Analysis of a new partial integro-differential equation with mixed fractional operators

doi:10.1016/j.chaos.2019.06.005

Chaos, Solitons & Fractals

Volume 127, October 2019, Pages 257-271

https://doi.org/10.1016/j.chaos.2019.06.005 Get rights and content

Abstract

We have introduced a new partial integro-differential equation with mixed fractional operators. The differential operator can be taken as Caputo while the integral is consider to be Caputo–Fabrizio or the Atangana–Baleanu integral. We presented the well poseness of the new class of partial differential equation. We presented the conditions which the existence and uniqueness are obtained. We presented the derivation of exact solution under some conditions. We suggested a numerical scheme that will be used to solve such mathematical equations. We presented some illustratives examples.

Introduction

Integro-differential equations while being very important class of mathematical equations have not been intensively studied in the last years. One of the challenges in studying this class of equation is perhaps to obtain a closed-form solution as this can be sometimes very challenging. We shall recall that integro-differential equations can capture many physical occurrences arising in science and engineering for instance, the Kirchhoff second law [1], [2]. The Wilson-Cowan model, for this model, the activity of interacting and excitatory neurons can be depicted using the system of integro-differential equations.

We can also mention the model of a radiation energy intensity along the ray of a vector ${\vec{I}}_{L}$ in the emitting, absorbing and scattering medium $γ {\vec{I}}_{L}$ can be very accurately described by an integro-differential equation of radiation transfer. We have also the Prandf equation for air circulation in the vicinity of the plane wring [3], [4]. We can mention the heat conduction model that accounts for a finite velocity of heat transport in the medium, including a relaxation time τ [20], [21], [22], [23], [24]. One can quote many more, however, most of these problems are described using classical differential and integral operators which have not being successful in modelling very complex problems. While fractional calculus is a booming subject in the last years, the derivatives based on power law has witness great success in the field of modelling, the field has witness a great revolution [5], [6], [7], [8], [9], [10], [11], [12]. Recently as some already well-established theories were questioned and new differential operators and integrations were suggested [13], [14], [15], [16], [17], [18], [19]. Also in literature, there have been some studies which include recent developments in the field of the fractional calculus as well as its applications, for example, model of spring pendulum in fractional sense [25], [26], [27], [28]. One of the limitations that was extended is perhaps the singularity problem and also the crossover property. Thus due to the revolution and suggestion of the new fractional calculus, one new mathematical equation can be constructed and be used to open new doors of investigation in theory and modelling.

In this paper, we introduce a new fractional integro-differential equation with mixed fractional operators for example we will consider the following equation $_{0}^{A B C} D_{t}^{α} u (x, t) = f (x, u, t) +_{0}^{C F} J_{t}^{α} K (x, u, t) + g (x, t)$ where $_{0}^{A B C} D_{t}^{α}$ is the Atangana–Baleanu fractional derivative and $_{0}^{C F} J_{t}^{α}$ is the Caputo–Fabrizio integral or we can take $_{0}^{C F} D_{t}^{α} u (x, t) = f (x, u, t) +_{0}^{A B} J_{t}^{α} K (x, u, t) + g (x, t) .$

This paper is organized as follows. In Section 2, we present some definitions that will be used in this article. In Section 3, we introduce and study a new fractional integro-differential equation with mixed operators and we give numerical scheme for this equation. In Section 3, we handle fractional integro-differential equation with differential and integral operators. In Section 5, we construct numerical scheme for this equation. In Section 6, we present some examples of the suggested equations and present their numerical approximations. Also we give numerical simulation with the considered problem for different values of fractional orders.

Section snippets

Preliminaries

In this section, we give some definitions used in this article.

Definition 1

Let f: $R^{+} \to R$ and $α \in (n - 1, n), n \in N$ . The left Caputo fractional derivative of order α of the function f is given by the following equality; $D_{0}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {(t - τ)}^{n - α - 1} f^{(n)} (τ) d τ, t > 0 .$ $- D^{β}$ f (t) is the right fractional Caputo derivative.

Definition 2

Let f(t) be continuous and differentiable on C¹[0,1], then the Caputo–Fabrizio derivative with fractional order 0 < α ≤ 1 is given as follows. $_{0}^{C F} D_{t}^{α} f (t) = \frac{M (α)}{1 - α} \int_{0}^{t} \frac{d f (τ)}{d τ} \exp (\frac{- α (t - τ)}{1 - α}) d τ .$

Definition 3

Let $f (t) \in W_{2}^{1} (0, T),$ a

Caputo–Atangana–Baleanu volterra equation

In this section, we introduce and study the following fractional integro-differential equation $\begin{matrix} _{0}^{C} D_{t}^{α} u (x, t) & = & g (x, t) + f (x, t, u) + \frac{1 - α}{A B (α)} K (x, t, u) \\ + \frac{α}{Γ (α) A B (α)} \int_{0}^{t} K (x, τ, u) {(t - τ)}^{α - 1} d τ \end{matrix}$ and $\begin{matrix} _{0}^{A B C} D_{t}^{α} u (x, t) & = & g (x, t) + f (x, t, u) \\ + \frac{1}{Γ (α)} \int_{0}^{t} K (x, τ, u) {(t - τ)}^{α - 1} d τ . \end{matrix}$

We shall start our investigation with Eq. (1). We reformulate Eq. (1) as follow $\begin{matrix} u (x, t) - u (x, 0) & = & \frac{1}{Γ (α)} \int_{0}^{t} g (x, τ) {(t - τ)}^{α - 1} d τ \\ + \frac{1}{Γ (α)} \int_{0}^{t} f (x, τ, u) {(t - τ)}^{α - 1} d τ \\ + \frac{1 - α}{A B (α)} \frac{1}{Γ (α)} \int_{0}^{t} K (x, τ, u) {(t - τ)}^{α - 1} d τ \\ + \frac{α}{A B (α)} \frac{1}{Γ (2 α)} \int_{0}^{t} K (x, τ, u) {(t - τ)}^{2 α - 1} d τ . \end{matrix}$

Theorem 8

If the functions g, f and K are bounded, then the

Caputo–Fabrizio volterra equation

In this section, we introduce the following fractional integro-differential equations $\begin{matrix} _{0}^{C} D_{t}^{α} u (x, t) & = & g (x, t) + f (x, t, u) + \frac{1 - α}{M (α)} K (x, t, u) \\ + \frac{α}{M (α)} \int_{0}^{t} K (x, τ, u) d τ \end{matrix}$ and $\begin{matrix} _{0}^{C F} D_{t}^{α} u (x, t) & = & g (x, t) + f (x, t, u) \\ + \frac{1}{Γ (α)} \int_{0}^{t} K (x, τ, u) {(t - τ)}^{α - 1} d τ . \end{matrix}$

For both cases, we present existence and uniqueness of the exact solution using a very preliminary technique of Banach fixed-point theorem. Additionally, we present for both cases derivation of the numerical technique suitable to solve them. We start with the first case where the derivative

Numerical method

In this section, we present the numerical method that can be used to solve Eq. (75). $\begin{matrix} u (x, t) & = & u (x, 0) + \frac{1 - α}{M (α)} g (x, t) + \frac{α}{M (α)} \int_{0}^{t} g (x, τ) d τ \\ + \frac{1 - α}{M (α)} f (x, t, u (x, t)) + \frac{α}{M (α)} \int_{0}^{t} f (x, τ, u (x, τ)) d τ \\ + \frac{1 - α}{M (α)} \frac{1}{Γ (α)} \int_{0}^{t} K (x, τ, u (x, τ)) {(t - τ)}^{α - 1} d τ \\ + \frac{α}{M (α)} \frac{1}{Γ (α + 1)} \int_{0}^{t} K (x, τ, u (x, τ)) {(t - τ)}^{α} d τ . \end{matrix}$

At $(x_{i}, t_{n + 1}),$ we have $\begin{matrix} u_{i}^{n + 1} & = & u_{i}^{0} + \frac{1 - α}{M (α)} g (x_{i}, t_{n + 1}) + \frac{α Δ t}{M (α)} \sum_{j = 0}^{n} g (x_{i}, t_{j}) \\ + \frac{1 - α}{M (α)} f (x_{i}, t_{n}, u_{i}^{n}) + \frac{α}{M (α)} \sum_{j = 0}^{n} f (x_{i}, t_{j}, u_{i}^{j}) \\ + \frac{1 - α}{M (α)} \frac{{(Δ t)}^{α}}{Γ (α + 2)} \sum_{j = 0}^{n} [\begin{matrix} K (x_{i}, t_{j}, u_{i}^{j}) {\begin{matrix} {(n - j + 1)}^{α} (n - j + 2 + α) \\ - {(n - j)}^{α} (n - j + 2 + 2 α) \end{matrix}} \\ - K (x_{i}, t_{j - 1}, u_{i}^{j - 1}) {\begin{matrix} {(n - j + 1)}^{α + 1} \\ - {(n - j)}^{α} (n - j + 1 + α) \end{matrix}} \end{matrix}] \\ + \frac{α}{M (α)} \frac{{(Δ t)}^{α + 1}}{Γ (α + 3)} \end{matrix}$

Illustrative examples and simulations

In this section, we present some examples of the suggested equations and present their numerical approximations. We consider the following examples $_{0}^{C} D_{t}^{α} u (x, t) = g (x, t) + \frac{\partial}{\partial x} u (x, t) + \frac{1 - α}{M (α)} \frac{\partial^{2}}{\partial x^{2}} u (x, t) + \frac{α}{M (α)} \int_{0}^{t} \frac{\partial^{2}}{\partial x^{2}} u (x, τ) d τ$ and $_{0}^{C F} D_{t}^{α} u (x, t) = g (x, t) + \frac{\partial}{\partial x} u (x, t) + \frac{1}{Γ (α)} \int_{0}^{t} \frac{\partial^{2}}{\partial x^{2}} u (x, τ) {(t - τ)}^{α - 1} d τ .$

We present numerical solution of the first equation using both scheme. Thus for Volterra case, we have $\begin{matrix} u_{i}^{n + 1} = u_{i}^{0} & + & \frac{{(Δ t)}^{- α}}{Γ (α + 1)} \sum_{j = 0}^{n} g_{i}^{j} δ_{n, j}^{α} + \frac{{(Δ t)}^{α}}{Γ (α + 2)} \sum_{j = 0}^{n} [\begin{matrix} \frac{u_{i + 1}^{j} - u_{i - 1}^{j}}{Δ x} {\begin{matrix} {(n - j + 1)}^{α} (n - j + 2 + α) \\ - {(n - j)}^{α} (n - j + 2 + 2 α) \end{matrix}} \\ - \frac{u_{i + 1}^{j - 1} - u_{i - 1}^{j - 1}}{Δ} \end{matrix} \end{matrix}$

Conclusion

The nature is above the understanding of making because mankind live within it, therefore he is part of it. Nevertheless due to continuous improvement by mankind within the environment he lives, it is required to him to understand, analysis and predict it. We have to confess that up to new mankind has really devoted his attention in modelling real world problems using mathematical equations that are constructed using differential and integral operators. One can find in the literature many

References (28)

A. Pedas et al.
Spline collocation for fractional weakly singular integro-differential equations
Appl Numer Math
(2016)
L. Zhu et al.
Numerical solution of nonlinear fractional-order volterra integro-differential equations by SCW
Commun Nonlinear Sci Numer Simul
(2013)
A. Atangana et al.
Fractional stochastic modeling: new approach to capture more heterogeneity
Chaos
(2019)
K.M. Owolabi et al.
Mathematical modeling and analysis of two-variable system with noninteger-order derivative
Chaos
(2019)
A. Atangana et al.
Capturing complexities with composite operator and differential operators with non-singular kernel
Chaos
(2019)
M. Khader et al.
On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method
Appl Math Modell
(2013)
D. Baleanu et al.
A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative
Adv Differ Equ
(2017)
M. Subaşı
An estimate for the solution of a perturbed nonlinear quantum mechanical problem
Chaos Soliton Fract
(2002)
Oldham KTS. The doctrine of description: Gustav Kirchhoff, classical physics, and the “purpose of all science” in...
C.R. Paul
Fundamentals of electric circuit analysis
(2001)

M. Planck

The theory of heat radiation

(1914)

M. Born et al.

Principles of optics: electromagnetic theory of propagation, interference and diffraction of light

(1999)

A. Atangana et al.

Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena

Eur Phys J Plus

(2018)

A. Atangana

Non validity of index law in fractional calculus: afractional differential operator with Markovian and non-Markovian properties

Physica A

(2018)

Cited by (0)

View full text

Analysis of a new partial integro-differential equation with mixed fractional operators

Abstract

Introduction

Section snippets

Preliminaries

Caputo–Atangana–Baleanu volterra equation

Caputo–Fabrizio volterra equation

Numerical method

Illustrative examples and simulations

Conclusion

Appl Numer Math

Commun Nonlinear Sci Numer Simul

Chaos

Chaos

Chaos

Appl Math Modell

Adv Differ Equ

Chaos Soliton Fract

Fundamentals of electric circuit analysis

The theory of heat radiation

Principles of optics: electromagnetic theory of propagation, interference and diffraction of light

Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena

Eur Phys J Plus

Non validity of index law in fractional calculus: afractional differential operator with Markovian and non-Markovian properties

Physica A