Analytical solutions of fractional wave equation with memory effect using the fractional derivative with exponential kernel

Cuahutenango-Barro, B.; Taneco-Hernández, M.A.; Lv, Yu-Pei; Gómez‐Aguilar, J.F.; Osman, M. S.; Jahanshahi, Hadi; Aly, Ayman A.

doi:10.1016/j.rinp.2021.104148

Cited by 29 publications

(4 citation statements)

References 24 publications

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“…To the best of our knowledge this is a new representation of the step functions, like the Heaviside function. The set formed by these generalised functions, solutions of second-order differential equations in the form of equation (22) in which the source is a Dirac function represent a subspace of ¢ D on which can be further utilised to expand the study for verifying its usefulness in other fields of mathematical physics that are not explicitly mentioned in this work [25][26][27]. The study of this subspace will be part of the continuation of this research work.…”

Section: Concluding Discussionmentioning

confidence: 99%

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Revisiting the fundamental equations of the mathematical physics in the space of generalised functions

Nanni¹

2021

Phys. Scr.

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A broad class of physical processes is described by second-order differential equations. The equation of waves, the diffusion equation and the stationary equation, which are found both in classical and quantum physics, belong to this class. In this article, these equations are reformulated under the assumption that the source determining the physical phenomenon acts instantaneously, using the formalism of generalised functions. Some particular cases of these equations are precisely solved by applying the Fourier transform. The obtained solutions form a subspace of the singular generalised functions’ space and are mentioned by the product between a step function and a function with at most a finite number of discontinuity points.

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Section: Concluding Discussionmentioning

confidence: 99%

“…| | Therefore, recalling equation (27), the solution of equation (31) is calculated by the following equation:…”

Section: ( ) ( )mentioning

confidence: 99%

Revisiting the fundamental equations of the mathematical physics in the space of generalised functions

Nanni¹

2021

Phys. Scr.

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“…Researchers are therefore motivated to present new methods and refine existing approaches. Various significant and powerful methods have been introduced such as Darboux transformation [14], Weierstrass elliptic functions methods [15,16], Bäcklund transformation [17], Lie group [18][19][20][21], Hirota's method [22,23], bifurcation method [24][25][26][27][28][29][30][31][32][33], and for distinct method, as shown in e.g., [34][35][36][37][38][39][40][41][42]. e analytical and numerical solutions for various types of nonlinear partial differential equations were investigated using traditional Lie symmetry approaches; for instance as shown in [43].…”

Section: Introductionmentioning

confidence: 99%

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

Elmandouh

Elbrolosy

2022

Journal of Mathematics

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The Ivancevic option pricing model comes as an alternative to the Black-Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well-known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright-like soliton, kinky-bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D-graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

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“…In the 20th century, fractional calculus, by regarding the historical values of the considered functions according to their order, was adopted as an important tool to model memory effects [2,3]. This resulted in significant and useful real-word applications of wave equations [4], chemical kinetics [5], optimal control of biochemical reactions [6], among many others [7].…”

Section: Introductionmentioning

confidence: 99%

Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels

Zine

Lotfi

Torres

et al. 2022

Axioms

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We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of new relations for nth-weighted generalized fractional integrals and derivatives. As an application, new mean value theorems for generalized weighted fractional operators are obtained. Direct corollaries allow one to obtain the recent Taylor’s and mean value theorems for Caputo–Fabrizio, Atangana–Baleanu–Caputo (ABC) and weighted ABC derivatives.

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Analytical solutions of fractional wave equation with memory effect using the fractional derivative with exponential kernel

Cited by 29 publications

References 24 publications

Revisiting the fundamental equations of the mathematical physics in the space of generalised functions

Revisiting the fundamental equations of the mathematical physics in the space of generalised functions

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels

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