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Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science. [ABSTRACT FROM AUTHOR]

The shear shallow water (SSW) model introduces an approximation for shallow water flows by including the effect of vertical shear in the system. Six non-linear hyperbolic partial differential equations with non-conservative laws make up this system. Shear, contact, rarefaction, and shock waves are all admissible in this model. We developed the finite-volume two-step scheme, the so-called generalized Rusanov (G. Rusanov) scheme, for solving the SSW model. This method is split into two stages. The first one relies on a local parameter that permits control over the diffusion. In stage two, the conservation equation is recovered. Numerous numerical instances were taken into consideration. We clarified that the G. Rusanov scheme satisfied the C-property. We also compared the numerical solutions with those obtained from the Rusanov, Lax-Friedrichs, and reference solutions. Finally, the G. Rusanov technique may be applied for solving a wide range of additional models in developed physics and applied science. [ABSTRACT FROM AUTHOR]

The perturbed chiral nonlinear Schrö dinger equation (PCNLSE) reflects the quantum actions such as quantum pictures of Bohm potential and internal self-potential properties. Indeed, this equation introduces the basics of the hidden variable theory in quantum mechanics. Two unified solver methods and exp(−φ(ξ)) -expansion technique applied to PCNLSE to present many solitonic solutions in an explicit and effective way. The behaviour of these solutions is of qualitatively different structural natures, relying on physical coefficient parameters. The application of three mathematical techniques to our model system provides us with several possible physical property solutions that account for the majority of many phenomena the model under study attempts to depict. The reported bright explosive envelopes, explosive solitons, periodic blow up, bright periodic envelope and huge solitary waves are highly applicable in plasma and nuclear physics, optical communications, electro-magnetic propagations, superfluid and in a lot other applied sciences. The results of this system's solitary structures are consistent with the characteristics of the nonlinear Schrödinger equation systems used to study dispersive modes and higher-order perturbed systems. For more details about the physical dynamical representation of the presented solutions, we have illustrated them with profile pictures using Mathematica and Matlab 18, to obtain complete configurations. The proposed approach can be applied to several equations arising in all applied sciences. [ABSTRACT FROM AUTHOR]

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ -weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O (h 2 + (Δ t) 2) . Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques. [ABSTRACT FROM AUTHOR]

This paper considers the one-dimensional model of heat conduction in solids at low temperature, the so called phonon-Bose model. The nonlinear model consists of a conservation equation for the energy density e and the heat flux Q with ∣Q∣ < e. We present a simple and accurate class of finite volume schemes for numerical simulation of heat flow in arteries. This scheme consists of predictor and corrector steps, the predictor step contains a parameter of control of the numerical diffusion of the scheme, which modulate by using limiter theory and Riemann invariant, the corrector step recovers the balance conservation equation, the scheme can compute the numerical flux corresponding the real state of solution without relying on Riemann problem solvers and it can thus be turned to order 1 in the regions where the flow has a strong variation and to order 2 in the regions where the flow is regular. The numerical test cases demonstrate high resolution of the proposed finite volume scheme (modified Rusanov) and confirm its capability to provide accurate simulations for heat flow under flow regimes with strong shocks. [ABSTRACT FROM AUTHOR]

The article aimed to develop an accurate approximation of the fractional derivative with a non-singular kernel (the Rabotnov fractional-exponential formula), and show how to use it to solve numerically the blood ethanol concentration system. This model can be represented by a system of fractional differential equations. First, we created a formula for the fractional derivative of a polynomial function tp using the Rabotnov exponential kernel. We used the shifted Vieta-Lucas polynomials as basis functions on the spectral collocation method in this work. By solving the specified model, this technique generates a system of algebraic equations. We evaluated the absolute and relative errors to estimate the accuracy and efficiency of the given procedure. The results point to the technique's potential as a tool for numerically treating these models. [ABSTRACT FROM AUTHOR]

We introduce some vital solutions for the dusty gas flow. Specifically, we implement the nonhomogeneous Riemann solver (NHRS) scheme in order to investigate the unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The principle of conservation of mass, momentum, and energy governs the dynamical behavior of a fluid. The Rusanov scheme is also implemented to examine the dusty gas flow model for comparison and validation. One of the important characteristics of the NHRS technique is its capability to compute the numerical flux corresponding to the real states of solution in the absence of a Riemann solution. The numerical results exhibit that the NHRS system outperforms the Rusanov scheme and validates the ability to simulate the dusty gas flow model with strong shocks and rarefaction waves accurately. Finally, more sophisticated models in applied science can be solved using the NHRS scheme. [ABSTRACT FROM AUTHOR]

In this work, we consider the model of shallow water equation with horizontal density gradients. We develop the modified Rusanov (mR) scheme to solve this model in one and two dimensions. Predictor and corrector are the two stages of the suggested scheme. The predictor stage is dependent on a local parameter (αn i+1/2) that allows for difiusion control. The balance conservation equation is recovered in the corrector stage. The proposed approach is well-balanced, conservative, and straightforward. Several 1D and 2D test cases are produced after presenting the shallow water model and the numerical technique. In the 1D case, we compared the proposed scheme with the Rusanov scheme, mR with constant and analytical solutions. The numerical simulation demonstrates the mR's great resolution and attests to its capacity to produce accurate simulations of the shallow water equation with horizontal density gradients. Our results demonstrate that the mR technique is a highly effective instrument for solving a variety of equations in applied science and developed physics. [ABSTRACT FROM AUTHOR]

The superthermality indices of positron (electron) contributions to the new nonlinear wave profiles characterized by the model as supersoliton, cnoidal, shocklike, and super-periodic wave propagation have been obtained in the superthermal ionosphere plasma pair. These profiles that exhibit critical behaviors are described by the modified KP equation. Furthermore, the novel solutions have been exposed and the super-solitary and period solutions are derived and explained using the Jacobi elliptic function expansion method for the Modified Korteweg–De Vries (MKdV) equation. These solutions may be useful in the space of ionosphere fluids. Indeed, the MKdV model is potent for the dynamics of fluid flow. The super propagating waves obtained under the influence of critical densities represent a source of forcing or collapsing energy waves in space plasma fluids. To check the impacts of superthermality on the progress of nonlinear formations, all the given solutions are deliberated by the superthermal factor in ionosphere fluid observations and applications. [ABSTRACT FROM AUTHOR]

We propose a class of finite volume algorithms that are both simple and efficient for solving numerically the shallow water equations with varying densities; shallow water flows in single and two layers are considered. In these flow regimes, variable horizontal or vertical density is taken into account. The shallow water equations for the hydraulic variables are coupled with a suspended sediment transport equation for the concentration variable to construct the model. To approximate the numerical solution of the models under consideration, a generalized Rusanov method is proposed; this method avoids solving Riemann problems during the time integration process, and it is simple and accurate. The proposed method is divided into two stages: predictor and corrector; the presented finite volume approach is well balanced, conservative, and simple. Non‐oscillatory and suitable for shallow water equations when Riemann problems are challenging to solve. Several test problems for single‐layer and two‐layer shallow water flows are applied to the proposed method. The numerical results show that the proposed finite volume approach has high resolution and that it is capable of simulating shallow water equations with variable density in flow regimes with strong shocks accurately. [ABSTRACT FROM AUTHOR]