Solutions Of Second-order Differential Equations Research Articles

Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples.

New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results.

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H 1 - and L ∞ -norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.

Periodic solutions to superlinear indefinite planar systems: A topological degree approach

We deal with a planar differential system of the form{u′=h(t,v),v′=−λa(t)g(u), where h is T-periodic in the first variable and strictly increasing in the second variable, λ>0, a is a sign-changing T-periodic weight function and g is superlinear. Based on the coincidence degree theory, in dependence of λ, we prove the existence of T-periodic solutions (u,v) such that u(t)>0 for all t∈R. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or ϕ-Laplacian-type differential operators).

Necessary existence conditions of Pω(Y0, Y1, λ0)-solutions of second-order differential equation with rapidly varying nonliniarity

We consider a differential equation of the second order of the general form y’’ = f(t, y, y’), where f : [a, ω] x ΔY0 x ΔY1 → R – a continuous function, −∞ < a < ω ≤ +∞, ΔYi – one-side neighborhood of Yi, Yi ∈ {0, ±∞} (i ∈ {0, 1}). Under certain conditions for the function f, this equation can be represented close to the two-term differential equation, namely y’’ = α0p(t)ϕ1(y’)(1 + o(1)) at t↑ω, where ϕ1 is a rapidly varying function at y’ → Y1. Found the necessary conditions for the existence of solutions for which lim<sup>t↑ω</sup> y(i) (t) + Yi (i ∈ {0,1}), lim<sup>t↑ω</sup> [y’(t)]2/y(t)y’’(t) = λ0, so called Pω(Y0, Y1, λ0)-solutions. This type of solution was previously presented in works by Evtukhov V. M., Belozerova M. O. when studying the two-term equation y’’ = α0p(t)ϕ0(y)ϕ1(y’), where α0 ∈ {−1, 1}, p : [a, ω[→]0, +∞[–continuous function, ϕi : ΔYi →]0, +∞[(i = 0, 1)–continuous regularly variables for z → Yi (i = 0, 1) functions of orders σi (i = 0, 1), and σ0 + σ1 ≠ 1. Further, in the studies of V. M. Evtukhov, A. G. Chernikova for equation y’’ = α0p(t)ϕ0(y) necessary and sufficient conditions are established existence, as well as asymptotic at t↑ω representations Pω(Y0, Y1, λ0)-solutions in the case when ϕ0 is a rapidly varying function at y → Y0.

Uniform Asymptotics of Solutions of Second-Order Differential Equations with Meromorphic Coefficients in a Neighborhood of Singular Points and Their Applications

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular.

On the stability of the zero solution of a periodic reversible second-order differential equation

The problem of the stability of the zero solution of the second-order differential equation describing the periodic perturbations of an oscillator with a nonlinear reducing force is studied. The problem in the autonomous case was solved by A.M. Lyapunov. The so called transcendental case when all members of the decomposition of the right part of the differential equation into series are to be taken into account, is considered. This case takes place for reversible differential equations, i. e. equations that do not change when time is replaced by the opposite value. The problem is solved by the methods of the KAM theory, according to which in any neighborhood of the equilibrium position at the origin of the phase plane there are periodic invariant two-dimensional tori that separate the three-dimensional configuration space. These tori are considered as two-dimensional periodic invariant surfaces covering the time axis from where the stability (non-asymptotic) of the zero solution followed. The problem to be solved is characterized by the fact that the unperturbed part of the equation contains a dissipative term (a term dependent on velocity) which has the same order of smallness as the restoring force. It is established that if a dissipative part of the perturbation is small enough then the unperturbed movement is stable according to Lyapunov.

Solution for Second-Order Differential Equation Using Least Square Method

Solution for Second-Order Differential Equation Using Least Square Method

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as , where r k is the kth root of the digamma function.

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator {mathcal {L}}equiv phi ,D^2+psi ,D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers lambda _n, ngeqslant 0, and a Sobolev bilinear form B(p,q)=∑k=0Nuk,p(k)q(k),N⩾0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {B}}(p,q)\\,=\\,\\sum _{k=0}^N\\left\\langle {{\\mathbf {u}}_k,\\,p^{(k)}\\,q^{(k)}}\\right\\rangle , \\quad N\\geqslant 0, \\end{aligned}$$\\end{document}where {mathbf {u}}_k, 0leqslant k leqslant N, are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation {mathcal {L}}[y]=lambda _n,y with respect to {mathcal {B}}. We show that such polynomials are orthogonal with respect to {mathcal {B}} if the Pearson equations D(phi ,{mathbf {u}}_k)=(psi +k,phi '),{mathbf {u}}_k, 0leqslant k leqslant N, are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

Periodic solutions for second-order difference equations with quadratic\u2013supquadratic condition

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.

Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit

A sudden application of sources results in time-varying currents and voltages in the circuit known as transients. This phenomenon occurs frequently during switching. A simple circuit constituting a resistor, an inductor, and a capacitor is termed an RLC circuit. It may be in parallel or series configuration or both. Different values of damping factors determine the different nature of the transient response. We applied different numerical solution methods such as explicit (forward) Euler method, third-order Runge-Kutta (RK3) method, and Butcher's fifth-order Runge-Kutta (BRK5) method to approximate the solution of second-order differential equation with initial value problem (IVP). We thoroughly compared the numerical solutions obtained by these methods with the necessary visualization and analysis of error. We also examined the superiority of these methods over one another and the appropriateness of numerical methods for different damping conditions is explored. With high accuracy of the approximation and thorough analysis of the observation, we found Butcher's fifth-order Runge-Kutta (BRK5) method to be the best numerical technique. Regarding the different values of damping factors, we considered the further possibility of discussion and analysis of this iterative method.

New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

Asymptotic Representations of Rapidly Varying Solutions of Second-Order Differential Equations with Rapidly and Regularly Varying Nonlinearities

We establish necessary and sufficient conditions for the existence of Pω (Y0 ,Y1, λ0)-solutions of one class of second-order differential equations with rapidly and regularly varying nonlinearities. This result is a natural generalization of the results of previous investigations carried out by V. М. Evtukhov and his school. We also established the asymptotic representations of these solutions and their first-order derivatives.

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n−1, and n−2 respectively. We show that for n≥3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

Penyelesaian Numerik Persamaan Diferensial Orde Dua Dengan Metode Runge-Kutta Orde Empat Pada Rangkaian Listrik Seri LC

One alternative to solve second order differential equations by numerical methods, specificallynon-liner differential equations is the Runge-Kutta fourth order method. The Runge-Kutta fourth ordermethod is a numerical method that has high degree of precision and accuracy when compared to othernumerical methods. In this paper we will discuss the numerical solution of second order differentialequations on LC series circuit problem using the Runge-Kutta fourth order method. The numericalsolution generated by the computational calculation using the MATLAB program, the strong current andcharge are obtaind from t = 0 and t =0,5 second and different step size values

Applications of Conley index theory on difference equations with non-resonance

Inspired by Chang (1981) and Long (1990), we consider the existence of periodic solutions for second-order difference equations with asymptotically linear nonlinearity via Conley index theory. Moreover, when we study the following one-dimensional autonomous difference equations with form Δ2xn−1+f(xn)=0,n∈Z,xn∈R,we can derive at least three periodic solutions under suitable assumptions.

Positive solutions for second-order differential equations with singularities and separated integral boundary conditions

We study the existence of positive solutions for second-order differential equations with separated integral boundary conditions. The nonlinear part of the equation involves the derivative and may be singular for the second and third space variables. The result ensures existence of a positive solution when the parameters are in certain ranges. The proof depends on general properties of the associated Green’s function and the Krasnosel’skii–Guo fixed point theorem applied to a perturbed Hammerstein integral operator. Both numerical and analytical examples are constructed to illustrate applications of the theorem to a group of equations. The result generalizes previous work.