Second-order Differential Equations Research Articles

Coincidence and common fixed points for F-Contractive mappings

The purpose of this article is to establish the existence and uniqueness of coincidence and common fixed point of discontinuous non-compatible faintly compatible pair of self maps in non-complete metric space without using containment requirement of range space of involved maps satisfying Ciric type F-contraction and Hardy-Roger type F-contraction. Some illustrative examples associated with pictographic validations are provided to demonstrate the main results and to show the genuineness of our results. We consider the application of our results to the study of a two-point boundary value problem related to second order differential equation, solve the two-point boundary value problem of the second-order differential equation arising in electric circuit equation, and also apply our results to Volterra type integral equation using Ciric type F-contraction as well as Hardy Roger type F-contraction. Mathematics subject classification: 47H10; 54H25; 54E50

Pikoe: A computer program for distorted-wave impulse approximation calculation for proton induced nucleon knockout reactions

pikoe, Proton-Induced KnockOut reaction calculation for Exclusive processes, is a Fortran 90 program that calculates triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for proton-induced nucleon knockout reactions in normal and inverse kinematics. The distorted-wave impulse approximation with the factorization approximation to the nucleon-nucleon transition matrix is adopted, and the distorted waves of the incoming proton and the outgoing two nucleons are calculated quantum mechanically. Kinematics of the reaction particles are treated in a relativistic manner, which gives the proper asymptotics of the three-body scattering wave in the plane-wave limit. Program summaryProgram Title: pikoeCPC Library link to program files:https://doi.org/10.17632/m594h58kck.1Licensing provisions: MITProgramming language: Fortran 90Supplementary material: See https://www.rcnp.osaka-u.ac.jp/~kazuyuki/pikoe/index.phpNature of problem: Calculation of triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for nucleon knockout reactions with proton beams/targets in normal/inverse kinematics.Solution method: Numerov method for solving single-channel second-order differential equations; multi dimensional integration with the Gauss-Legendre quadrature rule.Additional comments including restrictions and unusual features: The global optical potential by Koning and Deraloche [1] for nucleons at energies from 1 keV to 200 MeV, for (near-)spherical nuclei with mass number between 24 and 209, is implemented in the program. Tables of nucleon-nucleon (NN) elastic differential cross sections and NN transition amplitudes calculated with the NN effective interaction by Franey and Love [2] are provided as external files for pikoe (included in the package). Users are encouraged to prepare tables for the optical potentials and the cross sections of the elementary process according to the reaction system of interest by themselves. With external files thus prepared, pikoe can be applied to various kinds of knockout processes such as α or deuteron knockout by proton, α-induced knockout reactions, etc. Calculation of spin observables, the vector analyzing power in fact, however, is restricted to nucleon-induced nucleon knockout processes with the coplanar kinematics.

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

In this work, the geometric nature of solutions to two second-order differential equations, zy′′(z)+a(z)y′(z)+b(z)y(z)=0 and z2y′′(z)+a(z)y′(z)+b(z)y(z)=d(z), is studied. Here, a(z), b(z), and d(z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F(z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1+z. We construct several examples by the judicious choice of a(z), b(z), and d(z). The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion.

Discounted Risk-Sensitive Optimal Control of Switching Diffusions: Viscosity Solution and Numerical Approximation

This work considers the infinite horizon discounted risk-sensitive optimal control problem for the switching diffusions with a compact control space and controlled through the drift; thus, the the generator of the switching diffusions also depends on the controls. Note that the running cost of interest can be unbounded, so a decent estimation on the value function is obtained, under suitable conditions. To solve such a risk-sensitive optimal control problem, we adopt the viscosity solution methods and propose a numerical approximation scheme. We can verify that the value function of the optimal control problem solves the optimality equation as the unique viscosity solution. The optimality equation is also called the Hamilton–Jacobi–Bellman (HJB) equation, which is a second-order partial differential equation (PDE). Since, the explicit solutions to such PDEs are usually difficult to obtain, the finite difference approximation scheme is derived to approximate the value function. As a byproduct, the ϵ-optimal control of finite difference type is also obtained.

The Jacobi last multiplier, Lagrangian and Hamiltonian for Levinson–Smith type equations

We derive the Jacobi last multiplier for second-order ordinary differential equations of the Levinson–Smith type by using a combination of previous techniques employed for the Liénard-I and II classes of equations. This opens up the possibility for a Lagrangian or Hamiltonian description of the systems governed by the Levinson–Smith type of equations as well as simplifying the problem of finding first integrals of motion. The procedure has been illustrated by a number of suitable examples alongwith Kamke's equation with explicit time dependent coefficients.

Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

This research addresses analytical solutions for vibration performance of simply supported sandwich plate subjected to coupled mechanical-electric-magnetic-thermal loads. The plate is made of graphene reinforced metal matrix composite (GRMMC) core and magneto-electro-elastic (MEE) face sheets. Three graphene distribution patterns, FG-X, FG-O, and UD are considered for evaluating the reinforcement efficiency of graphene. The Reddy’s higher order shear deformation plate theory and Hamilton’s principle are employed to formulate the equations of motion of the sandwich plate. Then, using double trigonometric functions for variables and applying Bubnov–Galerkin procedure, the system of nonlinear second-order differential equations is constructed to reveal the natural frequency, frequency ratio, and dynamic response of the sandwich plate. The influences of graphene distribution patterns and volume fraction, magnetic and electric potentials, temperature change, and volume fraction of piezoelectric phase are discussed in details. This research provides basis for the design of sandwich composite structures, especially smart structures and devices.

On the solution of T−controllable abstract fractional differential equations with impulsive effects

In this research article, we delimitate the definition of mild solution for abstract fractional differential equations with state-dependent delay (AFDEw/SDD) of order α ∈(1,2) with impulsive effects and compare the solution to the second-order impulsive differential equations. Further, we obtain sufficient conditions of the existence of mild solution for instantaneous and non-instantaneous impulsive fractional functional differential inclusions with state-dependent delay (IFDIw/SDD) using the multi-valued fixed point theory and operator techniques. Furthermore, we study the trajectory controllability (T−controllability) of the AFDEw/SDD. At last, we present some examples to illustrate the sufficient conditions involving partial and ordinary derivatives.

Oscillation Theorems for Nonlinear Second-Order Delay Differential Equations with Some Sublinear Neutral Terms via Canonical Transform

Abstract New sufficient conditions for oscillation of a second-order nonlinear differential equation with some sublinear neutral terms are established via canonical transform and integral averaging method. Examples are provided to illustrate the significance and novelty of the presented results.

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples.

Harmonic-wavelet approach for response spectrum estimation of vehicle and bridge systems with uncertain parameters subjected to stochastic excitation

Vehicle and bridge systems (VBSs) are subjected to stochastic excitations. Significant research effort has been devoted to the stochastic response analysis of VBSs. In most previous studies, VBSs have been treated as deterministic systems with certain parameters. However, the real VBSs always have uncertain parameters, such as the mass and spring stiffness of the vehicles and elastic modulus of the bridge material, which significantly affect the response power spectrum (RPS) estimation. Moreover, the equations of motion of the VBSs are essentially second-order differential equations with time-dependent and random coefficient matrices, and the excitation-response spectral relation cannot be directly described using the frequency response functions. These properties render the available methods unsuitable for the RPS estimation of the VBSs when both structural uncertainty and stochastic excitations are considered. In the present study, a harmonic-wavelet approach is proposed to estimate the time-varying RPSs of the VBSs with random parameters. The second-order differential equations of motion of VBSs with random parameters are first transformed into linear algebraic equations with random coefficient matrices using a harmonic-wavelet analysis technique; then, the excitation-response spectral relation of VBSs is derived based on these linear algebraic equations. Finally, the RPSs of VBSs with uncertain parameters are solved using the perturbation technique. The accuracy of the proposed method was verified by comparing it with the Monte Carlo method using a numerical example of railway VBS. The numerical simulation results show that the parameter uncertainties significantly affect the RPSs of the VBSs and increase the amplitude of the time-varying RPSs. The uncertainty of vehicle stiffness was the most sensitive factor for vehicle RPSs, and the bridge elastic modulus was the most sensitive factor for the bridge RPSs.

A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving [formula omitted]: Application to Verhulst logistic growth model

Explicit exponentially-fitted two-derivative Runge–Kutta–Nyström method with single f-function and multiple third derivatives is proposed for solving special type of second-order ordinary differential equations with exponential solutions. B-series and rooted tree theory for the proposed method are developed for the derivation of order conditions. Then, we build frequency-dependent coefficients for the proposed method by integrating the second-order initial value problem exactly with solution in the linear composition of set functions eλt and e−λt with λ∈R. An exponentially-fitted two-derivative Runge–Kutta–Nyström method with three stages fifth order is derived. Linear stability and stability region of the proposed method are analyzed. The numerical tests show that the proposed method is more effective than other existing methods with similar algebraic order in the integration of special type of second-order ordinary differential equations with exponential solutions. Also, the proposed method is used to solve a famous application problem, Verhulst logistic growth model and the result shows the proposed method still works effectively for solving this model.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.

O razrushenii resheniya odnoy zadachi dlya sobolevskogo uravneniya s nekoertsitivnym istochnikom

We consider an initial–boundary value problem for a nonlinear Sobolev-type second-order differential equation with a noncoercive source describing the behavior of charges in a semiconductor plasma in an external field. The local solvability of this problem is proved, and upper bounds are obtained for the solution blow-up time.

Parametric interaction of modes in the presence of quadratic or cubic nonlinearity

The purpose of this work is a study of the dynamics of the systems of ordinary differential equations of the second order, which is obtained using the Lagrange formalism. These systems describe the parametric interaction of oscillators (modes) in the presence of a general quadratic or cubic nonlinearity. Also, we compare the dynamics of the systems of ordinary differential equations of the second order and dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models in order to determine the possibilities of the latter models when modeling coupled oscillators of the above type. Methods. The study is based on the numerical solution using the methods of the theory of the obtained analytically differential equations. Results. For both systems of second-order differential equations, is was presented a chart of in the parameter plane, a graphs of Lyapunov exponents at the value of the parameter that specifies the dissipation of oscillators, a time dependences of the generalized coordinates of oscillators and its amplitudes, portraits of attractors, a projection of the attractors on a phase planes of oscillators. A comparison with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models is carried out. These models are three-dimensional real approximations of the above systems obtained by the method of slowly varying amplitudes. Conclusion. The study of the constructed systems showed that in the parameter space there are regions corresponding to both various regular regimes, such as the equilibrium position, limit cycle, two-frequency tori, and chaotic regimes. For both systems, it was shown that the transition to chaos occurs as a result of a sequence of period doubling bifurcations of the tori. In addition, a comparison of the dynamics of the constructed systems with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models allows us to assert that if the Vyshkind–Rabinovich model predicts the dynamics of the corresponding initial system well enough, then the Rabinovich–Fabrikant model does not have such a property.

Existence Results for Systems of Nonlinear Second-Order and Impulsive Differential Equations with Periodic Boundary

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, in the general case, some Banach space. In this work, sufficient conditions for the existence of solutions for a system with parameters are obtained. The results are obtained by using fixed point theorems for operators on a cone. Our approach is based on Schaefer’s fixed point theorem more precisely. In addition, the existence of positive solutions is also investigated.

The most probable transition paths of stochastic dynamical systems: a sufficient and necessary characterisation

The most probable transition paths (MPTPs) of a stochastic dynamical system are the global minimisers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange (EL) equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterisation for the MPTPs of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the MPTPs are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager–Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler’s model, the first-order differential equations determining the MPTPs are shown analytically to imply the EL equations of the Onsager–Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results.

A Numerical scheme to Solve Boundary Value Problems Involving Singular Perturbation

The Wang-Ball polynomials operational matrices of the derivatives are used in this study to solve singular perturbed second-order differential equations (SPSODEs) with boundary conditions. Using the matrix of Wang-Ball polynomials, the main singular perturbation problem is converted into linear algebraic equation systems. The coefficients of the required approximate solution are obtained from the solution of this system. The residual correction approach was also used to improve an error, and the results were compared to other reported numerical methods. Several examples are used to illustrate both the reliability and usefulness of the Wang-Ball operational matrices. The Wang Ball approach has the ability to improve the outcomes by minimizing the degree of error between approximate and exact solutions. The Wang-Ball series has shown its usefulness in solving any real-life scenario model as first- or second-order differential equations (DEs).

Existence, uniqueness, boundedness and stability of periodic solutions of a certain second-order nonlinear differential equation with damping and resonance effects

Summary In this paper, some qualitative behaviors of solutions for certain second-order nonlinear differential equation with damping and resonance effects are considered. By employing Lyapunov’s direct method, a complete Lyapunov function was used to investigate the stability of the system. Krasnoselskii’s fixed point theorem was used to establish sufficient conditions that guaranteed the existence and boundedness of a unique solution. The results show that the equilibrium point was asymptotically stable. Furthermore, a test for periodicity was conducted using the Bendixson criterion, and the results showed that the solution of the second-order nonlinear differential equation is aperiodic, which extends some results from the literature.

Oscillatory behavior of second-order nonlinear noncanonical neutral differential equations

Abstract This paper discusses the oscillatory behavior of solutions to a class of second-order nonlinear noncanonical neutral differential equations. Sufficient conditions for all solutions to be oscillatory are given. Examples are provided to illustrate all the main results obtained.