New Challenges Arising in Engineering Problems with Fractional and Integer Order

Baskonus, Hacı Mehmet; Ruiz, Luis Manuel Sánchez; Ciancio, Armando

doi:10.3390/fractalfract5020035

Cited by 9 publications

(9 citation statements)

References 12 publications

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“…The theory of differential equations in abstract Banach spaces has been established by some authors from different viewpoints, for example, [24][25][26][27]. Here we have proved the existence of solutions x ∈ C(I, E) for the delay functional integral Equation (1) and its corresponding initial value problem (2) and (3) which have been studied in a reflexive Banach spaceE.…”

Section: Discussionmentioning

confidence: 80%

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

Elsayed

Omar

2022

Mathematics

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Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved.

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Section: Discussionmentioning

confidence: 80%

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

Elsayed

Omar

2022

Mathematics

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“…The local fractional partial derivative of ψ(ζ, θ) at θ = θ 0 can be defined in a similar way. The reader can be referred to [35][36][37][38][39] for more details about the local fractional calculus.…”

Section: Preliminariesmentioning

confidence: 99%

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

et al. 2022

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In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

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“…These highlights permit us to show wonders transversely different reality scales without having to break the problem down into smaller and smaller sections. The spectrum of a FO model's applicability on compound scales is constructed on the premise that fractional derivatives can produce key features of complex systems [15][16][17][18][19][20]. Additionally, the behaviour of certain derivatives is local, i.e., employing conventional approaches we can predict variations in neighbourhood of a point, however, we can state variations in interval when fractional derivative is taken into consideration.…”

Section: Fractional Calculus In Real-life Phenomenamentioning

confidence: 99%

New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method

et al. 2021

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In the present research, we established an efficient and novel algorithm for time fractional multidimensional partial differential equations arising from physics and engineering. Taking into account Caputo fractional derivative, this algorithm involves the fractional natural decomposition method ( ) FNDM , and the nonlinearity term decayed by utilizing the aforesaid method. The solution of the model is based on time dependent fractional-order equations such as ( )iii Sine-Gordon equation ( ) iv wave-like equations. By considering the FNDM algorithm, the analytic solutions of the aforesaid models are analyzed. The suggested approach employs to solve several models of real-world phenomena and the consequences demonstrate that the approach is reliable, explicit and viable. Moreover, closed form solutions are established in many cases, and exact solutions are derived in particular. Numerical simulations were carried out to ensure that the proposed methods are perfect and precise in terms of efficiency and effectiveness, as shown by the exact solutions resolving complex nonlinear problems. The comparative analysis for the projected method reveals innovative attributes of the hybrid fractional derivative in the discussed model.

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New Challenges Arising in Engineering Problems with Fractional and Integer Order

Abstract: Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...]

Cited by 9 publications

References 12 publications

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method

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