The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations

Tuz, Münevver

doi:10.3390/sym11010121

Cited by 7 publications

(5 citation statements)

References 24 publications

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“…On the other hand, it is easy to show by a direct calculation that x(t), which is given by (6), verifies the linear ψ-Hilfer FCB model (5) under the nonlinear boundary conditions.…”

Section: Preliminariesmentioning

confidence: 72%

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Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

et al. 2021

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In this paper, we establish sufficient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ-Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s fixed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the fixed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.

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“…On the other hand, it is easy to show by a direct calculation that x(t), which is given by (6), verifies the linear ψ-Hilfer FCB model (5) under the nonlinear boundary conditions.…”

Section: Preliminariesmentioning

confidence: 72%

“…Hence, the solution x(t) follows by applying c 1 and c 2 in (9). This implies that x(t) satisfies (6).…”

Section: Preliminariesmentioning

confidence: 97%

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

et al. 2021

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“…In this work we study the algebraic properties of the Euler-Bernoulli, the Rayleigh and of the Timoshenko-Prescott according to the admitted Lie point symmetries, for the source-free equation as also in the case where a homogeneous source term exists. The application of symmetry analysis for the Euler-Bernoulli equation is not new, there are various studies in the literature [6,12,16,29,32], however in this paper, we obtained some new results, as the reduction of Euler-Bernoulli form to perturbed form of Painlevé-Ince [20] equation, which is integrable and the third-order ode which falls into the category of equations studied by Chazy, Bureau and Cosgrove. Also, we show that the three beam equations of our study admit the same travelling-wave solution.…”

Section: Introductionmentioning

confidence: 97%

Similarity solutions and conservation laws for the Beam Equations: a complete study

Halder¹,

Paliathanasis²,

Leach³

2020

Preprint

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We study the similarity solutions and we determine the conservation laws of the various forms of beam equation, such as, Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of order second and third. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the above mentioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

show abstract

“…In this work, we study the algebraic properties of the Euler-Bernoulli, the Rayleigh and of the Timoshenko-Prescott according to the admitted Lie point symmetries, for the source-free equation as also in the case where a homogeneous source term exists. The application of the symmetry analysis for the Euler-Bernoulli equation is not new, there are various studies in the literature [5][6][7][8][9], however in this paper, we obtained some new results, as the reduction of the Euler-Bernoulli form to a perturbed form of Painlevé-Ince [10] equation, which is integrable and the third-order ode, which falls into the category of equations studied by Chazy, Bureau and Cosgrove. Also, we show that the three beam equations of our study admit the same travelling-wave solution.…”

Section: Introductionmentioning

confidence: 97%

Similarity Solutions and Conservation Laws for the Beam Equations: A Complete Study

Halder

Paliathanasis

2020

Acta Polytech

View full text Add to dashboard Cite

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

show abstract

The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations

Cited by 7 publications

References 24 publications

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

Similarity solutions and conservation laws for the Beam Equations: a complete study

Similarity Solutions and Conservation Laws for the Beam Equations: A Complete Study

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