Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph

Mehandiratta, Vaibhav; Mehra, Mani; Leugering, Günter

doi:10.3934/nhm.2021003

Cited by 14 publications

(5 citation statements)

References 54 publications

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“…Given its wide-ranging applications, studying graphs holds significant meaning. In recent years, significant progress has been made in both theoretical and practical aspects in the study of fractional differential equations on graphs [16][17][18][19][20][21][22][23]. Numerous scholars have dedicated their efforts to exploring mathematical models on graphs, describing them through ordinary differential equations or fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…A study was conducted by Mehandiratta et al [16] in 2021, focusing on fractional calculus applied to metric graph. The researchers examined a nonlinear fractional boundary value problem on a graph that included cycles (see Figure 1), and they successfully obtained results related to its existence and stability analysis:…”

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} left \\ array_{C} D_{0, t}^{γ} v_{j} (t) = g_{j} (t, v_{j} (t),_{C} D_{0, t}^{α} v_{j} (t)), 0 < t < π, j = 1,2, \\ array_{C} D_{0, x}^{γ} v_{3} (x) = g_{3} (x, v_{3} (x),_{C} D_{0, t}^{α} v_{3} (x)), 0 < x < l, \\ array v_{1} (0) = v_{2} (0) = v_{3} (0), v_{1} (π) = v_{2} (π), \\ array v_{1}^{^{'}} (0) + v_{2}^{^{'}} (0 ... \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

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Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

Wang,

Yuan

2024

Math Methods in App Sciences

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The present study examines the Langevin coupled boundary value problem with the fractional conformable derivative on the ethane graph. The main objective of this research is to establish the existence and uniqueness of solutions with the help of the fixed point theorem and to analyze the Ulam stability of the corresponding problem. Several examples are provided to demonstrate the obtained results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} left \\ array_{C} D_{0, t}^{γ} v_{j} (t) = g_{j} (t, v_{j} (t),_{C} D_{0, t}^{α} v_{j} (t)), 0 < t < π, j = 1,2, \\ array_{C} D_{0, x}^{γ} v_{3} (x) = g_{3} (x, v_{3} (x),_{C} D_{0, t}^{α} v_{3} (x)), 0 < x < l, \\ array v_{1} (0) = v_{2} (0) = v_{3} (0), v_{1} (π) = v_{2} (π), \\ array v_{1}^{^{'}} (0) + v_{2}^{^{'}} (0 ... \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

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Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

Wang,

Yuan

2024

Math Methods in App Sciences

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“…Then they established existence and uniqueness results by a fixed point theory. This work was extended to the existence and uniqueness of solutions of a nonlinear fractional boundary value problem on a circular ring with an attached edge in [28] by the same authors. The results are achieved by the Banach contraction principle and Krasnoselskii's fixed point theorem.…”

Section: Introduction and Problem Settingmentioning

confidence: 99%

Optimal control problems of parabolic fractional sturm liouville equations in a star graph

Leugering¹,

Mophou²,

Moutamal³

et al. 2021

Preprint

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In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm-Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler-Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm-Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.

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“…Then they established existence and uniqueness results by a fixed point theory. This work was extended to the existence and uniqueness of solutions of a nonlinear fractional boundary value problem on a circular ring with an attached edge in [31] by the same authors. The results are achieved by the Banach contraction principle and Krasnoselskii's fixed point theorem.…”

mentioning

confidence: 99%

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

Leugering¹,

Mophou²,

Moutamal³

et al. 2023

MCRF

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<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

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Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph

Cited by 14 publications

References 54 publications

Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

Optimal control problems of parabolic fractional sturm liouville equations in a star graph

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

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