An Exact Solution for a Nonautonomous Delay Differential Equation

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf-Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers-Huxley, generalized equation of Camassa-Holm, generalized equation of Swift-Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.

Previous article Next article On Ryabov’s Asymptotic Characteization of the Solutions of Quasi-Linear Differential Equations with Small DelaysRodney D. DriverRodney D. Driverhttps://doi.org/10.1137/1010058PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. Boffi and , R. Scozzafava, Sull'equazione funzionale lineare $f\sp{\prime} (x)=-A(x)f(x-1)$, Rend. Mat. e Appl. (5), 25 (1966), 402–410 MR0218702 0178.09602 Google Scholar[2] L. Bruwier, Sur l'équation fonctionelle $y^{(n)}(x)+a_{1}y^{(n-1)}(x+c)+\cdots +a_{n-1}y'(x+(n-1)c)+a_{n}y(x+nc)=0$, C. R. Congr. Nat. Sci. Bruxelles, (1930), 91–97 Google Scholar[3] Rodney D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401–426 10.1007/BF00281203 MR0141863 0105.30401 CrossrefISIGoogle Scholar[4] É. Goursat, A Course in Mathematical Analysis, Vol. I, Ginn, Boston, 1904 Google Scholar[5] Oskar Perron, Über Bruwiersche Reihen, Math. 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Fiz., 7 (1967), 836–851 0199.50801 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Smooth Inertial Manifolds for Neutral Differential Equations with Small DelaysJournal of Dynamics and Differential Equations, Vol. 170 | 19 April 2021 Cross Ref Existence of global solutions to nonlinear mixed-type functional differential equationsNonlinear Analysis, Vol. 195 | 1 Jun 2020 Cross Ref The Approximation of Invariant Sets in Infinite Dimensional Dynamical SystemsAdvances in Dynamics, Optimization and Computation | 21 July 2020 Cross Ref On the computation of attractors for delay differential equationsJournal of Computational Dynamics, Vol. 3, No. 1 | 1 Oct 2016 Cross Ref Infinite DimensionsMultiple Time Scale Dynamics | 11 November 2014 Cross Ref Traveling Wavefront Solutions for Reaction-Diffusion Equation with Small DelayFunkcialaj Ekvacioj, Vol. 54, No. 2 | 1 Jan 2011 Cross Ref Solitary waves for Korteweg–de Vries equation with small delayJournal of Mathematical Analysis and Applications, Vol. 368, No. 1 | 1 Aug 2010 Cross Ref Synchronization analysis of linearly coupled systems described by differential equations with a coupling delayPhysica D: Nonlinear Phenomena, Vol. 221, No. 2 | 1 Sep 2006 Cross Ref Inertial and slow manifolds for delay equations with small delaysJournal of Differential Equations, Vol. 190, No. 2 | 1 May 2003 Cross Ref Point initial value problem for linear functional differential equations in Banach spacesAequationes Mathematicae, Vol. 41, No. 1 | 1 Mar 1991 Cross Ref Behavior of solutions of differential equations with deviating argumentUkrainian Mathematical Journal, Vol. 37, No. 3 | 1 Jan 1986 Cross Ref Linear differential systems with small delaysJournal of Differential Equations, Vol. 21, No. 1 | 1 May 1976 Cross Ref Point Data Problems for Functional Differential EquationsDynamical Systems | 1 Jan 1976 Cross Ref The Equation x′ ( t ) = ax ( t ) + bx ( t − τ) With “Small” DelayThe American Mathematical Monthly, Vol. 80, No. 9 | 11 April 2018 Cross Ref SOME HARMLESS DELAYSDelay and Functional Differential Equations and their Applications | 1 Jan 1972 Cross Ref BibliographyDifferential and Integral Inequalities - Theory and Applications: Functional, Partial, Abstract, and Complex Differential Equations | 1 Jan 1969 Cross Ref Volume 10, Issue 3| 1968SIAM Review291-405 History Submitted:29 September 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1010058Article page range:pp. 329-341ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

Existence results of periodic solutions for non-autonomous differential delay equations with asymptotically linear properties

Polygons of differential equations for finding exact solutions

SECTION 1 - Classification and Solutions of First-Order Differential Equations

Uniformly attracting solutions of nonautonomous differential equations

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena. Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation. Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions. Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process. Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

In this paper, by using the Krasnoselskii fixed point theorem, we study the existence of one or multiple positive periodic solutions of a nonautonomous delay differential equation. We also give some examples to demonstrate our results.

We study the multiplicity of periodic solutions of nonautonomous delay differential equations which are asymptotically linear both at zero and at infinity. By making use of a theorem of Benci, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.

We are concerned here with the existence of monotonic and uniformly asymptotically stable solution of an initial-value problem of non-autonomous delay differential equations of arbitrary (fractional) orders.

Generalization of the simplest equation method for nonlinear non-autonomous differential equations