Sound & Vibration |

DOI: 10.32604/sv.2022.015882

ARTICLE

On Fractional Integro-Differential Equation with Nonlinear Time Varying Delay

1Department of Mathematics, Faculty of Science, Arish University, AL-Arish, Egypt

2Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt

3Basic Science Department, Higher Technological Institute, 10th of Ramadan City, Egypt

*Corresponding Author: A. M. Abdallah. Email: ahmedabdel3l@yahoo.com

Received: 21 January 2021; Accepted: 09 December 2021

Abstract: In this manuscript, we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay. Also, we demonstrate the stability analysis for these equations. Our paper provides a convergence of semi-analytical approximate method for these equations. It would be desirable to point out approximate results.

Keywords: Convergence; stability; fractional integro-differential equation

Nowadays, the study of differential equations with nonlinear time varying delay has attracted a wide-ranging interest [1–4]. Meanwhile, few research texts have been based on the theoretical aspects. It is worth to construct an accurate strategy that will facilitate the task of many scholars, followers and experts later. In mathematics, stability of the system is a mathematical property which deals with the behavior of the dynamic systems and corresponds to the convergence of solutions of differential, integro differential and fractional equations [5–12].

It considers as a suitable central role in the study of fractional system [13–15]. This work is perhaps the best straightforward technique to generalize the related papers [16–21].

In [21], Soliman et al. carried out a detailed and comprehensive analysis of fractional Volterra Fred-holm integro-differential equation with constant delay. Likewise, our present study motivated, constructed, continuously developed thus far parallels and properly cited (inspired) by the results of the above work [22–26]. It also includes analysis extensively, but differs significantly about the aforementioned work. It is an attempt to generalize different precedent works and in order to achieve a notable contribution with its counterparts. For more details, this context relied on the appropriate analysis of some linear and nonlinear problems described by fractional integro-differential equations with variable delays. More precisely, we receive a noticeable attention to the nonlinear integro-differential equations with nonlinear time varying delay as in [17]. So far, the strategy of the current research contains four beneficial items. At the first, existence and uniqueness of the solution of the proposed problem will be examined. Secondly, we will provide the readers with the stability and the convergence of this solution. At the end, we shall suggest a method and its convergence therein. This literature consists of five sections. Some basic definitions and theorems are introduced in the first section. Indeed, the suggested problem is discussed and analytical explanation of the proposed problem is detailed in section two. Some Important properties of the solution is mentioned in section three. A modified method is explained in section four. Finally, An experimental example is illustrated in section five.

1.1 Mathematical Tools and Theoretical Background

The most commonly notations, definitions and theorems are mentioned. The presented preliminaries are related to our paper.

Definition 1.1. The Banach space U = C([a, b], ℝ4) is the space of all real-valued continuous functions from :[a, b] → ℝ4, let U(t) = {u(t):u(t) ∈ C([a, b], ℝ4 and Dsu ∈ C([a, b], ℝ4, s ∈ (0, 1]} endowed with the norm ‖ . ‖; ‖ u ‖ = max{|u(t)| + |Dsu(t)|:t ∈ [a, b]}, Dsu denotes the Caputo derivative of fractional order s [27,28].

Definition 1.2. Let σ:U → U be a mapping on a Banach space (U, ‖ . ‖). The point u ∈ U is called a fixed point of σ with σu = u.

Definition 1.3. The mapping σ on a Banach space (U, ‖ . ‖) is called contractive if there exists C ∈ (0, 1), such that

Definition 1.4. For w ∈ C([a, b], ℝ4), the q−th Caputo fractional derivative of a function is defined by

where n = [q] + 1, m ∈ ℕ, [q] is Euler Gamma function for q and [q] denotes the integer part of the real number q.

Definition 1.5. The Mittag-Leffler type is defined as

at b = 1, a > 0 the previous equation becomes the classical Mittag-Leffler.

Definition 1.6. [27] The fractional equation

is Ulam-Hyers stable if there exists Cf such that for each ε > 0 and v ∈ C([a, b], ℝ4)

t is the independent variable.

Definition 1.7. [27] The fractional integro-equation

is Ulam-Hyers stable Rassias stable with respect to ν if there exists a positive constant

L > 0 with the following property: For each u(t) satisfying

then

Definition 1.8. The sequence

ε, N > 0 such that

Lemma 1.9. [26] (Gronwall’lemma)

Let u and v be nonnegative continuous functions on some interval t ∈ [a, b]. Also, let the function f(t) be positive, continuous and monotonically non-decreasing on [a, b] and u satisfies the inequality

Then, there holds the inequality

Lemma 1.10. [26] (Pachpatte’inequality) Let u, v and w be nonnegative continuous functions on ℝ+ and f(t) be a positive and non-decreasing continuous function, the inequality

holds, then

Theorem 1.11. [26] (Banach contraction mapping theorem)

Let (U, ‖ . ‖) be a Banach space, σ:U → U is an operator. If σ is contraction (contractive) mapping. Then σ has exactly one fixed point.

Theorem 1.12. [27] (Schauder fixed point theorem)

Let (U, ‖ . ‖) be a closed, convex and nonempty subset of a Banach space C[a, b], suppose that σ:U → U is a continuous mapping such that σ(U) is a relatively compact subset of C[a, b]. Then σ has at least one fixed point in U.

2 Analytic Explanation of the Problem

Throughout this paper, we will consider Caputo fractional integro-differential equation of the form:

through the initial condition:

where

0 < α < 1, 0 < τ < b, f:[a, b] × U × U × U → U is a continuous function, G, H:[a, b]2 × U → U → U are nonlinear Lipschitz continuous functions of u(t) and χ:U → ℝ4 is a continuous function.

Let us assume the following conditions:

(1) There exists a constant Cf > 0, for each u1, u2, v1, v2, w1, w2 ∈ U

(2) There exists a constant CG > 0

(3) There exists a constant CH > 0

(4) There exists a constant

(5) There exists a constant

Theorem 2.1. The Eq. (14) is equivalent to

Proof. Integrating two both sides of Eq. (14), we get

This leads to

Operate with

Then,

Differentiating we obtain

where

Let σ:U → U, for any u ∈ U. Now, we establish the following theorem for the fixed point σ.

Theorem 2.2. The operator σ maps U into itself and it is also continuous on [a, b].

Proof.

That is, σ maps U into itself. Also, A becomes uniformly bounded. Suppose a sufficiently small number n > 0,

In short,

Consequently, we thus conclude that

where t ∈ [a, b],

Then, σu(t) is continuous on [a, b]. Our approach for proving that σ is continuous, we suppose that un converge to u,

If we follow the conditions Eqs. (16)–(20), we reach

This is equivalent to

This shows that σun → σu.

2.1 Existence and uniqueness of the solution for Eq. (14)

In what follows, we will investigate the existence and uniqueness of solution for the fractional integro-differential equation with time-varying delay (variable delay).

Theorem 2.3. Suppose that the conditions Eqs. (16)–(20) hold, then the non-linear fractional integro-differential Eq. (14) has at least a unique solution u ∈ U.

Proof. By analogous proof to the continuity of σ operator.

For brevity,

In consequent, we have ‖ σu(t) − σv(t)‖ ≤ Y‖u(s) − v(t)‖, where t ∈ [a,b],

We conclude that σ is Lipschitz on U with Lipschitz constant Y. It is well known that σ is a fixed point as a consequence of Theorem 1.11., i.e., σ is a contractive mapping. Eq. (14) has immediately at least a unique solution u ∈ U.

Lemma 2.1. Assume that {u(t)} is a continuous function on [a, b], it satisfies

Further, ‖u(t1) − u(t2)‖ ≤ q. Then {u(t)} is equicontinuous on [a, b].

Proof. Without loss of generality, for t1, t2 ∈ [a, b] such that t1 < t2, we get

whenever t1 → t2, q > 0, where

Lemma 2.2. If the conditions Eqs. (16)–(20) satisfied, then the non-linear Eq. (14) has a unique solution provided

Our following attention is focused on checking the stability of the solution u(t) for Eq. (14) in the frame of Ulam-Hyers and Ulam-Hyers-Rassias.

2.2 Stability Analysis of the Solution for Eq. (14)

Theorem 2.4. Assume that the conditions Eqs. (16)–(20) hold. Then the non-linear fractional integro-differential Eq. (14) is Ulam-Hyers stable.

Proof. Let u ∈ U be a solution of Eq. (14), D(s) is a continuous and non negative function such that

Now, we are going to apply the integral operator

Equivalently

for v(t) ∈ U, it can be written as

The difference |u(t) − v(t)| is given as

or equivalent to

From Gronwall’s lemma Eqs. (10), (11) yields

where K > 0, R = Cf(CG + CH) (CG + CH) such that

In consequence, the problem (14) is stable in the sense of Ulam-Hyers. This completes the proof.

Theorem 2.5. Suppose that the conditions Eqs. (16)–(20) satisfied, P(t) ∈ U is an increasing function and

Proof. Let w ∈ U be a solution of the following inequality

Further, for any t ∈ [a, b], ε > 0. Assume that u ∈ U is the solution of (14). Now, integrate (14), that is

It can be easily noticed that

Hence,

It directly follows from Pachpatte’s lemma Eqs. (12)–(13) that

for C > 0 which ends the proof.

Let us extend our results to asymptotically stable solution. For that, we shall perform the absolute value for the solution of (14)

In fact, by means of Cauchy Schwartz inequality, we deduce

where

Now, we observe that |u(t)| → 0 whenever t → ∞. Therefore, the zero solution of (14) is said to be asymptotically stable.

3 Some Important Properties of the Solution

There is no doubt that there are various properties that characterize solutions. So, we will look at two features, namely continuous dependence of solution and estimates on the solution.

Theorem 3.1. For the two solutions u1(t), u2(t) of Eq. (14).

Proof. See proof of Theorem 4.1.

Theorem 3.2. Assume that the function f in Eq. (14) is Lipschitz function. If u(t) is a solution of Eq. (14), then

Proof. Also, the proof is similar to the proof of Theorem 4.1.

4 Modified Variational Iteration Method with Adomain Decomposition Method

The ongoing method is distinct and the results improve quickly. We will create the correct functional in the following form:

where λ is a Lagrange multiplier. The solution is defined by the infinite series

The nonlinear function can be written as [16,24]

Since An, Bn and Cn are the Adomain polynomials of u0, u1, …, un. Substitute (37)–(40) in (36), we have

For λ = −1, with another formula

5 Experimental and Numerical Examples

Here, we give examples (application situations for the applied fractional equation) which clarifying the gained results.

In this subsection, we shall present the numerical results gained by employing iterative methods namely modified variational iteration method with Adomain decomposition method

Example 1

Solution

For n = 0

By Adomain decomposition method, α = 0.5, we get A0 = B0 = C0 = 0. Hence u1 = t2 is also the exact (analytical) solution. Exactly, u0 = u2 = … = 0.

Example 2

Solution

By the same procedures of the previous example, we have u1 = t.

5.2 Graphical Representation of Solution for Eq. (43)

Firstly, let α approach to 0.5 and n = 1, the obtained solution for this case represent graphically in Fig. 1.

Finally, let α approach to 1 and n = 1, 2, 3. Approximate solutions for this case is obtained in Fig. 2.

Table 1 shows the analysis results for Eq. (43).

Figs. 1–2 indicate the difference to uapp at different values of n.

Fig. 1 is the relationship between t and the approximate solution uapp at α = 0.5, n = 1 only, otherwise, at other values of n we find that the approximate solution approaches zero.

Fig. 2 is the relationship between t and the approximate solution uapp at α = 1, n = 1, 2, 3.

We emphasize that the analysis of fractional integro-differential equations with delay attracts considerable attention by many scientists [29,30]. The beneficial contribution of this work was the discovery much of the tools in the analysis process for many equations. It should be noted that the choice of α plays a vital role in the results of the suggested problem and this was evident in the examples that were listed in our research. We find that when the approximate solution and the exact solution apply when a value of α is at a certain value of n, and other than this value of n, the approximate solution is 0.

Acknowledgement: The authors are very grateful to the editors and reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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