Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

Zhang, Haiyan; Wang, Yongqing; Xu, Jiafa

doi:10.1186/s13662-020-02860-1

Cited by 8 publications

(3 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Several models have been utilized to develop Optical Solitons (OSs) for communication purposes. The Nonlinear Schrödinger’s equation (NLSE) and its variant forms have been extensively employed in dispersive mediums across various scopes of applied mathematics and physics 1 – 5 . Solitons emerge due to a delicate equilibrium between group velocity dispersion (GVD) and nonlinearity within the medium.…”

Section: Introductionmentioning

confidence: 99%

Unveiling optical soliton solutions and bifurcation analysis in the space–time fractional Fokas–Lenells equation via SSE approach

Refaie Ali,

Alam,

Parven

2024

Sci Rep

View full text Add to dashboard Cite

The space–time fractional Fokas–Lenells (STFFL) equation serves as a fundamental mathematical model employed in telecommunications and transmission technology, elucidating the intricate dynamics of nonlinear pulse propagation in optical fibers. This study employs the Sardar sub-equation (SSE) approach within the STFFL equation framework to explore uncharted territories, uncovering a myriad of optical soliton solutions (OSSs) and conducting a thorough analysis of their bifurcations. The discovered OSSs encompass a diverse array, including bright-dark, periodic, multiple bright-dark solitons, and various other types, forming a captivating spectrum. These solutions reveal an intricate interplay among bright-dark solitons, complex periodic sequences, rhythmic breathers, coexistence of multiple bright-dark solitons, alongside intriguing phenomena like kinks, anti-kinks, and dark-bell solitons. This exploration, built upon meticulous literature review, unveils previously undiscovered wave patterns within the dynamic framework of the STFFL equation, significantly expanding the theoretical understanding and paving the way for innovative applications. Utilizing 2D, contour, and 3D diagrams, we illustrate the influence of fractional and temporal parameters on these solutions. Furthermore, comprehensive 2D, 3D, contour, and bifurcation analysis diagrams scrutinize the nonlinear effects inherent in the STFFL equation. Employing a Hamiltonian function (HF) enables detailed phase-plane dynamics analysis, complemented by simulations conducted using Python and MAPLE software. The practical implications of the discovered OSS solutions extend to real-world physical events, underlining the efficacy and applicability of the SSE scheme in solving time–space nonlinear fractional differential equations (TSNLFDEs). Hence, it is crucial to acknowledge the SSE technique as a direct, efficient, and reliable numerical tool, illuminating precise outcomes in nonlinear comparisons.

show abstract

Section: Introductionmentioning

confidence: 99%

Unveiling optical soliton solutions and bifurcation analysis in the space–time fractional Fokas–Lenells equation via SSE approach

Refaie Ali,

Alam,

Parven

2024

Sci Rep

View full text Add to dashboard Cite

show abstract

“…Numerous models have been employed to develop OSs to be used in the process of communication. Nonlinear Schrödinger's equation (NLSE) and its variant forms are implemented in dispersive mediums in various scopes of applied mathematics, physics and have been explored mathematically [1,2,3,4,5]. Solitons happen because of a perfect balance among group velocity dispersion (GVD) and nonlinearity in the topic.…”

Section: Introductionmentioning

confidence: 99%

Exploring Optical Soliton Solutions and Bifurcation Analysis in the Space- Time Fractional Fokas-Lenells (STFFL) Equation with Sardar Sub-Equation (SSE) Approach

Ali,

Alam,

Parven

2023

Preprint

View full text Add to dashboard Cite

The space-time fractional Fokas-Lenells (STFFL) equation is a mathematical model used in telecommunications and transmission technology to describe complex phenomena. It explains nonlinear pulse propagation in optical fibers. In this study, we use the Sardar sub-equation (SSE) approach with the STFFL equation to find unexplored optical soliton solutions (OSSs) and analyze their bifurcations. These OSSs include bright-dark, periodic, multiple bright-dark solitons, and various types. We use 2D, contour, and 3D diagrams to illustrate the impact of fractional and time parameters on these solutions. Additionally, we create 2D, 3D, contour, and bifurcation analysis diagrams to examine the nonlinear effects of the STFFL equation. We establish a Hamiltonian function (HF) for phase plane dynamics analysis and conduct simulations using Python and MAPLE software. The discovered OSS solutions have practical implications for real-world physical events. This study demonstrates the utility and reliability of the SSE scheme for solving time-space nonlinear fractional differential equations (TSNLFDEs).

show abstract

“…The technique of upper and lower solutions combined with the monotone iterative approach utilized effectively to derive sufficient conditions about the existence and uniqueness of solutions for the differential equations of integer order [20,21,22] and fractional order [23,24,25,26,27,28] subject to initial or boundary condition. The primary thought behind this method is to construct the monotonic sequences for the corresponding differential equations, where the initial approximations taken are upper and lower solutions.…”

Section: Introductionmentioning

confidence: 99%

On the Boundary Value Problems of Ψ -Hilfer Fractional Differential Equations

Mali,

Kucche

2020

Preprint

View full text Add to dashboard Cite

In the current paper, we derive the comparison results for the homogeneous and non-homogeneous linear initial value problem (IVP) for Ψ-Hilfer fractional differential equations. In the presence of upper and lower solutions, the obtained comparison results and the location of roots theorem utilized to prove the existence and uniqueness of the solution for the linear Ψ-Hilfer boundary value problem (BVP) through the linear non-homogeneous Ψ-Hilfer IVP. Assuming the existence of lower solution w 0 and upper solution z 0 , we establish the existence of minimal and maximal solutions for the nonlinear Ψ-Hilfer BVP in the line segment [w 0 , z 0 ] of the weighted space C 1− γ; Ψ J, R . Further, it demonstrated that the iterative Picard type sequences that began with lower and upper solutions respectively converges to a minimal and maximal solutions, and that started with any point on a line segment converge to the exact solution of nonlinear Ψ-Hilfer BVP. Finally, an example is provided in support of the main results we acquired.

show abstract

Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

Cited by 8 publications

References 42 publications

Unveiling optical soliton solutions and bifurcation analysis in the space–time fractional Fokas–Lenells equation via SSE approach

Unveiling optical soliton solutions and bifurcation analysis in the space–time fractional Fokas–Lenells equation via SSE approach

Exploring Optical Soliton Solutions and Bifurcation Analysis in the Space- Time Fractional Fokas-Lenells (STFFL) Equation with Sardar Sub-Equation (SSE) Approach

On the Boundary Value Problems of Ψ -Hilfer Fractional Differential Equations

Contact Info

Product

Resources

About