Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

Liu, Lily Li; Zhang, Xinqiu; Liu, Lishan; Wu, Yonghong

doi:10.1186/s13662-016-0876-5

Cited by 21 publications

(10 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations. Recently, fractional-order differential calculus and its applications have attached much attention [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Constructing Jacobi elliptic doubly periodic solutions of nonlinear PDEs with fractional derivatives is worthy of the study.…”

Section: Resultsmentioning

confidence: 99%

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Zhang

2018

Complexity

View full text Add to dashboard Cite

Under investigation in this paper is a more general time-dependent-coefficient Whitham-Broer-Kaup (tdcWBK) system, which includes some important models as special cases, such as the approximate equations for long water waves, the WBK equations in shallow water, the Boussinesq-Burgers equations, and the variant Boussinesq equations. To construct doubly periodic wave solutions, we extend the generalized F-expansion method for the first time to the tdcWBK system. As a result, many new Jacobi elliptic doubly periodic solutions are obtained; the limit forms of which are the hyperbolic function solutions and trigonometric function solutions. It is shown that the original F-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system, but the novel approach of this paper is valid. To gain more insight into the doubly periodic waves contained in the tdcWBK system, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations.

show abstract

Section: Resultsmentioning

confidence: 99%

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Zhang

2018

Complexity

View full text Add to dashboard Cite

show abstract

“…attracted much attention [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. How to extend the complex multirational exp-function ansatz (4) to such fractional PDEs and the NLS equations with varying coefficients in [25][26][27][28][29][30][31][32][33][34][35][36][37][38] is worthy of study.…”

Section: Complexitymentioning

confidence: 99%

Rational Waves and Complex Dynamics: Analytical Insights into a Generalized Nonlinear Schrödinger Equation with Distributed Coefficients

Zhang

2019

Complexity

View full text Add to dashboard Cite

In this paper, we first present a complex multirational exp-function ansatz for constructing explicit solitary wave solutions, N-wave solutions, and rouge wave solutions of nonlinear partial differential equations (PDEs) with complex coefficients. To illustrate the effectiveness of the complex multirational exp-function ansatz, we then consider a generalized nonlinear Schrödinger (gNLS) equation with distributed coefficients. As a result, some explicit rational exp-function solutions are obtained, including solitary wave solutions, N-wave solutions, and rouge wave solutions. Finally, we simulate some spatial structures and dynamical evolutions of the modules of the obtained solutions for more insights into these complex rational waves. It is shown that the complex multirational exp-function ansatz can be used for explicit solitary wave solutions, N-wave solutions, and rouge wave solutions of some other nonlinear PDEs with complex coefficients.

show abstract

“…So the subject of coupled systems is gaining much attention and importance. There are a large number of articles dealing with the existence or multiplicity of solutions or positive solutions for some nonlinear coupled systems with boundary conditions; for details, see [7,8,10,11,20,21,27,29,32,33,[35][36][37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

2020

View full text Add to dashboard Cite

show abstract

Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

Cited by 21 publications

References 18 publications

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Rational Waves and Complex Dynamics: Analytical Insights into a Generalized Nonlinear Schrödinger Equation with Distributed Coefficients

Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

Contact Info

Product

Resources

About