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The unstable time fractional Schrödinger model (UTFSM) is studied through the development of disturbances in marginally stable or unstable media. A modified Sardar-sub equation technique (MSSE) for a conformable fractional-order nonlinear evolution model is presented in this paper. The objective here is to construct new wave solutions for UTFSM. These solutions have particular relevance in quantum physics and assume several forms such as rational, exponential, trigonometric and hyperbolic functions, as well as combo solutions. This technique produces various shapes of dark, kink-type solitons and periodic solitary waves by setting proper parametric values. These discrete physical frameworks contribute to an understanding from analysis of unstable dynamical models. Additionally, we investigate modulation instability and stability analysis to ensure that obtained solutions are highly stable. The multiplicity of waves and solutions emphasizes how this technique can be used for different nonlinear fractional models in quantum physics and other areas. [ABSTRACT FROM AUTHOR]

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod's material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work. [ABSTRACT FROM AUTHOR]

The main objective of this study is to find novel wave solutions for the time-fractional generalized Rosenau-Kawahara-RLW equation, which occurs in unidirectional water wave propagation. The generalized Rosenau-Kawahara-RLW equation comprises three equations Rosenau equation, Kawahara equation, RLW equation and also p-th order nonlinear term. All these equations describe the wave phenomena especially the wave-wave and wave-wall interactions in shallow and narrow channel waters. The auxiliary equation method is employed to get the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-relativistic limit of the Dirac equation for relativistic spin-1/2 particles within the framework of the conformable fractional derivative (CFD) using the Foldy–Wouthuysen (FW) transformation. This approach leads to the derivation of a conformable fractional Schrödinger–Pauli equation. We propose and employ a conformable fractional version of the FW transformation, thoroughly examining its efficacy and behavior in the non-relativistic limit. Additionally, based on perturbation theory, we compute the energy shifts within the context of CFD and derive a conformable fractional fine structure of the hydrogen spectrum. [ABSTRACT FROM AUTHOR]

Summary: In this article, we study parameter estimation problems for fractional commensurate Hammerstein systems utilizing the conformable fractional derivative. Two algorithms are investigated: first, the Poisson moment functions (PMF) method, aiming to transfer the fractional derivative of the measurement signal into PMF using the fractional Laplace transform and convolution; second, a proposed new instrumental variable algorithm, which is based on the conformable fractional derivative. Both algorithms have been analyzed and shown to be consistent. A comprehensive complexity analysis is provided for each algorithm. Furthermore, a kind of special time‐varying systems are discussed under the conformable fractional derivative. Finally, an example is given to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains. [ABSTRACT FROM AUTHOR]

The article explores a linear set-valued differential equation featuring both conformable fractional and generalized conformable fractional derivatives. It presents conditions for the existence of solutions and provides analytical expressions for the shape of solution sections at different time points. Model examples are employed to illustrate the results.

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation. [ABSTRACT FROM AUTHOR]

This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering. [ABSTRACT FROM AUTHOR]

This paper strives to investigate the time fractional system that characterizes the ion sound wave influenced by the ponderomotive force induced by a high‐frequency field, as well as the Langmuir wave in plasma. Initially, based on the qualitative theory for planar integrable systems, four‐phase portraits are found in the (u,y)$$ \left(u,y\right) $$ phase plane under certain conditions on the physical parameters. These conditions are used to prove analytically the existence of solitary, kink (anti‐kink), periodic, super‐periodic, and unbounded wave solutions. The correspondence between the energy levels, phase orbits, and consequently the type of the solution is announced. We derived the bounded wave solutions associated with the phase orbits, which are shown to be consistent with the qualitative analysis of the types of solutions. Moreover, we studied the consistency between the obtained solutions by investigating the degeneracy of the solutions through the transmission between the phase orbits, or equivalently, through the dependence on the initial conditions. With the presence of perturbed periodic terms, the quasi‐periodic behavior and chaotic patterns are investigated. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Chiral nonlinear Schrödinger equation with time-dependent coefficients (FCNSE-TDCs) is considered. The mapping method is applied in order to get hyperbolic, elliptic, trigonometric and rational fractional solution. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as optics, plasma physics and nonlinear quantum mechanics. Finally, the influence of the time-dependent coefficients and the conformable fractional derivative order on the exact solutions of the FCNSE-TDCs is presented. [ABSTRACT FROM AUTHOR]

The article introduces a generalization of the Khalouta transform to conformable fractional order. Topics include the derivation and proof of fundamental properties and theorems related to the conformable fractional Khalouta transform, its application in solving fractional differential equations to demonstrate its efficiency and validity, and its relevance in modeling complex phenomena across fields like fluid dynamics, quantum mechanics, and wave theory.

The objective of this paper is to study the existence, uniqueness and stability of solutions for some nonlocal impulsive evolution problems involving conformable fractional derivative. The main results are based on fixed point theorems combined with the semigroup theory. As applications, we give an example to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of space. This exploration involved the utilization of Jensen's, Gröenwall-Bellman's, Hölder's, Burkholder-Davis-Gundy's inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes.

This paper discusses how to derive multidimensional conformable diffusion equation from continuous time random walks process by employing fractional Laplace transform and the conformable fractional derivative defined by Khalil et al in [10]. The results show that there exists intimate relationship between the stretched exponential function and conformable diffusion equation involving the conformable fractional derivative. The relationship is analogous to the relationship between the exponential function and usual diffusion equation involving usual derivative and the relationship between the Mittag-Leffler function and subdiffusion or slow diffusion equation involving Caputo fractional derivative. Multidimensional conformable FokkerPlanck equation is also derived here. The equation describes diffusion process influenced by external force fields. Conformable semigroup associated with the solution to the conformable diffusion equation has similar properties to the properties of the semigroup associated with the solution to usual diffusion except the semigroup property. For sufficiently large time, the graphs of the solution to the conformable diffusion equation and the subdiffusion equation involving Caputo fractional derivative are sufficiently close. Especially, for, the graphs are sufficiently close for . In presence of nonzero constant external force fields, the particle involved in the diffusion process moves faster toward the directions of the external force fields. [ABSTRACT FROM AUTHOR]

This study examines the analytic wave solutions of a highly dispersive perturbed complex Ginzburg–Landau equation (CGLE) with conformable fractional derivative and polynomial law of nonlinearity using the improved modified extended tanh-function method. The results show a wide range of solutions including (bright, dark, singular) solitons, Jacobi elliptic solutions, exponential solutions, and Weierstrass elliptic solutions. The obtained soliton solutions showcase diverse dynamics, encompassing different solitary waves and localized structures. The polynomial nonlinearity adds complexity to the dynamics, resulting in the emergence of new solitons with distinct characteristics. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

In this paper, a numerical approach is proposed for solving the one-dimensional space-time-fractional telegraph equations with variable coefficients and Robin's boundary conditions. The fractional derivatives are described in the conformable sense. Based on the Legendre collocation method, our problem is reduced to a linear system of second order differential equations and the Generalized-α method is applied to solve this system. With five examples, we present a comparative study between our algorithm and some numerical methods available in the literature. This algorithm gives an excellent approximation with a small number of collocation points. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using L2 and L∞ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model. [ABSTRACT FROM AUTHOR]

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

This study employs the Extended Kudryashov method to address the fractional-order generalized Burgers–Huxley equation, the fractional-order generalized Huxley equation, the fractional-order generalized Burgers equation, and the fractional-order generalized Fisher equation. Through this approach, exact solutions for these equations are derived. Furthermore, we visualize the obtained results in 3D using Maple 2019.2. The outcomes underscore the effectiveness and reliability of the Extended Kudryashov method as a clear and efficient tool for solving the specified equations. [ABSTRACT FROM AUTHOR]

The multi-scroll or multi-wing attractor generated by the chaotic system has a complicated structure. Based on the definition of conformal fractional-order, the Adomian decomposition method is used to solve the fractional-order multi-wing chaotic system. It is worth noting that this system has been found to have at least six multi-wing chaotic attractors with different shapes. The system parameters are used as bifurcation control parameters, and multiple bifurcation behaviors and multi-wing attractors can be observed. The results show that the 0.9-order chaotic system still has multiple stable phenomena, and some typical coexistence phenomena such as period and period, period and chaos, and chaos and chaos can all be found. Finally, an analog circuit was built on the Multisim simulation software, and the chaotic system successfully carry out on the DSP board. This research gives guidance on applying the conformable fractional-order theory to nonlinear systems. [ABSTRACT FROM AUTHOR]

Dengue fever is an intense feverish virus related disease dispatched with the nibble of Aedes mosquitoes carrying one of the four serotypes, which are symbolized as DEN-1, DEN-2, DEN-3, DEN-4. Almost fifty two percent people of our planet is at risk due to the dengue fever. Particularly, people of the tropical and subtropical countries as like Bangladesh, India, Pakistan, Nepal, Bhutan, Malaysia, etc. are in very risky position due to this fever. In Bangladesh, dengue fever happens nationally and has been endemic for more than two decades. Therefore, for realizing the dynamical conduct of this fever a perfect scientific model of disease transmission plays a noteworthy role and help us to prevent this disease effectively. From this context, in this article a novel six compartmental ASIR dengue transmission model have been established by using conformable fractional order derivatives. In the proposed model, the dynamics of human and mosquito populations have been constructed with six compartments. Here, the analytical stability of equilibria has been shown by disease free and endemic equilibrium points. Furthermore, a sensitivity analysis of the proposed model has been executed to evaluate the comparative rank of the model parameters for dengue transmission. The data of infected human population has been accumulated from different health institutions of Bangladesh which have been used to compute the infection rate of dengue cases in Bangladesh. Finally, a numerical simulation has been constructed for studying the dynamical behavior of dengue transmission with the help of Conformable fractional differential transformation technique. [ABSTRACT FROM AUTHOR]

This paper deals with some existence of solutions for some classes of coupled systems of conformable fractional differential equations with initial and boundary conditions in Banach and Fréchet spaces. Our results are based on some fixed point theorems. Some illustrative examples are presented in the last section. [ABSTRACT FROM AUTHOR]

This study investigates the application of the novel Kudryashov approach to a time-fractional nonlinear Schrödinger model featuring second-order spatiotemporal and group velocity dispersion coefficients. Various exact solutions for this model in optical fibers are established, utilizing hyperbolic and exponential functions. These solutions encompass diverse optical solitons, such as bright, singular, bell-shaped, mixed dark-bright, dark-bright, and wave solitons. To assess the significance of the time-fractional nonlinear Schrödinger model and illustrate the different forms of these innovative optical solutions, contour plots, three-dimensional plots, and two-dimensional plots are presented. Furthermore, the influence of the conformable fractional order derivative on a specific category of the new optical solutions is explored through illustrative graphs, emphasizing the impact of fractional parameters. The primary objective of this paper is to elucidate the significant influence of the conformable fractional derivative parameter on the Schrödinger equation, particularly in shaping various physical aspects of signal propagation in optical fiber. Understanding and manipulating this parameter provide opportunities for optimizing optical fiber systems for specific applications. Moreover, the proposed technique demonstrates its reliability as a tool for examining analytical solutions of fractional differential equations. The introduced Schrödinger model holds potential applications in the transmission of ultra-fast pulses through optical fibers. [ABSTRACT FROM AUTHOR]

This work explores the effect of fractional derivative on the fourth-order nonlinear Schrödinger equation with Kerr law nonlinearity, a highly significant equation in the study of wave propagation in dispersive media. By employing the improved modified extended tanh-function method, a variety of optical soliton solutions are derived. These solutions including dark solitons, bright solitons, and singular solitons. Moreover, singular periodic solutions and exponential solutions are raised. These solutions offer valuable insights into the dynamic behavior of nonlinear wave phenomena. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. Bright and dark solitons, pivotal components of our findings, play a critical role in fiber optics by facilitating the transmission of high-power optical signals with exceptional attributes such as shape preservation. These properties eliminate the need for external pulse compression, simplifying the design and operation of optical systems. The outcomes of this study contribute in advancing our knowledge of wave propagation in dispersive media and have practical implications for the development of efficient and robust optical communication technologies. [ABSTRACT FROM AUTHOR]

This article examines the nonlinear (2+1) complex conformable time fractional nonlinear Schrödinger equation and the soliton solutions that may be found by using the improved F-expansion method. Many novel solutions of concatenated model such as periodic wave, dark soliton, singular, hyperbolic, trigonometric and rational wave soliton solutions are retrieved using proposed method. The modulation instability of the selected model through stability analysis is also discussed. In order to display the retrieved soliton solutions graphically, 2D, 3D, density, and contour graphs have been utilized. The retrieved soliton solutions play a significant role in nonlinear optics. The results prove that the suggested approach is a very straightforward, concise and dynamic addition in literature. Also, these results are novel and provide invaluable insight into the fundamental physical phenomena. [ABSTRACT FROM AUTHOR]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focusses on the nonlinear fractional Kadomtsev-Petviashvili (FKP) equation in space-time, employing the conformable fractional derivative (CFD) approach. The main objective of this paper is to examine the application of the (G'/G)-expansion method in order to find analytical solutions to the FKP equation. The (G'/G)-expansion method is a powerful tool for constructing traveling wave solutions of nonlinear evolution equations. However, its application to the FKP equation remains relatively unexplored. By employing traveling wave transformation, the FKP equation was transformed into an ordinary differential equation (ODE) to acquire exact wave solutions. A range of exact analytical solutions for the FKP equation is obtained. Graphical illustrations are included to elucidate the physical characteristics of the acquired solutions. To demonstrate the impact of the fractional operator on results, the acquired solutions are exhibited for different values of the fractional order α, with a comparison to their corresponding exact solutions when taking the conventional scenario where α equals 1. The results indicate that the (G'/G)-expansion method serves as an efficient method and dependable in solving the nonlinear FKP equation. [ABSTRACT FROM AUTHOR]

In this study, two fractional differential equations describing the characteristics of three-dimensional nonlinear ion-acoustic waves (IAWs) in plasmas are considered with the conformable fractional derivative (CFD). 3+1-dimensional space–time fractional Jimbo–Miwa (JM) equation and 3+1-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV-ZK) equation are reduced into ordinary differential equations and an investigation into the exact solutions is carried out through the modified exponential function method (MEFM). These exact wave solutions are beneficial in interpreting the interior structure and characteristics of physical phenomena modeled by the aforementioned nonlinear evolution equations (NLEEs) arising in magnetized plasma. Various analytical methods were applied to such equations. Therefore, it is vital to determine the proper technique for the present problem. The proposed scheme has demonstrated a different spectrum of wave structures. The portrait of the exact solutions permits the exploration of plasmas, plasma models, and optics. The software system Mathematica is employed and the rational, exponential, trigonometric, and hyperbolic function solutions are acquired. The graphical results are presented for the physical nature of the solutions. The results are compared with the integer-order models.

In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.

Abstract In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

This study investigates the modified nonlinear time fractional Harry Dym equation, incorporating the conformable fractional derivative. Functioning as a mathematical framework for examining nonlinear phenomena in shallow water waves, particularly solitons, this model elucidates the intricate effects of dispersion and nonlinear steepening on wave dynamics. Employing a blend of analytical and numerical methodologies, the research aims to decipher the physical implications of the equation and its interconnectedness with other nonlinear evolution equations. The model delineates the evolution of a nonlinear wave in 1 + 1 dimensions (one spatial dimension x and time t ). The proposed methodology encompasses the G ′ G , 1 G expansion method, an analytical technique, alongside three numerical schemes utilizing B-spline methods. These methodologies facilitate the exploration of the equation’s behavior and enable precise computations of its solutions. The principal findings underscore the effective application of the proposed methodologies in resolving the modified nonlinear time fractional Harry Dym equation, furnishing valuable insights into its dynamics and significantly contributing to its physical interpretation. The significance of these discoveries lies in their contribution to the broader comprehension of nonlinear evolution equations and their pertinence across various scientific and engineering domains. This study provides novel insights into the modified nonlinear time fractional Harry Dym equation through a combined analytical and numerical approach. It advances the field of nonlinear dynamics and carries implications for analyzing analogous nonlinear evolution equations. These findings deepen our understanding of the equation’s physical interpretation and lay the groundwork for future explorations in related domains. [ABSTRACT FROM AUTHOR]

This study examines the stability and numerical simulation of the generalized fractional-order Fisher equation. The equation serves as a mathematical model describing population dynamics under the influence of factors such as natural selection and migration. We propose an implicit exponential finite difference method to solve this equation, considering the conformable fractional derivative. Furthermore, we analyze the stability of the method through theoretical considerations. The method involves transforming the problem into systems of nonlinear equations at each time since our method is an implicit method, which is then solved by converting them into linear equations systems using the Newton method. To test the accuracy of the method, we compare the results obtained with exact solutions and with those available in the literature. Additionally, we examine the symmetry of the graphs obtained from the solution to examine the results. The findings of our numerical simulations demonstrate the effectiveness and reliability of the proposed approach in solving the generalized fractional-order Fisher equation. [ABSTRACT FROM AUTHOR]

Ideally, the state variable follows a constant motion law over time. However, due to the finiteness of motion speed, almost all systems have time delay. In this paper, we investigate a new class of piecewise conformable fractional impulsive differential system with delay under two point inhomogeneous boundary condition. In the system, the motion laws of state variable vary at different time periods, and they interact with each other through time delay " τ " and time leading " - τ ", so as to be more realistic. By employing the well-know fixed point theorems, the sufficient conditions for the existence and uniqueness of solutions to the system are established. Under the conditions of ensuring the existence of the system's solutions, we conclude further that the system has Ulam–Hyers stability and Ulam–Hyers–Rassias stability by means of nonlinear functional analysis method. Finally, we give a feasible example to explain our result. [ABSTRACT FROM AUTHOR]

The extended (3+1)-dimensional Kadomtsev–Petviashvili equation is widely used in such domains as fluid mechanics, optics and so on. In this paper, we derive a new time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation based on the conformable fractional derivative for the first time. With the help of the Hirota bilinear method, we obtain the N-soliton, breather and lump solutions of the time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation. In addition, the semi-inverse variational method and the advanced e x p (ϕ (- ξ)) -expansion method are introduced to construct the exact solutions of this equation. By choosing suitable parameters, these solutions which can help solve issues in the marine science, fluctuation theory and other fields are presented through 3D graphics, contour and density plots. The findings of this work can further extend the study of fractional partial differential equations. [ABSTRACT FROM AUTHOR]

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in P-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena. [ABSTRACT FROM AUTHOR]

In this work, we investigate the condition of the given interval which ensures the existence and uniqueness of solutions for two‐point boundary value problems within conformable‐type local fractional derivative. The method of analysis is obtained by the principle of contraction mapping. Furthermore, benefiting from calculating the integral of the Green's function, we are able to improve a recent result by obtaining a sharper lower bound for an eigenvalue problem. Two examples are presented to clarify the obtained results. Finally, we present an open problem for the interested reader. [ABSTRACT FROM AUTHOR]

In the present paper, modified simple equation method (MSEM) is implemented for obtaining exact solutions of three nonlinear (3 + 1)-dimensional space-time fractional equation, namely three types of modified Korteweg-de-Vries (mKdV) equations. Here, the derivatives are of the type of conformable fractional derivatives. The solving process produces a system of algebraic equations which is possible to be easily with no need of using software for determining unknown coefficients. Results show that this method can supply a powerful mathematical tool to construct exact solutions of mKdV equations and it can be employed for other nonlinear (3 + 1) - dimensional space-time fractional equations. [ABSTRACT FROM AUTHOR]

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In this paper, we give some geometric approach for magnetic curves according to quasi model in ordinary space. Firstly, we compute conformable derivatives of Θ t q , Θ n q , Θ b q Lorentz forces. Moreover, we present recursion and normalization operators of magnetic fields according to the quasi model. Then, we give conformable derivatives for these operators. Finally, we construct conformable F–W derivatives for these curves according to the quasi model in space. [ABSTRACT FROM AUTHOR]

Fractional differential equations are being used to define numerous physical phenomena instead of conventional ordinary or partial differential equations. The secret is to get more generalization of the analysis, hence its solutions. The applications vary between electrical circuit modeling, oscillating circuits, shallow water behavior, viscous fluids, solution transportation, control theory and ultrafast optics. Sine–Gordon Expansion (SGE) Method is being employed hereafter to fully investigate the space–time conformable fractional Vakhnenko–Parkes equation and its modified version. Based on SGE method, the fractional models are transformed into an equivalent ordinary differential equation. The solutions are very important in studying the behavior of ultrafast optics in fibers or waveguides and the wave propagation throughout fiber optics. The solutions are formatted as solitons and are illustrated graphically. The illustrations show that the traveling waves inside fiber optics behave like solitons with traveling peak values according to the wave velocity. One other mode of propagation is to travel in kink shapes. Additionally, the ultrafast optical waves may propagate in shockwaves mode if the waves propagate at ultra-high speeds. [ABSTRACT FROM AUTHOR]

Medical imaging, the process of visual representation of different organs and tissues of the human body, is employed for monitoring the normal as well as abnormal anatomy and physiology of the body. Imaging which can provide healthcare solutions ensuring a regular measurement of various complex diseases plays a critical role in the diagnosis and management of many complex diseases and medical conditions, and the quality of a medical image, which is not a single factor but a composite of contrast, artifacts, distortion, noise, blur, and so forth, depends on several factors such as the characteristics of the equipment, the imaging method in question as well as the imaging variables chosen by the operator. The medical images (ultrasound image, X-rays, CT scans, MRIs, etc.) may lose significant features and become degraded due to the emergence of noise as a result of which the process of improvement pertaining to medical images has become a thought-provoking area of inquiry with challenges related to detecting the speckle noise in the images and finding the applicable solution in a timely manner. The partial differential equations (PDEs), in this sense, can be used extensively in different aspects with regard to image processing ranging from filtering to restoration, segmentation to edge enhancement and detection, denoising in particular, among the other ones. In this research paper, we present a conformable fractional derivative-based anisotropic diffusion model for removing speckle noise in ultrasound images. The proposed model providing to be efficient in reducing noise by preserving the essential image features like edges, corners and other sharp structures for ultrasound images in comparison to the classical anisotropic diffusion model. Furthermore, we aim at proving the viscosity solution of the fractional diffusion model. The finite difference method is used to discretize the fractional diffusion model and classical diffusion models. The peak signal-to-noise ratio (PSNR) is used for the quality of the smooth images. The comparative experimental results corroborate that the proposed, developed and extended mathematical model is capable of denoising and preserving the significant features in ultrasound towards better accuracy, precision and examination within the framework of biomedical imaging and other related medical, clinical, and image-signal related applied as well as computational processes.