No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Existence and uniqueness of solutions for a class of fractional differential coupled system with integral boundary conditions
Abstract
In this paper, we prove the existence and uniqueness of solutions for a coupled system of fractional differential equations with integral boundary conditions. Our analysis relies on a generalized coupled fixed point theorem in the space of the continuous functions defined on [0,1]. An example is also presented to illustrate the obtained results.
... In fact, we show that our modified exponential technique is capable of preserving the non-negativity, the boundedness and the monotonicity of the approximations. It is important to point out that these analytical characteristics of the solutions are indeed present in many mathematical models [13,14], especially in some traveling-wave solutions of nonlinear partial differential equations [22,9]. This note is sectioned as follows. ...
In this work, we propose an exponential-type discretization of the well-known Fisher's equation from population dynamics. Only non-negative, bounded and monotone solutions are physically relevant in this note, and the discretization that we provide is able to preserve these properties. The method is a modified explicit exponential technique which has the advantage of requiring a small amount of computational resources and computer time. It is worthwhile to notice that our technique has the advantage over other exponential-like methodologies that it yields no singularities. In addition, the preservation of the properties of non-negativity, boundedness and monotonicity are distinctive features of our method. As consequences of the analytical properties of the technique, the method is capable of preserving the spatial and the temporal monotonicity of solutions. Qualitative and quantitative numerical simulations assess the convergence properties of the finite-difference scheme proposed in this manuscript.
... It should be noted that problems for a class of fractional differential system and for the non-line differential equations with integral conditions was investigated in works [26][27][28][29][30] and BVPs for the mixed type equations involving the Caputo and the Riemann-Liouville fractional differential operators were investigated by many authors, see for instance [31][32][33]. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 BVPs with discounting matching conditions for the loaded equations with fractional derivative have not been investigated yet. ...
This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations.
... These methods often make the dimension of original system increases, forming a new coupled system. In order to understand the ultimate effect of chaos control and synchronization, we not only need to know the dynamic behavior of original system, but also need to discuss the one of new coupled system (see [14]- [18]). ...
In this paper, we establish the existence result for nonlinear fractional differential equations with integral boundary conditions at resonance by means of coincidence degree theory. As applications, three examples are presented to illustrate the main results.
We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions CD alpha u(t) + f(t, u(t)) = 0, 0 < t < 1, u (0) = u ''(0) = 0, u(1) = lambda integral(1)(0) u(s)ds, where 2 < alpha < 3, 0 < lambda < 2 and D-C(alpha) is the Caputo fractional derivative and f : [0, 1] x [0, infinity) -> [0, infinity) is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.
In this paper, we consider the existence of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. Our analysis relies on known Guo-Krasnoselskii fixed point theorem.
This paper is concerned with the second-order singular Sturm-Liouville integral boundary value problems [GRAPHICS] where lambda > 0; h is allowed to be singular at t = 0, 1 and f(t, x) may be singular at x = 0. By using the fixed point theory in cones, an explicit interval for lambda is derived such that for any lambda in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and non-singular cases.
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.
The existence of positive solutions for a class of fractional equations involving Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green’s function, we obtain the existence of positive solution and establish iterative sequence for approximating the solution.
By constructing suitable operators, we investigate the existence of solutions for a coupled system of fractional differential equations with integral boundary conditions at resonance. Our analysis relies on the coincidence degree theory due to Mawhin. An example is given to illustrate our main result.
MSC:
34A08, 70K30, 34B10.
In this paper, we prove the existence and uniqueness of solutions for systems of fractional differential equations with antiperiodic boundary conditions by applying some standard fixed point principles.
Analyses concerning arbitray Banach spaces were conducted. In a Banach space, if X is an arbitrary Banach principles; Introduction of each then a selfmap that satisfies the Banach contraction principle. It was shown that for Hilbert spaces, weakly contractive maps possess a unique fixed point.
This paper studies a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Applying the Schauder fixed point theorem, an existence result is proved for the following system where α,β,p,q,η,γ satisfy certain conditions.
In this work we discuss a boundary value problem for a coupled differential system of fractional order. The differential operator is taken in the Riemann–Liouville sense and the nonlinear term depends on the fractional derivative of an unknown function. By means of Schauder fixed-point theorem, an existence result for the solution is obtained. Our analysis relies on the reduction of the problem considered to the equivalent system of Fredholm integral equations.
Fractional Differential Equations
- I Podlubny
I. Podlubny, Fractional Differential Equations, 198,
Academic Press, New York, USA 1999.
- A A Kilbas
- H M Srivastava
- J J Trujillo
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory
and Applications of Fractional Differential Equations, 204,
North-Holland Mathematics Studies, Elsevier Science B.V.,
Amsterdan, The Netherlands, 2006.