Unique solvability of pseudohyperbolic equations with singular right-hand sides

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.

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

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.

Principles of Construction of Adjoint Operators in Non-Linear Problems Dual Spaces and Adjoint Operators Construction of Adjoint Operators Based on Using the Lagrange Identity Definition of Adjoint Operators Based on Using Taylor's Formula Operators of the Class D and their Adjoint Operators Properties of Adjoint Operators Constructed on the Basis of Various Principles General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators Properties of Operators of the Class D Properties of Adjoint Operators Constructed with the Use of the Taylor Formula Solvability of Main and Adjoint Equations in Non-Linear Problems Main and Adjoint Equations. Problems Solvability of the Equation F(u) = y Solvability of the Equation A(u)v = y Solvability of the Equation A(u)v = y Solvability of the Equation A*(u)w = p Solvability of the Equation A*(u)w = p Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems Definitions. Non-Linear Equations and Operators. Conservation Laws Transformation of Equations Adjoint Equations Relationship between Different Adjoint Operators General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws Construction of Adjoint Operators with Prescribed Properties The Noether Theorem, Conservation Laws and Adjoint Operators On Some Applications of Adjoint Equations Perturbation Algorithms in Non-Linear Problems Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations Spectral Method in Perturbation Algorithms Justification of the N-th Order Perturbation Algorithms Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory Some Problems of Transport Theory The N-th Order Perturbation Algorithms for an Eigenvalue Problem A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion Statement of the Problem. Basic Assumptions. Operator Formulation Transformation of the Problem. Properties of the Non-Linear Operator Adjoint Equation An Algorithm for Computing the Functional The Problem on Chemical Exchange Processes Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil Mathematical Models of Mass Transfer in Soil Formulation of a Non-Linear Mathematical Model Transformation of the Problem. Properties of the Non-Linear Operator Perturbation Algorithm. Adjoint Equation Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient An Algorithm for Solving the Problem Applications of Adjoint Equations in Science and Technology Adjoint Equations in Data Assimilation Problems Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics Shape Optimization Using Adjoint Equation Approaches Global Transport of Pollutants Problems of Climate Change Sensitivity in Various Regions of the World

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 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.

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.

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.

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.

This paper gives some sufficient conditions for every solution of delay differential equation $$ \multline \buildrel \ldots \over x (t) +f(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t),\ddot{x}(t-r)) \\ +b(t)g(x(t-r),\dot{x}(t-r)) +c(t)h(x(t)) \\ =p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t)) \endmultline $$ to be bounded.

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.

We derive Lyapunov-type inequalities for third-order half-linear difference equations. The results are significant as the existing work on Lyapunov-type inequalities for difference equations is limited only to even order cases. As the techniques for even order difference equations are not applicable to odd order difference equations, the methods used here are novel. The resulting inequalities utilize and . New results for linear difference equations are obtained as a special case and are further extended to more general linear equations. Moreover, brief discussions are also given for third-order backward difference equations and dynamic equations on time scales.

The quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third-order linear differential equation. We perform a semiclassical study of this third-order equation, generalizing the familiar JWKB theory for the second-order Schrodinger equation, and deriving explicit (albeit approximate) formulas for the eigenvalues within this semiclassical context. A central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion (generated by a complex polynomial Hamiltonian function) in the complexified position and momentum planes.