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This work aims to derive new inequalities that improve the asymptotic and oscillatory properties of solutions to fourth-order neutral differential equations. The relationships between the solution and its corresponding function play an important role in the oscillation theory of neutral differential equations. Therefore, we improve these relationships based on the modified monotonic properties of positive solutions. Additionally, we set new conditions that confirm the absence of positive solutions and thus confirm the oscillation of all solutions of the considered equation. We finally explain the importance of the new inequalities by applying our results to some special cases of the studied equation, as well as comparing them with previous results in the literature.

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on a few earlier findings. The extended quantum Zakharov–Kuznetsov has practical applications in dealing with quantum electronpositron–ion magnetoplasmas, warm ions, and hot isothermal electrons in the presence of uniform magnetic fields, which makes the solutions obtained useful in analyzing a number of intriguing physical phenomena. We plot our data in MATLAB and display various 3D and 2D graphical representations to explain how the stochastic term and fractional derivative influence the exact solutions of the FSEQZKE.

In this paper, we analyze the q-iterative schemes to determine the roots of nonlinear equations by applying the decomposition technique with Simpson’s 13-rule in the setting of q-calculus. We discuss the convergence analysis of our suggested iterative methods. To check the efficiency and performance, we also compare our main outcomes with some well known techniques existing in the literature.

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liquid in fluid mechanics, the acquired solutions may be applied to explain a lot of fascinating physical phenomena. To address how Brownian motion effects the exact solutions of the SBLMPE, we present some 2D and 3D diagrams.

The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular to a D-shaped ellipse. The natural geometry that describes such a shape is the prolate elliptical one, i.e., the cap-cyclide coordinate system. When written in this geometry, the Grad–Shafranov equation can be solved in terms of the general Heun function. In this paper, we obtain the complete analytical solution of the Grad–Shafranov equation in terms of the general Heun functions and compare the result with the limiting case of the standard toroidal geometry written in terms of the Fock functions.

In this paper, we construct an analytical solution of the coupled Burgers’ equation, using the homotopy analysis method, which is a semi-analytical method, the approximate solution obtained by this method is convergent for different values of the convergence control parameter ℏ, the optimal value of ℏ corresponding with the minimum error to be determined by the residual. The results obtained by the present method are compared with other obtained solutions by different numerical methods.

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m.

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations.

The stochastic fractional (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation (SFHFSCE), which is driven in the Stratonovich sense by a multiplicative Wiener process, is considered here. The analytical solutions of the SFHFSCE are attained by utilizing the Jacobi elliptic function method. Various kinds of analytical fractional stochastic solutions, for instance, the elliptic functions, are obtained. Physicists can utilize these solutions to understand a variety of important physical phenomena because magnetic solitons have been categorized as one of the interesting groups of non-linear excitations representing spin dynamics in semi-classical continuum Heisenberg systems. To study the impact of the Wiener process on these solutions, the 3D and 2D surfaces of some achieved exact fractional stochastic solutions are plotted.

In this work, new criteria were established for testing the oscillatory behavior of solutions of a class of even-order delay differential equations. We follow an approach that depends on obtaining new monotonic properties for the decreasing positive solutions of the studied equation. Moreover, we use these properties to provide new oscillation criteria of an iterative nature. We provide an example to support the significance of the results and compare them with the related previous work.

In this paper, we focus on the stochastic fractional Kundu–Mukherjee–Naskar equation perturbed in the Stratonovich sense by the multiplicative Wiener process. To gain new elliptic, rational, hyperbolic and trigonometric stochastic solutions, we use two different methods: the Jacobi elliptic function method and the (G′/G)-expansion method. Because of the significance of the Kundu-Mukherjee equation in a magnetized plasma, the obtained solutions are useful in understanding many remarkable physical phenomena. Furthermore, we show the effect of the multiplicative Wiener process on the obtained solutions of the Kundu–Mukherjee–Naskar equation.

In this work, we investigate the oscillatory properties of the neutral differential equation (r(l)[(s(l)+p(l)s(g(l)))′]v)′+∑i=1nqi(l)sv(hi(l))=0, where s≥s0. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation.

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He’s semi-inverse method. Since the shallow water equation has many uses in ocean engineering, including river irrigation flows, tidal waves, tsunami prediction, and weather simulations, the solutions discovered can be utilized to represent a wide variety of exciting physical events. We create many 2D and 3D graphs to demonstrate how the beta-derivative and Brownian motion affect the analytical solutions of the SSWWE.

This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space E3. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space E3 is given as corollary. Moreover, our results are illustrated via some examples.

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and the Jacobi elliptic function method are utilized to attain new hyperbolic, trigonometric, elliptic, and rational fractional solutions. In addition, we generalize some previous results. The acquired solutions are beneficial in analyzing definite intriguing physical phenomena because the FFE equation is crucial for explaining various phenomena in optics, fluid mechanics and ocean engineering. To demonstrate how the M-truncated derivative affects the analytical solutions of the FFE, we simulate our figures in MATLAB and show several 2D and 3D graphs.

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted.

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions.

Equiform geometry is considered an extension of other geometries. Furthermore, an equiform frame is a generalization of the Frenet frame. In this study, we begin by defining the term “equiform parameter (EQP)”, “equiform frame”, and “equiform formulas (EQF)” in regard to the Minkowski three-space. Second, we define spacelike normal curves (SPN) in Minkowski three-space and present a variety of descriptions of these curves with equiform spacelike (EQS) or equiform timelike (EQN) principal normals in Minkowski three-space. Third, we discuss the implications of these findings. Finally, an example is given to illustrate our theoretical results.

In this work, we develop enhanced Hille-type oscillation conditions for arbitrary-time, second-order quasilinear functional dynamic equations. These findings extend and improve previous research that has been published in the literature. Some examples are given to demonstrate the importance of the obtained results.

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example.

A class of stochastic fractional diffusion equations with polynomials is considered in this article. This equation is used in numerous applications, such as ecology, bioengineering, biology, and mechanical and chemical engineering. As a result, it is critical to obtain exact solutions to this equation. To obtain these solutions, the tanh-coth method is utilized. Furthermore, we clarify the impact of noise on solution stabilization by simulating our solutions.

This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space E3 using the quasi-frame. In particular, we discuss some geometric properties and some characterizations of parameter curves of these surfaces in E3.

The theory of convexity has a rich and paramount history and has been the interest of intense research for longer than a century in mathematics. It has not just fascinating and profound outcomes in different branches of engineering and mathematical sciences, it also has plenty of uses because of its geometrical interpretation and definition. It also provides numerical quadrature rules and tools for researchers to tackle and solve a wide class of related and unrelated problems. The main focus of this paper is to introduce and explore the concept of a new family of convex functions namely generalized exponential type m-convex functions. Further, to upgrade its numerical significance, we present some of its algebraic properties. Using the newly introduced definition, we investigate the novel version of Hermite–Hadamard type integral inequality. Furthermore, we establish some integral identities, and employing these identities, we present several new Hermite–Hadamard H–H type integral inequalities for generalized exponential type m-convex functions. These new results yield some generalizations of the prior results in the literature.

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

This work is devoted to providing new kinds of deterministic and stochastic solutions of one of the famous nonlinear equations that depends on time, called the Schrödinger–Hirota equation. A new and straightforward methodology is offered to extract exact wave solutions of the stochastic nonlinear evolution equations (NEEs) with generalized differential conformable operators (GDCOs). This methodology combines the features of GDCOs, some instruments of white noise analysis, and the generalized Kudryashov scheme. To demonstrate the usefulness and validity of our methodology, we applied it to extract diversified exact wave solutions of the Schrödinger–Hirota equation, particularly in a Wick-type stochastic space and with GDCOs. These wave solutions can be turned into soliton and periodic wave solutions that play a main role in numerous fields of nonlinear physical sciences. Moreover, three-dimensional, contour, and two-dimensional graphical visualizations of some of the extracted solutions are exhibited with some elected functions and parameters. According to the results, our new approach demonstrates the impact of random and conformable factors on the solutions of the Schrödinger–Hirota equation. These findings can be applied to build new models in plasma physics, condensed matter physics, industrial studies, and optical fibers. Furthermore, to reinforce the importance of the acquired solutions, comparative aspects connected to some former works are presented for these types of solutions.

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation (a0t((ut+a1tug0t)′)β)′+a2tuβg1t=0, where t≥t0, are established. Using Riccati substitution and comparison with an equation of the first-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature.

In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.

In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example is presented in order to illustrate the significance of the results.

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given.

The third and fourth pseudo-Chebyshev irrational functions of half-integer degree are defined. Their definitions are connected to those of the first- and second-kind pseudo-Chebyshev functions. Their orthogonality properties are shown, with respect to classical weights.