key: cord-0974255-us8nm4kn
authors: Moumen Bekkouche, M.; Mansouri, I.; Ahmed, A. A. Azeb
title: Numerical solution of fractional boundary value problem with caputo-fabrizio and its fractional integral
date: 2022-02-04
journal: J Appl Math Comput
DOI: 10.1007/s12190-022-01708-z
sha: 00a08da3f87e6a97193574fcdb192e44c47431c2
doc_id: 974255
cord_uid: us8nm4kn

In this article, we investigate the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo-Fabrizio type. In order to study this problem we used a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo-Fabrizio, therefore, so we transformed the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and taking sufficient conditions existence and uniqueness of this solution is proven based on the results obtained. The analytical study is followed by a complete numerical study.

In recent years, the mathematical models that involving fractional order derivative have become dominant in many fields due to their remarkable importance as being more accurate than the classical order models [14, 24, 25] . In order to advancement of fractional calculus, many researchers have focused to investigate the solution s of nonlinear differential equations involves several fractional differential operators B M. Moumen Bekkouche moumen-med@univ-eloued.dz I. Mansouri mansouri-ikram@univ-eloued.dz A. A. Azeb Ahmed azizazeb@gmail.com like: Riemann-Lioville, Hilfer, and Caputo . . . ect (see [3, 26, 28] ). However, these operators have a power law kernel and have limitations in modeling physical problems. To overcome this difficulty in 2015, a new fractional derivation approach known as Caputo and Fabrizio (C-F) [8] was developed, characterized by without singular kernel, which gave it importance in modeling a certain class of real-world problem, which follow the exponential decay law.

In recent times, several authors contributed to the development and simulation of C-FFDE, to know some applications presented for example, the fractional C-F derivative has been employed for the description of many complex biological systems, including the rabies model [4] , the Cancer treatment by radiotherapy model [9] , the mathematical model for the transmission of Covid-19 [6] , and as well as a new model of human liver [5] , as it has been used in many other fields, such as further extend the groundwater flow model [1] , . . . ect. To know more characteristics and application (see [2, 7, 11-13, 16, 31] ).

In this paper, We study the existence and uniqueness of the solution of the fractional differential equation boundary value problem with the Caputo-Fabrizio, as follows:

where 1 < ρ < 2 is a real number, q is the potential function, and f : [0, 1] → R is continuous. and D (ρ) is the new fractional derivative, and we introduce a new definition of its fractional integral with some properties, using this fractional integral upon problem (1) to obtain an equivalent linear Volterra-Fredholm integral equations of second kind. Finally, by the means of some theorems, the existence and uniqueness of solutions are obtained, and we introduce an algorithm for finding a numerical solution of this problem class.

For the convenience of the reader, we present here the necessary definitions and lemmas from fractional calculus theory. These definitions can be found in the recent literature. 

where γ ∈ [0, 1] and a ∈ ]−∞, t), v ∈ H 1 (a, b), b > a, and M(γ ) is a normalization function such that M(0) = M(1) = 1.

Let n ≥ 1, and γ ∈ [0, 1] the fractional derivative D (γ +n) f of order (n + γ ) is defined by

The formula :

, and it's as an inverse of the conformable fractional derivative of Caputo of order (n + γ ).

In the following, we suppose the function M(γ ) = 1.

Given q ∈ C[0, 1], and 1 < ρ < 2, the solution of

Proof We may apply Lemma 1 to Convert Eq. (4) to an equivalent integral equation

Hence, the unique solution of problem (4) is

The proof is complete.

The classical approach to proving the existence and uniqueness of the solution of (5) is the Picard method. This consists of the simple iteration for n = 1, 2,

with v 0 (t) = g(t). For ease of manipulation, it is convenient to introduce

with u 0 (t) = g(t). On subtracting from (6), the same equation with n replaced by n − 1, an we see that

Also, from (7) v n (t) = n i=0 u i (t)

The following theorem uses this iteration to prove the existence and uniqueness of the solution under quite restrictive conditions, namely that L(x, τ ), F(t, τ ) and g(t) are continuous. Proof Choose λ 1 , λ 2 and λ 3 such that

We first prove by induction that

If we assume that (10) is true for n − 1, then from (8)

Since (10) is obviously true for n = 0, it holds for all n. This bound makes it obvious that the sequence v n (t) in (9) converges, and we can write

We now show that this v(t) satisfies Eq. (5). The series (11) is uniformly convergent since the terms u i (t) are dominated by λ 1 (λ 2 t) i /i! + λ 1 λ i 3 . Consequently, we can interchange the order of integration and summation in the following expression to obtain

Each of the u i (t) is clearly continuous. Therefore v(t) is continuous, since it is the limit of a uniformly convergent sequence of continuous functions.

To show that v(t) is the only continuous solution, suppose there exists another continuous solutionṽ(t) of (5) Then (12) since f (t) andf (t) are both continuous, there exists a constant C such that

Substituting this into (12) |v(t) −ṽ(t)| ≤ C(λ 2 t + λ 3 ), 0 ≤ t ≤ 1 and repeating the step shows that |v(t) −ṽ(t)| ≤ C (λ 2 t) n n! + λ n 3 , 0 ≤ t ≤ 1, for any n.

For a large enough n, the right-hand side is arbitrarily small, therefore, we must have 

are continuous, and |F(t, τ )| = |t(γ τ − 1)q(τ )| ≤ |q(τ )| < 1, ∀t, τ ∈ [0, 1], which means that integral Eq. (5) possesses a unique continuous solution for 0 ≤ t ≤ 1. Therefore, there is a unique continuous solution of the fractional boundary value problem (1) for 0 ≤ t ≤ 1.

In this section, we introduce an algorithm for finding a numerical solution of linear Volterra-Fredholm integral equations of the second kind, the methods based upon trapezoidal rule. For all N ∈ N, Here the interval [0, 1] in to N equal sub-intervals,

The formula of the numerical integration is:

we apply this formula in Eq. (5), and we obtain:

Finally, we get a system of N − 1 equations, which is:

In this part, we give three numerical examples to illustrate the above methods to solve the Volterra-Fredholm linear integral equations of the second kind. The exact solution is known and used to show that the numerical solution obtained with our methods is correct. We used MATLAB to solve these examples.

Example 1 Consider the following fractional boundary value problem:

where ρ = 1.5, q(t) = 1 − t 4 20 , f (t) = t 4 (3e −t − t − 1), with the exact solution v(t) = e −t + t + 1 2 .

Example 2 Consider the following fractional boundary value problem:

Numerical solution of fractional boundary value problem… where ρ = 1.5, q(t) = 1 − t 10 10000

, f (t) = te −t − t 10 10000

− 1 e −t − 1 2

with the exact solution v(t) = e −t + 1 2 .

Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer

A new derivative with normal distribution kernel: theory, methods and applications

Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu

On the mathematical model of Rabies by using the fractional Caputo-Fabrizio derivative

A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative

A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio

Approximating solution of Fabrizio-Caputo Volterra's model for population growth in a closed system by homotopy analysis method

A new definition of fractional derivative without singular kernel

Cancer treatment model with the Caputo-Fabrizio fractional derivative

Existence and uniqueness for a problem involving Hilfer fractional derivative

New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications

Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel

Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel

Applications of Fractional Calculus in

Analytical and numerical methods for Volterra equations

Properties of a new fractional derivative without singular kernel

Fractional calculus models of complex dynamics in biological tissues

Optimal control problems driven by time-fractional diffusion equations on metric graphs: optimality system and finite difference approximation

An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems

Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: A study of fractional calculus on metric graph

Analytical and Numerical Study for an Fractional Boundary Value Problem with conformable fractional derivative of Caputo and its Fractional Integral

Analytical and numerical study of a nonlinear Volterra integro-differential equations with conformable fractional derivation of Caputo

On the solvability fractional of a boundary value problem with new fractional integral

A new fractional integral associated with the Caputo-Fabrizio fractional derivative

Fractional differential equations in electro chemistry

Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

Analysis of differential equations involving Caputo-Fabrizio fractional operator and its applications to reaction-diffusion equations

New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives

A First Course in Integral Equations

Linear and Nonlinear Integral Equations

A new fractional derivative without singular kernel

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Consider the following fractional boundary value problem:where ρ = 1.75, q(t) = (1 − t)/2, and f (t) = −100te 2t . In this case, we don't know the exact solution.

In this article, we proved the existence and uniqueness of the fractional boundary value problem with use of the minimum of hypotheses that ensure this, and using numerical methods and programming by Matlab to solve the problem.