Three-dimensional Nonlinear Equation Research Articles

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

Analyzing epidemics and designing policies with a modified SIR model

This research paper aims to study the impact of COVID-19 on the economy, where we have extended the standard epidemiological model, the Susceptible–Infected–Recovered (SIR) model. In the economy’s dynamics, we have incorporated aggregate demand and aggregate supply and further set up a system of three-dimensional nonlinear differential equations, which is more likely the Labor–Infected–Quarantine (LIQ) model, an extension of the SIR model. Here, we have derived the steady-state condition and analyzed the model’s stability. In addition, we have done a comparative static analysis of parameters with some policy implications. We aim to achieve the twin objective, i.e., controlling the pandemic and reviving the economy under the set of the policy mix.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

An energy-stable finite element method for nonlinear Maxwell's equations

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass.

Solutions of Three Dimensional Nonlinear Klein-Gordon Equations by Using Quadruple Laplace Transform

This study focuses on solving three-dimensional non-linear Klein-Gordon equations of four variables by using the quadruple Laplace transform method coupled with the iterative method. This study was designed in order to show the quadruple Laplace transform with an iterative method for solving three-dimensional nonlinear Klein-Gordon equations. The quadruple Laplace transform with the iterative method was aimed at getting analytical solutions of three-dimensional nonlinear Klein-Gordon equations. Exact solutions obtained through the iterative method have been analytically evaluated and presented in the form of a table and graph. The analytical solutions of these equations have been given in terms of convergent series with a simply calculable system, and the nonlinear terms in equations can easily be solved by the iterative method. Illustrative examples are also provided to demonstrate the applicability and efficiency of the method. The result renders the applicability and efficiency of the applied method. Finally, the quadruple Laplace transform and iterative method is an excellent method for the solution of nonlinear Klein-Gordon equations.

Exact solutions of time-dependent oscillations in multipolar spherical vortices

Exact solutions of the time-dependent three-dimensional nonlinear vorticity equation for Euler flows with spherical geometry are provided. The velocity solution is the sum of a multipolar oscillatory function and a rigid cylindrical motion with swirl. The multipolar oscillation is a velocity mode whose radial and angular dependencies are given by the spherical Bessel functions and vector spherical harmonics, respectively. The local frequency of the velocity oscillations equals the angular speed of the rigid flow times the angular azimuthal wavenumber of the oscillating flow. The unsteady motion corresponds to inertial oscillations in multipolar flows with spatial azimuthal waves (non-vanishing azimuthal wavenumber) in the presence of a background flow with constant axial vorticity. In these nonlinear solutions, the curl of the Lamb vector has a linear dependence with the oscillation velocity, a property that makes it possible for the oscillating motion to satisfy different linear wave equations. Based on these inviscid time-dependent velocity modes, new exact solutions to the time-dependent Navier–Stokes equation are also provided.

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

Analytical study of nonlinear water wave equations for their fractional solution structures

This paper examines the three-dimensional nonlinear time-fractional water wave equations for their analytical wave solutions. These are the equations of the names [Formula: see text]-Zakharov–Kuznetsov–Burgers equation and [Formula: see text]-Yu–Toda–Sasa–Fukuyama equation. The obtained wave solutions are in the form of kink, periodic and singular waves by utilizing the [Formula: see text]-expansion approach. The aforesaid solutions are verified and demonstrated graphically via symbolic soft computations.

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus {mathbb {T}}^3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on {mathbb {T}}^3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Phi ^4_3-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

The Solution of Non-Linear Equations System Containing Interpolation Functions by Relaxing the Newton Method

Many world phenomena lead to nonlinear equations systems. For some applications, the non-linear equations which construct the non-linear equations system are interpolation functions. However, the interpolation functions are usually not represented as mathematical expressions but as computer programs in specific programming languages. The research proposed using the relaxed Newton method to solve the non-linear equations system that contained interpolation functions. The interpolation functions were represented in the R programming language. Then, the experiment used the Spline interpolation function to construct a two-dimensional non-linear equations system. Eleven initial guesses, maximum of ten-time iterations, and 10-7 precision were applied. The solution of the non-linear equations system and the iteration needed on each initial guess were observed. The experiment shows that the proposed method works well for solving the non-linear equations system constructed by Spline interpolation functions. By observing the initial guesses used in the experiment, there are four possible results: true solution, spurious solution, false solution, and no solution. Applying 11 initial guesses have five initial guesses resulting in true solutions, one initial guess in spurious solution, three initial guesses in false solutions, and one initial guess in no solution. The discussions imply that this method can be generalized to the three-dimensional non-linear equations system or higher dimensions.

A Note on a Three-Dimensional Bäcklund Transformation

The Backlund transformation (BT) for a three-dimensional nonlinear wave equation and its nonlinear superposition formula are studied in this note. We prove that the three dimensional Backlund transformation obtained by Leibbrandt, et al. can be decomposed into three two-dimensional BTs. Some results on the N-dimensional Liouville equation are also discussed in the article.

Analysis of Local Stability and Bifurcation in Cervical Cancer and Immune Cells Interaction

The dynamics of cervical cancer cells and immune cell interaction are modeled into a three-dimensional nonlinear differential equation system. It is a representation of the interaction model between one prey and two predators, with cervical cancer cells as prey and two types of immune cells as predators. The nature between these predators is a supportive relationship. Based on the mathematical model is obtained that there are six steady states in which three of them exist and the others do not exist for biological cases. Based on the analysis of stability near these steady states there has a bifurcation point about the steady states. By modifying some parameters value, there is a possibility to heal the cervical cancer sufferers.

On SOR-like iteration methods for solving weakly nonlinear systems

ABSTRACT In this paper, we introduce a class of SOR-like iteration methods for solving the systems of the weakly nonlinear equation, which is by reformulating equivalently the weakly nonlinear equation as a two-by-two block nonlinear equation. Two types of the global convergence theorems are given under suitable choices of the involved splitting matrix and parameter. Numerical results for the three-dimensional nonlinear convection-diffusion equation and the linear complementarity problem show that the proposed iteration methods are feasible and efficient for solving the weakly nonlinear equations.

Mathematical Modeling of Antenna-Mast Structures With Aerodynamic Effects

At the present stage of development of the systems for wireless transmission of information, including providing high-quality communication channels of the transport systems, the important issue is to increase the coverage area of the electromagnetic waves emitted by base stations. The area of the region of the coating increases with the increase in installation height of transmitter, and at the same time, communication structures with height more than 75 meters it is necessary to additionally determine the aerodynamic coefficients of the wind load through purging models of buildings in wind tunnels or on the recommendations developed by specialized organizations. Such events can significantly increase the final cost of the project for the construction of antenna-mast structures, so relevant and timely is the development of mathematical models of the behavior of antenna-mast structures under wind influence, which would determine the aerodynamic coefficients at the gathering of loads on the designed or operated facility, and to adequately predict the behavior and condition of the property at different stages of the life cycle. In the work as the equations that define the dynamic behavior of the design used three-dimensional non-stationary nonlinear equations of fluid dynamics considering viscous resistance of the type of Navier-Stokes equations, the solution of which it is proposed to use a numerical analytical method, which allows to represent and calculate the desired functions in the form of discrete values. The results obtained agree well with the results of using verified software and computing systems, as well as with analytical solutions to test problems. At the present day, it is important to study the behavior of traditional constructions under various operating conditions of the rolling stock, including an increase in the frequency of dynamic effects and the superposition of effects from different loads and oscillations to achievement of the high-frequency range. High-frequency vibrations in the elements of the railway track and the rolling stock are more likely to appear during the traffic on artificial structures, such as bridges with different design diagrams and made from both reinforced concrete and steel. In such cases, the forced and free oscillation frequencies of different composite constructions of transport infrastructure are superimposed.

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

The solution of the three-dimensional nonlinear wave equation −U″TT + U″XX + U″YY + U″ZZ = f(eT, eX, eY, eZ, U) by means of the method of matched asymptotic expansions is considered. Here e is a small positive parameter and the right-hand side is a smoothly changing source term of the equation. A formal asymptotic expansion of the solution of the equation is constructed in terms of the inner scale near a typical butterfly catastrophe point. It is assumed that there exists a standard outer asymptotic expansion of this solution suitable outside a small neighborhood of the catastrophe point. We study a nonlinear second-order ordinary differential equation (ODE) for the leading term of the inner asymptotic expansion depending on three parameters: u″xx = u5 − tu3 − zu2 − yu − x. This equation describes the appearance of a step-like contrast structure near the catastrophe point. We briefly describe the procedure for deriving this ODE. For a bounded set of values of the parameters, we obtain a uniform asymptotics at infinity of a solution of the ODE that satisfies the matching conditions. We use numerical methods to show the possibility of locating a shock layer outside a neighborhood of zero in the inner scale. The integral curves found numerically are presented.

Dynamical behavior of a system of three-dimensional nonlinear difference equations

In this paper, we study the boundedness, persistence, and periodicity of the positive solutions and the global asymptotic stability of the positive equilibrium points of the system of difference equations \t\t\txn+1=A+xn−1zn,yn+1=A+yn−1zn,zn+1=A+zn−1yn,n=0,1,…,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+1}=A+\\frac{x_{n-1}}{z_{n}},\\qquad y_{n+1}=A+ \\frac{y_{n-1}}{z _{n}},\\qquad z_{n+1}=A+\\frac{z_{n-1}}{y_{n}},\\quad n=0,1,\\ldots , $$\\end{document} where Ain ( 0,infty ) and the initial conditions x_{i}, y_{i}, z_{i}in ( 0,infty ) , i=-1,0.

Reaction-Diffusion Models with Delay: Some Properties, Equations, Problems, and Solutions

The delay reaction-diffusion models used in thermal physics, chemistry, biochemistry, biology, ecology, biomedicine, and control theory were reviewed. New exact solutions were obtained for several classes of one- and three-dimensional nonlinear equations with distributed parameters, in which the kinetic functions involve a delay. The qualitative features of these equations related to nonsmoothness and potential instability of solutions (these features should be taken into account in the mathematical modeling of the corresponding processes) were discussed. The properties of delay reaction-diffusion equations were described, which allow exact solutions to be obtained and multiplied. The key principles of construction, selection, and use of the test problems of the reaction-diffusion type were formulated, which can be used for evaluating the accuracy of rough analytical and numerical methods for solving the delay equations.

A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations

AbstractThis paper employs a differential quadrature scheme for solving non-linear partial differential equations. Differential quadrature method (DQM), along with modified cubic B-spline basis, has been adopted to deal with three-dimensional non-linear Brusselator system, enzyme kinetics of Michaelis-Menten type problem and Burgers’ equation. The method has been tested efficiently to three-dimensional equations. Simple algorithm and minimal computational efforts are two of the major achievements of the scheme. Moreover, this methodology produces numerical solutions not only at the knot points but also at every point in the domain under consideration. Stability analysis has been done. The scheme provides convergent approximate solutions and handles different cases and is particularly beneficial to higher dimensional non-linear PDEs with irregularities in initial data or initial-boundary conditions that are discontinuous in nature, because of its capability of damping specious oscillations induced by high frequency components of solutions.

Numerical simulations of a third-grade fluid flow on a tube through a contraction

Based on a director theory approach related to fluid dynamics we reduce the nonlinear three-dimensional equations governing the axisymmetric unsteady motion of a non-Newtonian incompressible third-grade fluid to a one-dimensional system of ordinary differential equations depending on time and on a single spatial variable. From this new system we obtain the unsteady equation for the mean pressure gradient and the wall shear stress both depending on the volume flow rate, Womersley number and viscoelastic parameters over a finite section of a straight, rigid and impermeable tube with variable circular cross-section. We present some numerical simulations of unsteady flows regimes through a tube with a contraction using a nine-directors theory.