Unique solutions for a new coupled system of fractional differential equations

Zhai, Chengbo; Jiang, Ruiting

doi:10.1186/s13662-017-1452-3

Cited by 215 publications

(15 citation statements)

References 34 publications

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“…In consideration of the fact that the existence of unique solutions or multiple solutions of integral boundary value problem have been widely considered in recent decades as it focused on building diversiform fractional-order models, for more recent results, we can see [3,10,19,20,23,24,[26][27][28]33]. For example, Zhao et al [33] discussed equation (1) under integral boundary condition…”

Section: Introductionmentioning

confidence: 99%

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

Ren

Zhai

2019

NAMC

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In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.

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Section: Introductionmentioning

confidence: 99%

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

Ren

Zhai

2019

NAMC

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“…Recently, the study of coupled systems involving fractional differential equations appeared in the literature [4,9,10,17,18,32,33]. Much of the work has been considered on finite intervals; however, a study of boundary value problems on unbounded domain is well under way.…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, some remarkable results on the existence and multiplicity of solutions for fractional differential equations have been discussed widely on finite intervals [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Instead, it is relatively rare for work to be done related to existence results on infinite intervals [34][35][36][37][38][39][40][41][42][43][44][45][46][47].…”

Section: Introductionmentioning

confidence: 99%

A coupled system of fractional differential equations on the half-line

Zhai

Ren

2019

Bound Value Probl

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In this paper, we consider a new fractional differential system on an unbounded domain D α u(t) + ϕ(t, v(t), D γ 1 v(t)) = 0, t ∈ [0, +∞), α ∈ (2, 3], D β v(t) + ψ(t, u(t), D γ 2 u(t)) = 0, t ∈ [0, +∞), β ∈ (2, 3], subject to the conditions I 3-α u(t)| t=0 = 0, D α-2 u(t)| t=0 = h 0 g 1 (s)u(s) ds, D α-1 u(+∞) = Mu(ξ) + a, I 3-β v(t)| t=0 = 0, D β-2 v(t)| t=0 = h 0 g 2 (s)v(s) ds, D β-1 v(+∞) = Nv(η) + b. The nonlinear terms ϕ and ψ are dependent on the fractional derivative of lower order γ i ∈ (0, 1), i = 1, 2, which creates additional complexity to verify the existence of solutions. Moreover, a proper choice of Banach space allows the solutions to be defined on the half-line. From some standard fixed point theorems, sufficient conditions for the existence and uniqueness of solutions to boundary value problems are developed. Finally, the main result is applied to an illustrative example.

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“…FPDEs have gained a great significance and popularity over the last few years for their powerful potential applications, mainly in mathematical physics, biology, and engineering [3][4][5]. Recently, researchers are taking interest to investigate the exact solutions of nonlinear FPDEs [6][7][8][9][10][11]. Many effective methods have been introduced in order to acquire exact solutions of nonlinear FPDEs such as hyperbolic function method [12], extended hyperbolic tangent method [13][14][15][16], the sub-equation method [17], homotopy perturbation technique [18], exponential rational function method [19], and homotopy analysis method [20].…”

Section: Introductionmentioning

confidence: 99%

New exact solutions of fractional Cahn–Allen equation and fractional DSW system

2018

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This work explores the new exact solutions of nonlinear fractional partial differential equations (FPDEs). The solutions are obtained by adopting an effective technique, the first integral method (FIM). The Riemann-Liouville (R-L) derivative and conformable derivative definitions are used to deal with fractional terms in FPDEs. The proposed method is applied to get exact solutions for space-time fractional Cahn-Allen equation and coupled space-time fractional (Drinfeld's Sokolov-Wilson system) DSW system. The suggested technique is easily applicable and effectual, which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.

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Unique solutions for a new coupled system of fractional differential equations

Abstract: In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions

Cited by 215 publications

References 34 publications

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

A coupled system of fractional differential equations on the half-line

New exact solutions of fractional Cahn–Allen equation and fractional DSW system

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