Search Results (11)

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

This article investigates the impact of Arrhenius energy and the radius of a nanoparticle subject to an irregular heat source on tangent-hyperbolic nanofluid flow over a stretching sheet with nonlinear radiation. The convective boundary effect, higher-order slip, and micropolarity are all included for a water-based $Cu$ nanofluid. The present study investigates the significance of a nanoparticle’s radius under inclined MHD conditions. The thermally convective flow of the nanofluid is optimized for the heat-transfer rate using the response surface technique. The modeled governing equations are converted into a system of first-order ODEs using the proper similarity transformations, and the BVP5C algorithm—a finite-difference-based solver—is then used to solve these ODEs numerically. Microrotation, thermal boundary-layer thickness, and the skin-friction coefficient all decrease as the nanoparticle radius increases. The thermal layer thickens as the Biot number increases. As the higher-order slip parameter coefficient increases, the results indicate that the skin friction and local Nusselt number fall. Using tables, figures, contour plots, and surface plots, the effects of several influencing factors on the rates of heat and mass transfer, as well as on the skin-friction factor, are demonstrated. The study uses “Response Surface Methodology” (RSM) in conjunction with “Analysis of Variance” (ANOVA) to optimize the most important factors, which are probably the magnetic parameter and the nanoparticle radius that control the flow and heat-transfer properties. Additionally, with a Nusselt number $R^2$ value of $99.96$, indicating an excellent fit, the suggested model exhibits amazing precision. The reliability and efficiency of the estimated model are assessed using the residual versus fitted plot. Full article

This study investigates the dynamics of optical solitons for the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE) enriched with $\beta$-derivatives. By employing an extended direct algebraic method (EDAM), we successfully derive explicit soliton solutions that illustrate the intricate interplay between nonlinearities and variable coefficients. Our approach facilitates the transformation of the complex NLS into a more manageable form, allowing for the systematic exploration of diverse solitonic structures, including bright, dark, and singular solitons, as well as exponential, polynomial, hyperbolic, rational, and Jacobi elliptic solutions. This diverse family of solutions substantially expands beyond the limited soliton interactions studied in conventional approaches, demonstrating the superior capability of our method in unraveling new wave phenomena. Furthermore, we rigorously demonstrate the robustness of these soliton solutions against various perturbations through comprehensive stability analysis and numerical simulations under parameter variations. The practical significance of this work lies in its potential applications in advanced optical communication systems. The derived soliton solutions and the analysis of their dynamics provide crucial insights for designing robust signal carriers in nonlinear optical media. Specifically, the management of variable coefficients and fractional-order effects can be leveraged to model and engineer sophisticated dispersion-managed optical fibers, tunable photonic devices, and ultrafast laser systems, where controlling pulse propagation and stability is paramount. The presence of $\beta$-fractional derivatives introduces additional complexity to the wave propagation behaviors, leading to novel dynamics that we analyze through numerical simulations and graphical representations. The findings highlight the potential of the proposed methodology to uncover rich patterns in soliton dynamics, offering insights into their robustness and stability under varying conditions. This work not only contributes to the theoretical foundation of nonlinear optics but also provides a framework for practical applications in optical fiber communications and other fields involving nonlinear wave phenomena. Full article

In this paper, we investigate closed-form meromorphic solutions of the fifth-order Sawada-Kotera (fSK) equation and (3+1)-dimensional generalized shallow water (gSW) equation. The study of high-order and high-dimensional differential equations is pivotal for modeling complex nonlinear phenomena in physics and engineering, where higher-order dispersion, dissipation, and multidimensional dynamics govern system behavior. Constructing explicit solutions is of great significance for the study of these equations. The elliptic, hyperbolic, rational, and exponential function solutions for these high-order and high-dimensional differential equations are achieved by proposing the extended complex method. The planar dynamics behavior of the (3+1)-dimensional gSW equation and its phase portraits are analyzed. Using computational simulation, the chaos behaviors of the high-dimensional differential equation under noise perturbations are examined. The dynamic structures of some obtained solutions are revealed via some 2D and 3D graphs. The results show that the extended complex method is an efficient and straightforward approach to solving diverse differential equations in mathematical physics. Full article

The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice. Full article

This research represents the first theoretical investigation about the vibration behavior of circular organic solar cells. Therefore, the vibration response of asymmetric circular organic solar cells that represent a perfect renewable energy source is demonstrated. For this purpose, the differential quadrature method (DQM) is employed. The organic solar cell is modeled as a laminated plate consisting of five layers of Al, P3HT:PCBM, PEDOT:PSS, ITO, and Glass. This cell is rested on a Winkler–Pasternak elastic foundation and assumed to be exposed to various types of hygrothermal loadings. There are three different kinds of temperature and moisture variations that are taken into account: uniform, linear, and nonlinear distribution throughout the cell’s thickness. The displacement field is presented based on a new inverse hyperbolic shear deformation theory considering only two unknowns. The motion equations including hygrothermal effect and plate–foundation interaction are established within the framework of Hamilton’s principle. The DQM is utilized to solve these equations. In order to ensure the accuracy of the proposed theory, the present results are compared with those reported by other higher-order theories. A comprehensive parametric illustration is conducted on the impacts of different parameters involving the geometrical configuration, elastic foundation parameters, temperature, and moisture concentration on the deduced eigenfrequency of the circular organic solar cells. Full article

A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method $\left({\phi }_S=1.618\dots \right)$ to super-quadratic $\left({\phi }_T={\phi }_S+1=2.618\dots \right)$ and the quadratic convergence of the Newton method $\left({\phi }_N=2\right)$ to cubic $\left({\phi }_T={\phi }_N+1=3\right)$ in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases. Full article

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests. Full article

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp$\left(-\Phi \left(\eta \right)\right)$-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions. Full article

In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in $\Omega \times (t>0)$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ and $N\ge 2$. We prove that there exists a safe time interval of existence in the solution $[0,T]$, with T being a lower bound of the blowup time $t^{*}$. Moreover, we find an explicit lower bound of $t^{*}$, assuming the coefficients are positive constants. Full article

We describe the numerical implementation of a phase-resolving, nonlinear spectral model for shoaling directional waves over a mild sloping beach with straight parallel isobaths. The model accounts for non-linear, quadratic (triad) wave interactions as well as shoaling and refraction. The model integrates the coupled, nonlinear hyperbolic evolution equations that describe the transformation of the complex Fourier amplitudes of the deep-water directional wave field. Because typical directional wave spectra (observed or produced by deep-water forecasting models such as WAVEWATCH III™) do not contain phase information, individual realizations are generated by associating a random phase to each Fourier mode. The approach provides a natural extension to the deep-water spectral wave models, and has the advantage of fully describing the shoaling wave stochastic process, i.e., the evolution of both the variance and higher order statistics (phase correlations), the latter related to the evolution of the wave shape. The numerical implementation (a Fortran 95/2003 code) includes unidirectional (shore-perpendicular) propagation as a special case. Interoperability, both with post-processing programs (e.g., MATLAB/Tecplot 360) and future model coupling (e.g., offshore wave conditions from WAVEWATCH III™), is promoted by using NetCDF-4/HD5 formatted output files. The capabilities of the model are demonstrated using a JONSWAP spectrum with a cos^2s directional distribution, for shore-perpendicular and oblique propagation. The simulated wave transformation under combined shoaling, refraction and nonlinear interactions shows the expected generation of directional harmonics of the spectral peak and of infragravity (frequency <0.05 Hz) waves. Current development efforts focus on analytic testing, development of additional physics modules essential for applications and validation with laboratory and field observations. Full article