Fibonacci collocation method for solving a class of nonlinear differential equations

What follows is a discussion on the application of a digital computer to solve analytically a class of nonlinear ordinary differential equations. The asymptotic method of Krylov, Bogoliubov and Mitropolsky is employed to obtain analytical solutions. In addition, a procedure has been developed through which the initial amplitude and phase can be expressed as functions of initial displacement and velocity without the need of integration. Since the algorithms are analytical in nature, the truncated power series capability of the symbolic algebraic system ALTRAN has been used to generate analytical solutions automatically.

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

We establish asymptotic representations for the solutions of one class of nonlinear differential equations of the second order with rapidly and regularly varying nonlinearities.

On a class of nonlinear differential equations of astrophysics

Nonlinear differential equations arise widely in applied mathematics, physics, and engineering, yet many conventional analytical and numerical methods remain limited in their ability to handle strong nonlinearities efficiently and accurately. This paper presents a novel computational framework based on the Modified Laplace–Adomian Polynomial Method (LAPM) for solving nonlinear differential equations. The proposed method integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method, enabling complex nonlinear terms to be decomposed into rapidly convergent Adomian polynomials. This integration simplifies the solution procedure, reduces computational complexity, and preserves high accuracy. The performance of LAPM was validated using several benchmark nonlinear and linear differential equations, and the results demonstrated superior convergence speed, precision, and stability compared with traditional methods. The study concludes that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient approach for solving a broad class of nonlinear differential equations. This work contributes to the advancement of computational methods by offering a robust alternative for the analysis of differential equation models encountered in mathematics, physics, and engineering.

We apply the Fourier‐least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

This paper investigates the error between the approximate and exact solutions of a class of nonlinear differential equations which possess unique solution on a certain interval for any admissible initial conditions. The differential equations are assumed to be approximated by well-posed truncated Taylor series up to a certain order obtained about certain, in general non-periodic, sampling points t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ϵ [t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</sub> ] for i = 0,1,..., J of the solution. Two simulation examples are also provided. The first one is concerned with a predefined periodic sampling law while the second one generates new sampling points according to a constant amplitude difference sampling criterion.

A three operator split-step method covering a larger set of non-linear partial differential equations

We establish asymptotic properties of some types of solutions of one class of essentially nonlinear nonautonomous differential equations of the second order and find necessary and sufficient conditions for the existence of these solutions.

The stability of the trivial equilibrium solution of a class of non-linear differential equations with a non-stationary absolutely integrable random coefficient is studied. The mean square stability of random systems is discussed and stability theorems of Bertram and Sarachik (1959) are stated. It is shown that the equilibrium null solution is mean square stable under some mild conditions. Some examples are also considered.

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.

In this paper, a class of nonlinear difference equations with time-varying delays is considered. Based on a generalized discrete Halanay inequality, some sufficient conditions for the attracting set and the global asymptotic stability of the nonlinear difference equations with time-varying delays are obtained.

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.

An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations

We study a class of nonlinear difference equations admitting a 1-Gevrey formal power series solution which, in general, is not 1- (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.