Open Journal of mathematical Sciences (OMS)

In this investigation, we aim to investigate the novel exact solutions of nonlinear partial differential equations (NLPDEs) arising in electrical engineering via the -expansion method. New acquired solutions are kink, particular kink, bright, dark, periodic combined-dark bright and combined-dark singular solitons, and hyperbolic functions solutions. The achieved distinct types of solitons solutions contain critical applications in engineering and physics. Numerous novel structures (3D, contour, and density plots) are also designed by taking the appropriate values of involved parameters. These solutions express the wave show of the governing models, actually.

A continuous two-step block method with two hybrid points for the numerical solution of first order ordinary differential equations is proposed. The approximate solution in form of power series and its first ordered derivative are respectively interpolated at the point \(x=0\) and collocated at equally spaced points in the interval of consideration. The application of the method involves using the main scheme derived together with the additional schemes simultaneously to obtain the solution to the problem at the grid points. The analysis of the method and the results obtained from the examples considered show that the method is consistent, zero-stable, convergent and of high accuracy.

In this paper, we use the \(\varphi ^{6}\)-model expansion method to construct the traveling wave solutions for the reaction-diffusion equation. The method of \(\varphi ^{6}\)-model expansion enables the explicit retrieval of a wide variety of solution types, such as bright, singular, periodic, and combined singular soliton solutions. Kink-type solitons, also known as topological solitons in the context of water waves, are another type of solution that can be explicitly retrieved. This study’s results might enhance the equation’s nonlinear dynamical properties. The method proposes a practical and efficient method for solving a sizable class of nonlinear partial differential equations. The dynamical features of the data are explained and highlighted using exciting graphs.

In this paper, we introduce the notion of interval neutrosophic ideals in subtraction algebras. Also, introduce the intersection and union of interval neutrosophic sets in subtraction algebras. We prove intersection of two-interval neutrosophic ideals is also an interval neutrosophic ideal. Some exciting properties and results based on such an ideal are discussed. Moreover, we define the homomorphism and homomorphism of interval neutrosophic sets. We prove the image of an interval neutrosophic subalgebra is also an interval neutrosophic sub-algebra.

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one extra free parameter to them. In particular, we generalize generating functions for the continuous \(q\)-ultraspherical/Rogers, little \(q\)-Laguerre/Wall, and \(q\)-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and \(q\)-integrals.

We prove that if the frame \(S\) is decaying surface non-radiating, in the sense of Definition 1, then if \(\left(\rho,\overline{J}\right)\) is analytic, either \(\rho=0\) and \(\overline{J}=\overline{0}\), or \(S\) is non-radiating, in the sense of [1]. In particularly, by the result there, the charge and current satisfy certain wave equations in all the frames \(S_{\overline{v}}\) connected to \(S\) by a real velocity vector \(\overline{v}\), with \(|\overline{v}|<c\).