Second-order Nonlinear Differential Equations Research Articles

Solution of Klein-Gordon Equation in F(R) Theory of Gravity

The theory, as a modification of the general relativity theory, is frequently employed as an alternative theory of gravity and offers a promising avenue for addressing the challenges of formulating a quantum gravity theory. In this study, by applying the separation method of time, radial and angular variables, we derived the general solution of the Klein-Gordon equation in a curved space-time using modified Schwarzschild metric. We modified Ricci scalar form in Einstein’s action principle as a general function of Ricci scalar and formulated the general Schwarzschild metric. The solution of the time function was analytically obtained in exponential form, and the solution of the angular function in terms of Legendre polynomial depends on azimuthal and magnetic quantum numbers. The radial function in terms of a non-linear second-order differential equation was solved by a numerical method using Python. The solutions described the gravitational effect for a light particle on the area gravitationally has a strong interaction, represented by a spherically symmetric metric. For small (in Schwarzschild radius), the results analytically show that the gravitational effect in this region is massive. It follows that even light would be drawn into a black hole and unable to escape. For further research, it is expected to extend the Klein-Gordon equation in relativistic quantum mechanics to modified general relativity theory. This theory offers a different way of looking at the effects of gravity in quantum field theory.

On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.

Oscillating cosmic evolution andconstraints on big bang nucleosynthesisin the extended Starobinsky model

We investigate the cosmic evolutions in the extended Starobinsky model (eSM) obtained by adding one RabRab term to the Starobinsky model. We discuss the possibility of various cosmic evolutions with a special focus on the radiation-dominated era (RDE). Using simple assumptions, a second-order non-linear differential equation describing the various cosmic evolutions in the eSM is introduced. By solving this non-linear equation numerically, we show that the various cosmic evolutions, such as the standard cosmic evolution (a ∝ t 1/2) and a unique oscillating cosmic evolution, are feasible due to the effects of higher-order terms introduced beyond Einstein's gravity. Furthermore, we consider big bang nucleosynthesis (BBN), which is the most important observational result in the RDE, to constrain the free parameters of the eSM. The primordial abundances of the light elements, such as 4He, D, 3He, 7Li, and 6Li by the cosmic evolutions are compared with the most recent observational data. It turns out that most non-standard cosmic evolutions can not easily satisfy these BBN constraints, but a free parameter of the viable models with the oscillating cosmic evolution is shown to have an upper limit by the constraints. In particular, we find that the free parameter is most sensitive to deuterium and 4He abundances, which are being precisely measured among other elements. Therefore, more accurate measurements in the near future may enable us to distinguish the eSM from the standard model as well as other models.

Об устойчивости в целом по вероятности решений нелинейных дифференциальных уравнений второго порядка, возмущенных белым шумом. V

Настоящая статья является продолжением предыдущей статьи и представляет собой пятую, завершающую часть работы авторов. В первых четырех частях работы были приведены предварительные сведения из теории вероятностей, теории случайных процессов, теории стохастических дифференциальных уравнений и теории устойчивости стохастических дифференциальных уравнений. В работе рассматриваются нелинейные автономные дифференциальные уравнения второго порядка при случайных возмущениях их параметров. В качестве возмущений рассматриваются гауссовский процесс «белого шума» и случайный процесс из более широкого класса без специальных предположений о виде процесса. В статье получены новые достаточные условия асимптотической устойчивости по вероятности решений нелинейных стохастических уравнений второго порядка. Даны вероятностные оценки пребывания случайной траектории в ограниченной области фазовой плоскости. В качестве примера рассматривается гармонический осциллятор, возмущенный белым шумом. This paper is a continuation of the previous paper and presents the fifth part of the authors’ work. In the first four parts of the work, we presented some preliminaries from probability theory, stochastic processes, theory of stochastic differential equations and stability theory of stochastic differential equations. In the work, nonlinear autonomous second-order differential equations under random perturbations are considered. Here, in the paper we give new sufficient conditions for stochastic asymptotic stability of solutions of the nonlinear second-order stochastic differential equations. The probabilistic estimations of the remaining of a random trajectory in a bounded region of the phase plane are given. As an example, harmonic oscillator perturbed by white noise random process is considered

Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material

The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m², and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σ_rr = (-1.2973e-05)kg/m/s². We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with a singularly perturbed Laguerre-type weight

We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large n n asymptotic expansions of the recurrence coefficients. In addition, we also obtain a system of differential-difference equations for the recurrence coefficients.

Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements

Abstract We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one. We show that an optimal ellipticity condition, relating the nonlinear operator to the contraction constant of the shifted double-layer integral operator, is guaranteed by choosing a particular linear combination. These results generalize those obtained by Of and Steinbach [Is the one-equation coupling of finite and boundary element methods always stable?, ZAMM Z. Angew. Math. Mech. 93 (2013), 6–7, 476–484] and [On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems, Numer. Math. 127 (2014), 3, 567–593], and by Steinbach [A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), 4, 1521–1531], where the simple sum of the two coupling equations has been considered. Numerical examples confirm the theoretical results on the sharpness of the presented estimates.

Linearisation of a second-order nonlinear ordinary differential equation

We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f(v)∂v. The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3,R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

Modeling and analysis of dynamic load factor in a fail-safe crane mechanism equipped with a hydraulic equalizing cylinder

In the design and manufacturing of sensitive crane mechanisms, in particular, those utilized in nuclear facilities, safety is of utmost importance. A few designs are based on a hydraulic equalizing cylinder where the purpose is mainly to dampen the dynamic load factor generated by the sudden failure of the wire rope system. To the best of the authors' knowledge, design equations and engineering models for such an intricate dynamic mechanism are not reported anywhere in the literature. Toward achieving this goal, a model is proposed for the dynamic behavior of a hydraulic equalizing cylinder when coupled to a single-failure-proof cross-reeved crane hoist. The constitutive equations involve the time-dependent incompressible Navier-Stokes equations. These equations are coupled with the dynamics of the lower block and the elastic wire rope. A nonlinear second-order ordinary differential equation is obtained, which is solved analytically. This yields a closed-form relation for the dynamic load factor. The validity of the new model is investigated using experimental studies, which are presented in this work. The normalized root mean square error for the dynamic load factor obtained by the model compares favorably to the experimental results, where the maximum observed error is 3.9 percent. The results confirm that the current model can be used as a design equation by practitioners.

Analysis of Von Kármán Swirling Flows Due to a Porous Rotating Disk Electrode

The study of Von Kármán swirling flow is a subject of active interest due to its applications in a wide range of fields, including biofuel manufacturing, rotating heat exchangers, rotating disc reactors, liquid metal pumping engines, food processing, electric power generating systems, designs of multi-pore distributors, and many others. This paper focusses on investigating Von Kármán swirling flows of viscous incompressible fluid due to a rotating disk electrode. The model is based on a system of four coupled second-order non-linear differential equations. The purpose of the present communication is to derive analytical expressions of velocity components by solving the non-linear equations using the homotopy analysis method. Combined effects of the slip λ and porosity γ parameters are studied in detail. If either parameter is increased, all velocity components are reduced, as both have the same effect on the mean velocity profiles. The porosity parameter γ increases the moment coefficient at the disk surface, which monotonically decreases with the slip parameter λ. The analytical results are also compared with numerical solutions, which are in satisfactory agreement. Furthermore, the effects of porosity and slip parameters on velocity profiles are discussed.

Numerical solution of second-order nonlinear partial differential equations originating from physical phenomena using Hermite based block methods

A Hermite based block method (HBBM) is proposed for the numerical solution of second-order non-linear elliptic partial differential equations (PDEs). The development of the method was accomplished through the methodology of interpolation and collocation procedures. The method’s analysis reveals that it satisfied the requirements for a numerical technique to be convergent. The implementation of the method is extensively discussed. Five numerical examples originating from physical phenomena are presented, and the applicability and accuracy of the HBBM are established by comparing them with the existing methods; the haar wavelet collocation method, the modified cubic B-spline collocation method, and the modified decomposition method. The proposed methods of HBBM are more accurate, stable, and convergent

The radial slump of a gravity current in a confined porous layer

The flow of a gravity current widely occurs in many geophysical, environmental and industrial processes. We focus on the overlooked process of radial slump of a gravity current in confined porous layers, which is related to the post-injection flow dynamics of many subsurface displacement processes, initiated by fluid injection through vertical wells. Specifically, we describe the time evolution of the interface shape and location of the propagating front, governed by a second-order nonlinear partial differential equation (PDE). By solving a nonlinear eigenvalue problem, we identify a self-similar solution to describe the dynamics of confined flows at early times, when the interface attaches to both the top and bottom boundaries of the porous layer. As time progresses, the flow gradually becomes unconfined, as gravity-driven slump causes the interface to detach from one boundary; the interface evolution at late times is then described by a self-similar solution of a nonlinear diffusion equation. Time transition from the early-time confined towards late-time unconfined self-similar solutions is also verified by comparing the self-similar solutions and numerical solutions of the governing PDE for both the interface shape and the frontal location. Interestingly, two special vertical locations have also been identified where the best agreement appears between the PDE numerical solutions and the early-time self-similar solutions for the interface shape, which is a novel feature of the radial flow, and the degree of agreement is not impacted by the time evolution.

Exact Closed-Form Solution for the Oscillator with a New Type of Mixed Nonlinear Restitution Force

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity of any order. The mathematical model of the oscillator is a homogenous strong nonlinear second-order differential equation with an integer- or non-integer-order mixed term. In the paper, an analytical procedure for solving this new type of strong nonlinear equation is developed. The approximate solution is assumed as the perturbed version of the exact solution in the form of a sine Ateb function. As a result, it is obtained that the amplitude, period, and frequency of vibration depend not only on the coefficient and order of nonlinearity, but also on the initial velocity. The procedure is tested on two examples: oscillator perturbed with small linear damping and small linear displacement functions. The analytically obtained results are compared with the exact numerical ones and show good agreement. It is concluded that the mathematical model and also the procedure developed in the paper would be convenient for prediction of motion for this type of oscillator without necessary experimental testing.

Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

In this study, we present the stability analysis of a fully explicit finite difference method (FDM) for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear partial differential equation (PDE), which describes the antiphase boundaries of the binary phase separation. In the presented stability analysis, we consider the explicit Euler method for the temporal derivative and second-order finite difference in the space direction. The explicit scheme is fast and accurate because it uses a small time step, however, it has a temporal step constraint. We analyze and compute that the explicit time step constraint formula guarantees the discrete maximum principle for the numerical solutions of the AC equation. The numerical stability of the explicit scheme automatically holds when we use the time satisfying the discrete maximum principle. The computational numerical experiments demonstrate the stability, discrete maximum principle, and accuracy of the explicit scheme for the constrained time step. Furthermore, it is shown that the time step obtained is not severe restriction when we consider the temporal accuracy.

Моделювання динаміки колісних транспортних засобів під час руху шляхом із нерівностями

The paper outlines the main provisions of the methodology for analytical research of the individual road irregularities impact on the longitudinal-angular oscillations of the wheeled vehicle sprung body. A flat system of three bodies (front, rear bridges, sprung part), whose relative motion is in the vertical plane, was chosen as the physical model for the research. Its peculiarity is that the sprung and unsprung parts interact with each other as elastic shock absorbers with non-linear characteristics of the restoring force. As for road irregularities, it is considered that they are described by smooth functions and the tires are in constant contact with the supporting surface during the wheeled vehicle movement. A mathematical model of the sprung part dynamics was built, which is a second-order nonlinear differential equation with the following feature: its right-hand side is a piecewise continuous function. Based on physically justified assumptions, the differential equation solution was constructed, which describes the relative longitudinal-angular oscillations of the sprung part. It is based on the idea of using: the special periodic Ateb-functions to construct the solution of differential equations with exponential nonlinearity; the construction of asymptotic approximations using the specified functions for new classes of differential equations. Taken together, the above made it possible to obtain differential equations in the standard form, which describe the amplitude-frequency characteristics of the sprung part oscillations. As for the individual irregularities influence on the sprung part dynamics, the amplitude of the longitudinal-angular oscillations of the exit from the irregularity is smaller: at higher vehicle speed; for suspension systems with a regressive change law of the restoring force of elastic shock absorbers; at smaller values of their static deformation for the progressive characteristic of the suspension system (and at larger values for the regressive one). Analytical dependences were obtained, calculations were made and dependences were constructed that describe the amplitude and frequency of these oscillations caused by the irregularities parameters and motion speed.The obtained results can be a basis not only for evaluating the quality of the suspension system, but also for developing algorithms for managing the suspension stiffness in order to improve operational characteristics.

On a Nonlinear Second-Order Ordinary Differential Equation

We consider a nonlinear second-order ordinary differential equation of a special form whose particular case arises when constructing exact solutions of the nonlinear heat equation with a power-law coefficient. Conditions are obtained for the parameters under which the equation admits a single integration. A number of examples of constructing exact solutions expressed in terms of elementary functions or in terms of the Lambert function are given.

A Novel Seventh-Order Implicit Block Hybrid Nyström-Type Method for Second- Order Boundary Value Problems

This paper introduces a novel approach for solving second-order nonlinear differential equations, with a primary focus on the Bratu problem, which holds significant importance in diverse scientific areas. Existing methods for solving this problem have limitations, prompting the development of the Block Hybrid Nystrom-Type Method (BHNTM). BHNTM utilizes the Bhaskara points derived, using the Bhaskara cosine approximation formula. The method seeks a numerical solution in the form of a power series polynomial, efficiently determining coefficients. The paper discusses BHNTM’s convergence, zero stability, and consistency properties, substantiated through numerical experiments, highlighting its accuracy as a solver for Bratu-type equations. This research contributes to the field of numerical analysis by offering an alternative, effective approach to tackle complex second-order nonlinear differential equations, addressing critical challenges in various scientific domains.

Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

<p style='text-indent:20px;'>We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.</p>

Multiple nontrivial periodic solutions to a second-order partial difference equation

<abstract><p>In this article, applying variational technique as well as critical point theory, we establish a series of criteria to ensure the existence and multiplicity of nontrivial periodic solutions to a second-order nonlinear partial difference equation. Our results generalize some known results. Moreover, numerical stimulations are presented to illustrate applications of our major findings.</p></abstract>