Research

42 pages, 2461 KiB

Open AccessArticle

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

by Donal O’Regan, Safoura Rezaei Aderyani, Reza Saadati and Mustafa Inc

Axioms 2024, 13(2), 92; https://doi.org/10.3390/axioms13020092 - 30 Jan 2024

Cited by 1 | Viewed by 748

In this paper, we apply two different methods, namely, the

\frac{G^{'}}{G}

-expansion method and the

\frac{G^{'}}{G^{2}}

-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation [...] Read more.

In this paper, we apply two different methods, namely, the

\frac{G^{'}}{G}

-expansion method and the

\frac{G^{'}}{G^{2}}

-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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14 pages, 9095 KiB

Open AccessArticle

An Approach for Approximating Analytical Solutions of the Navier-Stokes Time-Fractional Equation Using the Homotopy Perturbation Sumudu Transform’s Strategy

by Sajad Iqbal and Francisco Martínez

Axioms 2023, 12(11), 1025; https://doi.org/10.3390/axioms12111025 - 31 Oct 2023

Viewed by 819

Abstract

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the [...] Read more.

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the SuT with the homotopy perturbation method using He’s polynomials. This approach proves to be powerful and practical for solving various linear and nonlinear fractional partial differential equations (FPDEs) in scientific and engineering fields. We demonstrate the efficiency and simplicity of this method through examples, showcasing its ability to approximate solutions for FPDEs. Additionally, we compare the numerical results obtained using this technique for different values of alpha, showing that as the value moves from a fractional order to an integer order, the solution becomes increasingly similar to the exact solution. Furthermore, we provide the tabular representations of the solution for each example. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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32 pages, 22977 KiB

Open AccessArticle

Exact Solutions of the Stochastic Conformable Broer–Kaup Equations

by Humaira Yasmin, Yusuf Pandir, Tolga Akturk and Yusuf Gurefe

Axioms 2023, 12(9), 889; https://doi.org/10.3390/axioms12090889 - 18 Sep 2023

Viewed by 811

Abstract

In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist [...] Read more.

In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist of hyperbolic, trigonometric, rational trigonometric, rational hyperbolic, and rational function solutions, respectively. This shows that the proposed methods are competent and sufficient. In addition, it is aimed to better understand the physical properties by drawing two- and three-dimensional graphics of the exact solutions according to different parameter values. When these exact solutions obtained by two different methods are compared with the solutions attained by other methods, it can be said that these two methods are competent. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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17 pages, 330 KiB

Open AccessArticle

Existence and Asymptotic Stability of the Solution for the Timoshenko Transmission System with Distributed Delay

by A. Braik, Kh. Zennir, E. I. Hassan, A. H. A. Alfedeel and Safa M. Mirgani

Axioms 2023, 12(9), 833; https://doi.org/10.3390/axioms12090833 - 28 Aug 2023

Viewed by 638

Abstract

In the present paper, a transmission problem of the Timoshenko beam in the presence of distributed delay is considered. Under appropriate assumptions, we prove the well-posedness by using the semi-group theory. Furthermore, we study the asymptotic behavior of solutions using the multiplier method. [...] Read more.

In the present paper, a transmission problem of the Timoshenko beam in the presence of distributed delay is considered. Under appropriate assumptions, we prove the well-posedness by using the semi-group theory. Furthermore, we study the asymptotic behavior of solutions using the multiplier method. We investigate the techniques and ideas used by the second author to extend the recent results. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

18 pages, 595 KiB

Open AccessArticle

A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator

by Abdul Hamid Ganie, Fatemah Mofarreh and Adnan Khan

Axioms 2023, 12(6), 609; https://doi.org/10.3390/axioms12060609 - 19 Jun 2023

Cited by 9 | Viewed by 955

Abstract

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves [...] Read more.

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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13 pages, 542 KiB

Open AccessArticle

An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order p

by Lazhar Bougoffa, Smail Bougouffa and Ammar Khanfer

Axioms 2023, 12(5), 497; https://doi.org/10.3390/axioms12050497 - 19 May 2023

Cited by 2 | Viewed by 999

Abstract

Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem [...] Read more.

Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem

\frac{d}{d x} [(1 + δ_{1} y^{p}) \frac{d y}{d x}] + 2 x (1 + δ_{2} y^{p}) \frac{d y}{d x} = \frac{4}{S t e} β (x), 0 < x < λ,

with

y (0) = 1, y (λ) = 0,

where

λ > 0

is a solution to the nonlinear equation

y^{'} (λ) = - \frac{2 λ}{S t e}

, where

δ_{i} > - 1, i = 1, 2,

p > 0,

and

S t e

is the Stefan number, which represents a phase-change problem with a nonlinear temperature-dependent thermal parameters (i.e., thermal conductivity and specific heat) on

(0, λ)

. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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10 pages, 707 KiB

Open AccessArticle

Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative

by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed

Axioms 2023, 12(5), 466; https://doi.org/10.3390/axioms12050466 - 12 May 2023

Cited by 15 | Viewed by 1071

Abstract

In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, [...] Read more.

In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, we generalize some previous results. As the Boiti–Leon–Manna–Pempinelli equation is a model for an incompressible fluid, the solutions obtained may be utilized to represent a wide variety of fascinating physical phenomena. We construct a large number of 2D and 3D figures to demonstrate the impact of the M-truncated derivative on the exact solution of the BLMPE-MTD. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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10 pages, 357 KiB

Open AccessArticle

Analysis of Generalized Bessel–Maitland Function and Its Properties

by Talha Usman, Nabiullah Khan and Francisco Martínez

Axioms 2023, 12(4), 356; https://doi.org/10.3390/axioms12040356 - 05 Apr 2023

Cited by 1 | Viewed by 1264

Abstract

In this article, we introduce the generalized Bessel–Maitland function (EGBMF) using the extended beta function and some important properties obtained. Thus, we first show interesting relationships of this function with Laguerre polynomials and the Whittaker functions. We also introduce and prove some properties [...] Read more.

In this article, we introduce the generalized Bessel–Maitland function (EGBMF) using the extended beta function and some important properties obtained. Thus, we first show interesting relationships of this function with Laguerre polynomials and the Whittaker functions. We also introduce and prove some properties of the derivatives associated with EGBMF. In this sense, we establish a result relative to the extended fractional derivatives of Riemann–Liouville. Furthermore, the Mellin transform of this function is evaluated in terms of the generalized Wright hypergeometric function, and its Euler transform is also obtained. Finally, we derive several graphical representations using the Gauss quadrature and the Laguerre–Gauss quadrature methods, which show that the numerical and theoretical simulations are consistent. The results derived from this research can be potentially useful in applications in several fields, in particular, physics, applied mathematics, and engineering. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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15 pages, 317 KiB

Open AccessArticle

Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

by Benoumran Telli, Mohammed Said Souid, Jehad Alzabut and Hasib Khan

Axioms 2023, 12(4), 339; https://doi.org/10.3390/axioms12040339 - 30 Mar 2023

Cited by 16 | Viewed by 1132

Abstract

This study establishes the existence and stability of solutions for a general class of Riemann–Liouville (RL) fractional differential equations (FDEs) with a variable order and finite delay. Our findings are confirmed by the fixed-point theorems (FPTs) from the available literature. We transform the [...] Read more.

This study establishes the existence and stability of solutions for a general class of Riemann–Liouville (RL) fractional differential equations (FDEs) with a variable order and finite delay. Our findings are confirmed by the fixed-point theorems (FPTs) from the available literature. We transform the RL FDE of variable order to alternate RL fractional integral structure, then with the use of classical FPTs, the existence results are studied and the Hyers–Ulam stability is established by the help of standard notions. The approach is more broad-based and the same methodology can be used for a number of additional issues. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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