Second-order Nonlinear Differential Equations Research Articles

Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning

This study addresses the problem of predicting convergence outcomes in the Duffing equation, a nonlinear second-order differential equation. The Duffing equation exhibits intriguing behavior in both undamped free vibration and forced vibration with damping, making it a subject of significant interest. In undamped free vibration, the convergence result oscillates randomly between 1 and −1, contingent upon initial conditions. For forced vibration with damping, multiple variables, including initial conditions and external forces, influence the vibration patterns, leading to diverse outcomes. To tackle this complex problem, we employ the fourth-order Runge–Kutta method to gather convergence results for both scenarios. Our approach leverages machine learning techniques, specifically the Long Short-Term Memory (LSTM) model and the LSTM-Neural Network (LSTM-NN) hybrid model. The LSTM-NN model, featuring additional hidden layers of neurons, offers enhanced predictive capabilities, achieving an impressive 98% accuracy on binary datasets. However, when predicting multiple solutions, the traditional LSTM method excels. The research encompasses three critical stages: data preprocessing, model training, and verification. Our findings demonstrate that while the LSTM-NN model performs exceptionally well in predicting binary outcomes, the LSTM model surpasses it in predicting multiple solutions.

The Recurrence Coefficients of Orthogonal Polynomials with a Weight Interpolating between the Laguerre Weight and the Exponential Cubic Weight

In this paper, we consider the orthogonal polynomials with respect to the weight w(x)=w(x;s):=xλe−N[x+s(x3−x)],x∈R+, where λ>0, N>0 and 0≤s≤1. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients αn(s) and βn(s). We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when q:=n/N is fixed as n→∞.

A New Class of Nonlinear Resonance Networks Modeled by Levinson–Smith and Liénard Equations

We propose a new class of nonlinear resonance networks which are modeled by standard second-order nonlinear differential equations of the Levinson-Smith type or its subset, the Liénard type. In particular, we modify series and parallel RLC resonance networks such that the passive resistor in these networks is composed of a fixed part and a variable voltage-or current-controlled part. The variable resistor is then made controllable by one of the circuit’s state-space variables leading to an embedded self feedback control mechanism. A possible discrete component circuit realizing the proposed concept is presented along with its simulations and experimental verification. The filter response behavior of one of the modified series resonance circuits is also experimentally verified using a Field Programmable Analog Array.

Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions.

Oscillation Analysis Algorithm for Nonlinear Second-Order Neutral Differential Equations

Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independent variables. The nonlinear second-order neutral differential equations studied in this paper are a class of quadratic differentiable equations that include delay terms. According to the t-value interval in the differential equation function, a basis is needed for selecting the initial values of the differential equations. The initial value of the differential equation is calculated with the initial value calculation formula, and the existence of the solution of the nonlinear second-order neutral differential equation is determined using the condensation mapping fixed-point theorem. Thus, the oscillation analysis of nonlinear differential equations is realized. The experimental results indicate that the nonlinear neutral differential equation can analyze the oscillation behavior of the circuit in the Colpitts oscillator by constructing a solution equation for the oscillation frequency and optimizing the circuit design. It provides a more accurate control for practical applications.

Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay

This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.

Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets

ABSTRACT In this paper, we present an effective method under Taylor wavelets and collocation technique to find an approximate solution of linear and non-linear second-order singular value differential equations. Application of this method on various problems confirms the efficiency, easy applicability, and rapid computation. The obtained solution is compared with some other existing numerical solutions. For many problems, this method gives a solution same as the exact solution of the problem which confirms the effectiveness and accuracy of this method. Additionally, we include graphs and figures to demonstrate that the Taylor wavelet method offers better accuracy for a number of problems. MATLAB software is used to process relative data and execute calculations.

Wavelets approach for the solution of nonlinear variable delay differential equations

Abstract In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. the Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

On the existence of a positive solution to a boundary value problem for one nonlinear functional-differential equation of the second order

This article considers a boundary value problem for one non-linear second-order functional differential equation on the segment [0, 1] with an integral boundary condition at one of the ends of the segment. Using the well-known Go-Krasnoselsky theorem, sufficient conditions for the existence of at least one positive solution to the problem under consideration are established. A non-trivial example is given, illustrating the fulfillment of the conditions for the unique solvability of the problem posed.

Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, the closed-form solutions can be established. The obtained solutions have a periodical behavior. These geometrical properties allow reducing the initial system to a second-order nonlinear differential equation. The latter equation is solved analytically using the OHPM procedure. The validation of the OHPM method is presented for three cases of the physical parameters. The advantages of the OHPM technique, such as the small number of iterations (the efficiency), the convergence control (in the sense that the semi-analytical solutions are approaching the exact solution), and the writing of the solutions in an effective form, are shown graphically and with tables. The accuracy of the results provides good agreement between the analytical and corresponding numerical results. Other dynamic systems with similar geometrical properties could be successfully solved using the same procedure.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.

Asymptotic Behavior of the Solutions of Essentially Nonlinear Nonautonomous Second-Order Differential Equations Close to Linear Functions

Asymptotic Behavior of the Solutions of Essentially Nonlinear Nonautonomous Second-Order Differential Equations Close to Linear Functions

Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term

We obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to \(0\) or \(\infty\). Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examplesthat cannot be included in earlier works.

Thermal performance of stretching/shrinking fully wet porous cylindrical and conical pin fin structures by differential transformation approach

In this paper, the thermal behavior of cylindrical and conical pin fin structures operating in a convective–radiative environment has been examined. The fully wet porous fin structures are assumed to be mounted with a stretching/shrinking mechanism similar to a conveyer belt. The conical pin fin has been comparatively analyzed with the cylindrical one for three distinct cases of stretching, stagnant, and shrinking. Further, the Darcy model has been employed to analyze the fluid–solid interactions. The differential transformation method (DTM) has been used to solve the resulting second-order nonlinear ordinary differential equations. On the thermal performance of conical and cylindrical pin fin structures, the effects of the stretching/shrinking parameter, Peclet number, wet porous parameter, convective parameter, and radiative parameter have been established graphically. The conical pin fin is found to aid the heat transmission process better than the cylindrical one. The work is also helpful in the realm of microelectronics, particularly in the development of micro-pin-fin structures.

On the uniqueness of limit cycles in second-order oscillators

In this paper we shall prove a theorem to guarantee the uniqueness of limit cycles for second-order nonlinear differential equations, and give some sufficient conditions for the existence and nonexistence of the limit cycles, respectively. As applications of these results, we will prove the uniqueness of limit cycles for smooth or nonsmooth van der Pol-Rayleigh oscillators, and give their global dynamics.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

ADEQUATE SOLUTIONS OF JERK OSCILLATORS CONTAINING VELOCITY TIMES ACCELERATION-SQUARED: HAQUE’S APPROACH WITH MICKENS’ ITERATION METHOD

Haque’s Approach with Mickens’ Iteration Method is used to find the exact analytic solution of the nonlinear equation involving velocity times acceleration squared. A truncated Fourier series is used in different rhythms with different repetition steps. Our results are very close to the exact results and our results are comparatively closer to the exact results than others. Our solution method is obtained around the second-order angular frequency using Newton’s method. For some third-order (jerk) differential equations with cubic nonlinearities and nonlinear second-order differential equations; Mickens’ iteration method is used to determine the exact analytical approximate periodic solution. A numerical experiment of general differential equations with third-order, one-dimensional, autonomous, quadratic, and cubic nonlinearity has uncovered several algebraically simple equations involving the shaking of time-dependent acceleration that contain chaotic solutions.

Time-Optimal Trajectory Design for Heading Motion of the Underwater Vehicle

Underwater vehicles are a powerful tool that can assist oceanologists with measuring the state of oceans on a large scale. The heading control is essential for the underwater vehicle to follow a specific path. This study describes the general decoupled dynamics of underwater vehicles, which is a nonlinear second-order differential equation considering linear and quadratic damping hydrodynamics. A novel aspect of this study is the development of a new analytical solution for the second-order nonlinear differential equation, which involves the heading motion of the underwater vehicle. In this study, the time-optimal trajectory is formulated as the closed-form solution for the heading dynamics of the underwater vehicle. The concept of this trajectory is based on the shortest arrival time when the maximum force from the thrusters is applied to the underwater vehicle. Finally, a simulation of the time-optimal trajectory and evaluation of the robustness of the controller were demonstrated in order to verify the effectiveness of the proposed trajectory for controlling underwater vehicles.