Bounded solutions to nonlinear delay differential equations of third order

Stability and boundedness in delay system of differential equations of third order

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.

In this study we investigate the existence and approximation of the periodic solutions for nonlinear system of differential equations of third order with boundary conditions. The numerical-analytic method has been used to study the periodic solutions of the ordinary differential equations that were introduced by Samoilenko. Also these investigation lead us to the improving and extending the above method and the results of Butris.

In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi‐delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov‐Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright © 2014 John Wiley & Sons, Ltd.

Solution of nonlinear partial differential equations from base equations

This chapter is concerned with the study of oscillatory and asymptotic behaviour of nonoscillatory solutions of nonhomogeneous third-order differential equations of the form $$x^{\prime\prime\prime} + a(t) x^{\prime\prime} + b(t)x^{\prime} + c(t) x = f \bigl(t,x,x^{\prime},x^{\prime\prime}\bigr), $$ where a, b and c∈C([σ,∞),R) and f:[σ,∞)×R 3→R, σ∈R. z-type oscillation criteria has been used in this chapter to study the nonoscillation of solutions of the considered equation. As an application, some sufficient conditions have been given for the nonoscillation of different mathematical models in Engineering.

In this paper, it is considered a quasi-linear strongly-damped wave equation defined by a non-linear partial differential equation of third order. The equation describes motions of viscoelastic solids. We study the conservation laws of this equation. By applying the multiplier method of Anco and Bluman to the equation, we find the multipliers. Consequently, we obtain a complete classification of conservation laws. Moreover, we use the Lie-group theory to analyse the symmetries of the equation. From the Lie symmetries, all the reductions are determined. Afterwards, we construct exact solutions with physical interest: travelling wave solutions.

This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results.

In this paper, we present some new oscillation criteria for nonlinear neutral difference equations of the form ?(b(n)?(a(n)?z(n))) + q(n)x?(?(n)) = 0 where z(n) = x(n) + p(n)x(?(n)),? > 0, b(n) > 0, a(n) > 0, q(n) ? 0 and p(n) > 1. By summation averaging technique, we establish new criteria for the oscillation of all solutions of the studied difference equation above. We present four examples to show the strength of the new obtained results.

Over the years, the teaching of differential equations in the engineering programs at the University of Antofagasta, Chile, has faced various challenges, particularly regarding the learning of the method of undetermined coefficients. This method, used to find particular solutions in third-order linear differential equations, has been identified as one of the most complex topics for students. The difficulty lies primarily in determining the general solution of these equations, which requires a high level of understanding and advanced mathematical skills. To address these challenges and improve the learning process, innovative technological tools such as GeoGebra applets have been developed. These applications allow students to interact dynamically with the concepts and procedures of the method of undetermined coefficients, facilitating their understanding and application. The results obtained after implementing these tools have been highly positive, showing a significant improvement in students' learning and motivation. Additionally, this work incorporates the use of artificial intelligence, specifically through DeepSeek. This AI has proven to be an effective tool for assisting in the determination of particular solutions, which not only simplifies the resolution process but also fosters students' interest and active participation in their tasks and studies. The integration of these technologies not only strengthens learning but also modernizes the teaching of differential equations, aligning it with the current demands of engineering education.

In this paper, we consider the solvability of the Cauchy problem for pseudohyperbolic equations (partial differential equations of third order). For the case in which the right-hand side is a generalized function (distribution) of finite order, we establish a theorem on the unique solvability for a sufficiently general pseudohyperbolic operator. The method of proof is based on a specially constructed “scale” of a priori inequalities for the direct and adjoint operators.

Bu çalışmada Lyapunov’un ikinci metodu kullanılarak üçüncü mertebeden gecikmeli bir diferansiyel denklemin çözümlerinin sınırlılığı ve sürdürebilirliği ile ilgili yeni sonuçlar elde edilmiştir. Bu çalışma literatürde üçüncü mertebeden gecikmeli ve gecikmesiz diferansiyel denklemler üzerine iyi bilinen bazı sonuçları kapsamış ve daha ileri götürmüştür. Ayrıca çalışmada elde ettiğimiz sonuçların daha iyi anlaşılması için bir örnek verilmiştir.

A direct method for the data evolution in the field of Small-Angle Scattering experiments including the discussion of a special procedure for the reduction of truncation errors is presented. The interface distribution function IDF, introduced by Ruland & Steibeck, can be considered as a starting point of data evaluation. The correlation function and the interface distribution are connected to a differential equation of third order.

We prove that there are no $6$-cycles of the form $x_{n+3}=x_i f(x_j,x_k),$ with $i,j,k\in\{n,n+1,n+2\}$ pairwise distinct, whenever $f:(0,\infty)\times (0,\infty)\rightarrow (0,\infty)$ is a continuous symmetric function, that is, $f(x,y)=f(y,x)$, for all $x,y>0$. Moreover, we obtain all the $6$-cycles of potential form and present some open questions relative to the search of $p$-cycles whenever symmetry does not hold.

This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Pad ´ e (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.