Narrow Results

The fractional mKDV–Zakharov–Kuznetsov (FmKDV-ZK) equation with M-truncated derivatives (MTDs) is considered. To obtain a novel rational, trigonometric, hyperbolic, and elliptic, solutions for FmKDV-ZK, we use two different methods including the modified mapping technique and generalized Riccati equation method. These methods are effective and simple. Also, they give us different kinds of solutions such as elliptic, trigonometric, hyperbolic, and rational solutions. The FmKDV-ZK equation is widely used in engineering, geophysics, meteorology, ocean dynamics, and plasma physics, therefore the obtained solutions may be applied to a wide range of important physical phenomena. Moreover, we display a wide range of solutions in two-dimensional and three-dimensional spaces to examine the effects of the MTDs operator. We conclude that the surface expands as the derivative order decreases.

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved $tan(\frac{\mathrm{\Psi }(\eta )}{2})$-expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent.

The optical soliton solutions to the fractional nonlinear Schrödinger (NLS) equation in the presence of nonlinear oscillating coefficient with Beta and M-truncated derivatives are studied by applying a complex wave transformation that converts the fractional NLS equation to an ordinary differential equation. The optical solution structures are attained with the use of the Sardar sub-equation (SSE) method. The NLS equation is an important nonlinear complex model which governs the propagation of an optical pulse in a birefringent optical fiber. The fractional NLS equation is used in optical telecommunication, high-energy physics, gas dynamics, electrodynamics and ocean engineering. The graphical presentation of the attained results is also discussed in detail.

The nonlinear Kadoma equation with M-truncated derivatives (NLKE-MTD) is taken into consideration here. By using generalized Riccati equation method (GRE method) and Jacobi elliptic function method, new hyperbolic, rational, trigonometric and elliptic solutions are discovered. Because the NLKE is widely employed in optics, fluid dynamics and plasma physics, the resulting solutions may be used to analyze a wide variety of important physical phenomena. The dynamic behaviors of the different derived solutions are interpreted using 3D and 2D graphs to explain the effects of M-truncated derivatives. We may conclude that the surface moves to the right as the order of M-truncated derivatives increases.

The KdV–Zakharov–Kuznetsov equation is an important and interesting mathematical model in plasma physics, which is used to describe the effect of magnetic field on weak nonlinear ion-acoustic waves. A fractional KdV–Zakharov–Kuznetsov equation in the $M$-truncated derivative sense is investigated. By taking into account the fractional ${tanh}_{\delta }$ method and fractional $sin{e}_{\delta }$–$\mathrm{\text{cosin}}{e}_{\delta }$ method, larger numbers of a new type of solitary wave solutions are obtained. The dynamic characteristics of these new solitary wave solutions are elaborated by sketching some three-dimensional (3D) and two-dimensional (2D) figures. The study reveals that the proposed two methods are very powerful to solve fractional evolution equations.

In ocean engineering, the long- and short-wave interaction system represents a crucial nonlinear evolution equation that elucidates the resonant interaction phenomenon between ocean waves. In this study, we describe the fractional long and short-wave interaction (FLSWI) system employing the M-truncated derivative. Subsequently, we employ the extended fractional ${tanh}_{\chi }-{coth}_{\chi }$ and the fractional $csc{h}_{\chi }$ methods to address the FLSWI system. These two approaches yields novel and intriguing soliton solutions. To further elucidate the derived soliton solutions, three-dimensional visualizations are constructed and analyzed.

In this work, the fractional Gross–Pitaevskii equation and fractional Radhakrishnan–Kundu Lakshmanan equation are investigated. A large number of new soliton solutions are obtained by implementing a simple and effective mathematical method, which is called the fractional unified solver method. Moreover, the 3D and 2D graphs are plotted to elaborate on the dynamic characteristics of these acquired solutions.

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry.