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In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to (3 + 1) -dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota's simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving the suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, fluid mechanics and engineering sciences can benefit from this study. [ABSTRACT FROM AUTHOR]

The coupled-Higgs equation, which depicts a system of preserved scalar nucleons cooperating with unbiased scalar mesons, is investigated using the ϕ 6 -model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m → 0 and m → 1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial differential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions. [ABSTRACT FROM AUTHOR]

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section. [ABSTRACT FROM AUTHOR]

In the current study, we consider the generalized Pochhammer-Chree equation with a term of order n. Based on the (1=G 0)-expansion method and with the aid of symbolic computation, we construct some distinct exact solutions for this nonlinear model. Various exact solutions are produced to the studied equation including singular solutions and periodic wave solutions. In addition, 2D, 3D, and contour plots are graphed for all obtaining solutions via choosing the suitable values for the involved parameters. All gained solutions verify the governing equation. [ABSTRACT FROM AUTHOR]

In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when ff = 1, we obtain the same results in the standard calculus. [ABSTRACT FROM AUTHOR]

In this study, the discrete fractional nabla calculus operator is used to investigate the k‐hypergeometric differential equation for both homogeneous and nonhomogeneous states. To solve the guiding equation, we implement certain classical transformations and also constrain the parameters needed to determine them valued. In order to achieve these results, some equipment like the Leibniz rule, the index law, the shift operator, and the power rule are set out in the frame of the discreet fractional calculus. We use all of these tools to the governing equation for homogeneous and nonhomogeneous situations. The major benefit of the fractional nabla operator is that it implements singular differential equations and converts them into fractional order equations. As a result, several new exact fractional solutions of the given equation are constructed. [ABSTRACT FROM AUTHOR]

In the current study, we instigate the four-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation. The NLmZK equation guides the attitude of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Two different methods are used, namely the sine-Gordon expansion method (SGEM) and the (1 / G ′) -expansion method to the proposed model. We have successfully constructed some topological, non-topological, and wave solutions. In addition, the 2D, 3D, and contour graphs of the solutions are also plotted under the choice of appropriate values of the parameters. [ABSTRACT FROM AUTHOR]