 10 years of the duration, Euler extended it to the case of the three-dimensional wave equations [19]. Indeed, there are various generalizations and associated developments.

Mathematically, the wave equations can be realized as a hyperbolic partial differential equation [6] that typically concerns a time variable $t$, and one or more spatial variables ${\text{x}}_{1} ,{\text{x}}_{2} , \ldots , {\text{x}}_{\text{n}}$ for a scalar function $u = u\left( {x_{1} ,x_{2} , \ldots ,x_{n} ,t} \right)$ whose values model the displacement of a wave in $n -$ dimensional background Euclidean space ${\mathbb{R}}^{n}$, see [19] for an introduction of typical partial differential equations in applied mathematics and engineering mathematics.

For a given scalar function $u:{\mathbb{R}}^{n + 1} \to {\mathbb{R}}$ as mentioned in the introduction section, the wave equation can be expressed as per the following expression

$$ \frac{{ \partial^{2} {\text{u}}}}{{\partial {\text{t}}^{2} }} = {\text{a}}^{2} \nabla^{2} {\text{u}}, $$

where $a$ is a fixed constant that defines the propagation speed of the wave and $\nabla^{2}$ denotes the standard Laplacian operator that reads as per the representation

$$ \nabla^{2} = \hat{i} \frac{{\partial^{2} }}{{\partial {\text{x}}^{2} }} + \hat{j}\frac{{\partial^{2} }}{{\partial {\text{x}}^{2} }} + \hat{k}\frac{{\partial^{\text{r}} }}{{\partial {\text{x}}^{2} }} . $$

1.3 Heat equation

In reference to the above Schrödinger and wave equations, the heat equation governs the temperature distribution of an object in a given domain of the space–time. This partial differential equation that describes the flow of the heat energy, wherefore the behavior of the temperature is worth examining in an enlarged setting of the fractional calculus. In particular, the heat equation in (1 + 1) dimensional space-times is given [19] as

$$ \frac{{\partial {\text{T}}}}{{\partial {\text{t}}}} = {\text{K}}\frac{{\partial^{2} {\text{T}}}}{{\partial {\text{x}}^{2} }}, $$

where ${\text{K}}$ is a positive constant defining the thermal diffusivity. In this case, we can think of a one-dimensional rectangular thin wire of its length ${\text{x}}$ at a given temperature profile. Ignoring the wire’s width and height dimensionalities, we may perform the analysis of temperature flows as follows. Let one of the ends along the length of the wire is set at 0 °C, whereas the other end is set to vary about 0 °C. The heat equation can be derived from the Fourier law and conservation of the energy, see [19] for associated details. Herewith, the heat equation is of the fundamental importance in varied fields of scientific research and developments.