On a nonlinear mixed-order coupled fractional differential system with new integral boundary conditions

Abstract: We present the criteria for the existence of solutions for a nonlinear mixed-order coupled fractional differential system equipped with a new set of integral boundary conditions on an arbitrary domain. The modern tools of the fixed point theory are employed to obtain the desired results, which are well-illustrated by numerical examples. A variant problem dealing with the case of nonlinearities depending on the cross-variables (unknown functions) is also briefly described.


Introduction
It is well known that the classical boundary conditions cannot describe certain peculiarities of physical, chemical, or other processes occurring within the domain. In order to overcome this situation, the concept of nonlocal conditions was introduced by Bicadze and Samarskiȋ [1]. These conditions are successfully employed to relate the changes happening at nonlocal positions or segments within the given domain to the values of the unknown function at end points or boundary of the domain. For a detailed account of nonlocal boundary value problems, for example, we refer the reader to the articles [2][3][4][5][6] and the references cited therein.
Computational fluid dynamics (CFD) technique directly deals with the boundary data [7]. In case of fluid flow problems, the assumption of circular cross-section is not justifiable for curved structures. The idea of integral boundary conditions serves as an effective tool to describe the boundary data on arbitrary shaped structures. One can find application of integral boundary conditions in the study of thermal conduction, semiconductor, and hydrodynamic problems [8][9][10]. In fact, there are numerous applications of integral boundary conditions in different disciplines such as chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. [11][12][13]. Also, integral boundary conditions facilitate to regularize ill-posed parabolic backward problems, for example, mathematical models for bacterial self-regularization [14]. Some recent results on boundary value problems with integral boundary conditions can be found in the articles [15][16][17][18][19] and the references cited therein.
The non-uniformities in form of points or sub-segments on the heat sources can be relaxed by using the integro multi-point boundary conditions, which relate the sum of the values of the unknown function (e.g., temperature) at the nonlocal positions (points and sub-segments) and the value of the unknown function over the given domain. Such conditions also find their utility in the diffraction problems when scattering boundary consists of finitely many sub-strips (finitely many edge-scattering problems). For details and applications in engineering problems, for instance, see [20][21][22][23].
The subject of fractional calculus has emerged as an important area of research in view of extensive applications of its tools in scientific and technical disciplines. Examples include neural networks [24,25], immune systems [26], chaotic synchronization [27,28], Quasi-synchronization [29,30], fractional diffusion [31][32][33], financial economics [34], ecology [35], etc. Inspired by the popularity of this branch of mathematical analysis, many researchers turned to it and contributed to its different aspects. In particular, fractional order boundary value problems received considerable attention. For some recent results on fractional differential equations with multi-point and integral boundary conditions, see [36,37]. More recently, in [38,39], the authors analyzed boundary value problems involving Riemann-Liouville and Caputo fractional derivatives respectively. A boundary value problem involving a nonlocal boundary condition characterized by a linear functional was studied in [40]. In a recent paper [41], the existence results for a dual anti-periodic boundary value problem involving nonlinear fractional integro-differential equations were obtained.
Motivated by aforementioned applications of nonlocal integral boundary conditions and fractional differential systems, in this paper, we study a nonlinear mixed-order coupled fractional differential system equipped with a new set of nonlocal multi-point integral boundary conditions on an arbitrary domain given by where c D χ is Caputo fractional derivative of order χ ∈ {ξ, ζ}, ϕ, ψ : [a, b] × R × R → R are given functions, p, q, δ i , x 0 , y 0 ∈ R, i = 1, 2, . . . , m.
Here we emphasize that the novelty of the present work lies in the fact that we introduce a coupled system of fractional differential equations of different orders on an arbitrary domain equipped with coupled nonlocal multi-point integral boundary conditions. It is imperative to notice that much of the work related to the coupled systems of fractional differential equations deals with the fixed domain. Thus our results are more general and contribute significantly to the existing literature on the topic. Moreover, several new results appear as special cases of the work obtained in this paper. We organize the rest of the paper as follows. In Section 2, we present some basic concepts of fractional calculus and solve the linear version of the problem (1.1). Section 3 contains the main results. Examples illustrating the obtained results are presented in Section 4. Section 5 contains the details of a variant problem. The paper concludes with some interesting observations.

Preliminaries
Let us recall some definitions from fractional calculus related to our study [53].
Definition 2.1. The Riemann-Liouville fractional integral of order α ∈ R (α > 0) for a locally integrable real-valued function of order α ∈ R, denoted by I α a + , is defined as where Γ denotes the Euler gamma function.
In the following lemma, we obtain the integral solution of the linear variant of the problem (1.1).
Then the unique solution of the system is given by a pair of integral equations

4)
and it is assumed that Proof. Applying the integral operators I ξ a + and I ζ a + respectively on the first and second fractional differential equations in (2.1), we obtain where c i ∈ R, i = 1, 2, 3 are arbitrary constants. Using the condition y(a) = 0 in (2.6), we get c 2 = 0. Making use of the conditions px(a) + qy x(s)ds in (2.6) after inserting c 2 = 0 in it leads to the following system of equations in the unknown constants c 1 and c 3 : Solving (2.7) and (2.8) for c 1 and c 3 and using the notation (2.5), we find that Inserting the values of c 1 , c 2 , and c 3 in (2.6) leads to the solution (2.2) and (2.3). One can obtain the converse of the lemma by direct computation. This completes the proof.

Main results
Let In view of Lemma 2.1, we define an operator T : X × X → X by: where (X × X, (x, y) ) is a Banach space equipped with norm (x, y) = x + y , x, y ∈ X, For computational convenience we put: Our first existence result for the system (1.1) relies on Leray-Schauder alternative [54].
Then there exists at least one solution for the system (1.
Let (x, y) ∈ P with (x, y) = νT (x, y). For any t ∈ [a, b], we have x(t) = νT 1 (x, y)(t), y(t) = νT 2 (x, y)(t). Then by (H 1 ) we have In consequence of the above inequalities, we deduce that which imply that Hence the set P is bounded. As the hypothesis of Leray-Schauder alternative [54] is satisfied, we conclude that the operator T has at least one fixed point. Thus the problem (1.1) has at least one solution on [a, b].
By using Banach's contraction mapping principle we prove in the next theorem the existence of a unique solution of the system (1.1).

A variant problem
In this section, we consider a variant of the problem (1.1) in which the nonlinearities ϕ and ψ do not depend on x and y respectively. In precise terms, we consider the following problem: x(s)ds, a < σ 1 < σ 2 < . . . < σ m < τ < . . . < b, where ϕ, ψ : [a, b] × R → R are given functions. Now we present the existence and uniqueness results for the problem (4.1). We do not provide the proofs as they are similar to the ones for the problem (1.1).

Conclusions
We studied the solvability of a coupled system of nonlinear fractional differential equations of different orders supplemented with a new set of nonlocal multi-point integral boundary conditions on an arbitrary domain by applying the tools of modern functional analysis. We also presented the existence results for a variant of the given problem containing the nonlinearities depending on the cross-variables (unknown functions). Our results are new not only in the given configuration but also yield some new results by specializing the parameters involved in the problems at hand. For example, by taking δ i = 0, i = 1, 2, . . . , m in the obtained results, we obtain the ones associated with the coupled systems of fractional differential equations in (1.1) and (4.1) subject to the boundary conditions:  Furthermore, the methods employed in this paper can be used to solve the systems involving fractional integro-differential equations and multi-term fractional differential equations complemented with the boundary conditions considered in the problem (1.1).