Journal articles: 'Delay differential equation'

1

Kulenović, M. R. S. "Oscillation of the Euler differential equation with delay." Czechoslovak Mathematical Journal 45, no. 1 (1995): 1–6. http://dx.doi.org/10.21136/cmj.1995.128506.

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2

Das, P. "Oscillation of odd order neutral delay differential equation." Czechoslovak Mathematical Journal 45, no. 2 (1995): 241–51. http://dx.doi.org/10.21136/cmj.1995.128520.

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3

Naoum, Riyadh, Abbas Al-Bayati, and Ann Al-Sawoor. "OSFESOR Code – The Delay Differential Equation Tool “Improving Delay Differential Equations Solver”." AL-Rafidain Journal of Computer Sciences and Mathematics 1, no. 2 (December 1, 2004): 199–217. http://dx.doi.org/10.33899/csmj.2004.164119.

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4

Cassidy, Tyler. "Distributed Delay Differential Equation Representations of Cyclic Differential Equations." SIAM Journal on Applied Mathematics 81, no. 4 (January 2021): 1742–66. http://dx.doi.org/10.1137/20m1351606.

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5

Busenberg, Stavros, and L. Thomas hill. "Construction of differential equation approximations to delay differential equations." Applicable Analysis 31, no. 1-2 (January 1988): 35–56. http://dx.doi.org/10.1080/00036818808839814.

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6

Chambers, LL G. "The delay differential equation." Mathematika 33, no. 1 (June 1986): 80–86. http://dx.doi.org/10.1112/s0025579300013899.

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7

Svoboda, Zdeněk. "Asymptotic properties of one differential equation with unbounded delay." Mathematica Bohemica 137, no. 2 (2012): 239–48. http://dx.doi.org/10.21136/mb.2012.142869.

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8

Hino, Yoshiyuki, and Taro Yoshizawa. "Total stability property in limiting equations for a functional-differential equation with infinite delay." Časopis pro pěstování matematiky 111, no. 1 (1986): 62–69. http://dx.doi.org/10.21136/cpm.1986.118265.

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9

Tunç, Cemil, and Osman Tunç. "On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order." Mathematics 10, no. 22 (November 13, 2022): 4235. http://dx.doi.org/10.3390/math10224235.

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In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovski functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new.

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10

Hu, Haijun, Li Liu, and Jie Mao. "Multiple Nonlinear Oscillations in a𝔻3×𝔻3-Symmetrical Coupled System of Identical Cells with Delays." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/417678.

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A coupled system of nine identical cells with delays and𝔻3×𝔻3-symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results.

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11

Kandala, Shanti S., Surya Samukham, Thomas K. Uchida, and C. P. Vyasarayani. "Spurious roots of delay differential equations using Galerkin approximations." Journal of Vibration and Control 26, no. 15-16 (January 13, 2020): 1178–84. http://dx.doi.org/10.1177/1077546319894172.

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The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

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12

Nakata, Kenta. "Integrable delay-difference and delay-differential analogs of the KdV, Boussinesq, and KP equations." Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 113505. http://dx.doi.org/10.1063/5.0125308.

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Delay-difference and delay-differential analogs of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the N-soliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delay-differential analog of the KP equation is proposed. We discuss its N-soliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delay-differential analogs of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.

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13

Miyazaki, R., and K. Tchizawa. "Bifurcation delay in a delay differential equation." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (November 2005): e2189-e2195. http://dx.doi.org/10.1016/j.na.2004.10.001.

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14

Wang, Shuyi, and Fanwei Meng. "Ulam Stability of n-th Order Delay Integro-Differential Equations." Mathematics 9, no. 23 (November 26, 2021): 3029. http://dx.doi.org/10.3390/math9233029.

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In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

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15

Rebenda, Josef, and Zuzana Pátíková. "Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays." Complexity 2020 (February 28, 2020): 1–12. http://dx.doi.org/10.1155/2020/2854574.

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An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

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16

Baculíková, Blanka, and Jozef Džurina. "Oscillation of the third order Euler differential equation with delay." Mathematica Bohemica 139, no. 4 (2014): 649–55. http://dx.doi.org/10.21136/mb.2014.144141.

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17

Pituk, Mihály. "Convergence to equilibria in a differential equation with small delay." Mathematica Bohemica 127, no. 2 (2002): 293–99. http://dx.doi.org/10.21136/mb.2002.134154.

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18

Chartbupapan, Watcharin, Ovidiu Bagdasar, and Kanit Mukdasai. "A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation." Mathematics 8, no. 1 (January 3, 2020): 82. http://dx.doi.org/10.3390/math8010082.

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The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.

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19

Domoshnitsky, Alexander, and Roman Koplatadze. "On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/168425.

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The following differential equationu(n)(t)+p(t)|u(σ(t))|μ(t) sign u(σ(t))=0is considered. Herep∈Lloc(R+;R+), μ∈C(R+;(0,+∞)), σ∈C(R+;R+), σ(t)≤t, andlimt→+∞⁡σ(t)=+∞. We say that the equation is almost linear if the conditionlimt→+∞⁡μ(t)=1is fulfilled, while iflim⁡supt→+∞⁡μ(t)≠1orlim⁡inft→+∞⁡μ(t)≠1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying PropertyAfor delay Emden-Fowler equations are obtained.

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20

Yeniçerioğlu, Ali Fuat, Vildan Yazıcı, and Cüneyt Yazıcı. "Asymptotic Behavior and Stability in Linear Impulsive Delay Differential Equations with Periodic Coefficients." Mathematics 8, no. 10 (October 16, 2020): 1802. http://dx.doi.org/10.3390/math8101802.

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We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study.

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21

Glagolev, Mikhail V., Aleksandr F. Sabrekov, and Vladimir M. Goncharov. "Delay differential equations as a tool for mathematical modelling of population dynamic." Environmental Dynamics and Global Climate Change 9, no. 2 (November 27, 2018): 40–63. http://dx.doi.org/10.17816/edgcc10483.

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The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University. Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture). In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson. Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system. This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function. Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.

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22

Kovalev, Anton V., Evgeny A. Viktorov, and Thomas Erneux. "Non-Spiking Laser Controlled by a Delayed Feedback." Mathematics 8, no. 11 (November 20, 2020): 2069. http://dx.doi.org/10.3390/math8112069.

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In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

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23

Soelaiman, Rully, and Yudhi Purwananto. "IMPLEMENTASI DELAY DIFFERENTIAL EQUATION PADA SOLVER ORDINARY DIFFERENTIAL EQUATION MATLAB." JUTI: Jurnal Ilmiah Teknologi Informasi 1, no. 1 (July 1, 2002): 79. http://dx.doi.org/10.12962/j24068535.v1i1.a99.

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Anokye, Martin, Henry Amankwah, Emmanuel Kwame Essel, and Irene Kafui Amponsah. "Dynamics of Equilibrium Prices With Differential and Delay Differential Equations Using Characteristic Equation Techniques." Journal of Mathematics Research 11, no. 4 (June 27, 2019): 1. http://dx.doi.org/10.5539/jmr.v11n4p1.

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This study compares differential model to delay differential model in terms of their qualitative behaviour with respect to equilibrium price changes using roots of characteristic equation techniques. The equilibrium states of both price adjustment models were simulated using inputs from same source. The study found that irrespective of initial prices set for the system, the current price of the differential model would always move monotonically towards the equilibrium price defined for the system. However, the current price of the delay- differential model will fluctuate and move away from the initial prices due to the delay parameter associated with the supply, then gradually decrease and turn towards the defined system equilibrium price. Results from the study also showed that current prices in the delay-differential model are not predictable at the initial stage due to the time delay parameter in the supply function of price. On the other hand, current prices in their counterpart models without delay are predictable, as they always converge to the equilibrium price points defined in the system. Since most economic and physical systems are time delay inherent, it is recommended that such systems are modeled using delay-differential equations to reflect realities of the phenomena.

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Saeed, Umer, and Mujeeb ur Rehman. "Hermite Wavelet Method for Fractional Delay Differential Equations." Journal of Difference Equations 2014 (July 2, 2014): 1–8. http://dx.doi.org/10.1155/2014/359093.

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We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

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Fišnarová, Simona, and Robert Mařík. "Oscillation of Half-Linear Differential Equations with Delay." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/583147.

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We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

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27

Opluštil, Zdeněk, and Jiří Šremr. "Some oscillation criteria for the second-order linear delay differential equation." Mathematica Bohemica 136, no. 2 (2011): 195–204. http://dx.doi.org/10.21136/mb.2011.141582.

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28

Kozyreff, G., and T. Erneux. "Singular Hopf bifurcation in a differential equation with large state-dependent delay." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2162 (February 8, 2014): 20130596. http://dx.doi.org/10.1098/rspa.2013.0596.

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We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

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29

Marriott, C., R. Vallée, and C. Delisle. "Analysis of a first-order delay differential-delay equation containing two delays." Physical Review A 40, no. 6 (September 1, 1989): 3420–28. http://dx.doi.org/10.1103/physreva.40.3420.

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30

Verdugo, Anael. "Linear Analysis of an Integro-Differential Delay Equation Model." International Journal of Differential Equations 2018 (2018): 1–6. http://dx.doi.org/10.1155/2018/5035402.

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This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of N delay differential equations (DDEs) coupled to N ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as N becomes large.

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31

Bel, Andrea, Romina Cobiaga, and Walter Reartes. "Periodic Orbits and Chaos in Nonsmooth Delay Differential Equations." International Journal of Bifurcation and Chaos 29, no. 10 (September 2019): 1950137. http://dx.doi.org/10.1142/s0218127419501372.

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In this paper, we present a method to find periodic solutions for certain types of nonsmooth differential equations or nonsmooth delay differential equations. We apply the method to three examples, the first is a second-order differential equation with a nonsmooth term, in this case the method allows us to find periodic orbits in a nonlinear center. The two remaining examples are first-order nonsmooth delay differential equations. In the first one, there is a stable periodic solution and in the second, the presence of a chaotic attractor was detected. In the latter, the method allows us to obtain unstable periodic orbits within the attractor. For large values of the delay, both examples can be seen as singularly perturbed delay differential equations. For them, an analysis is performed with an associated discrete map which is obtained in the limit of large delays.

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32

Naseem, Tahir. "Reduce Differential Transform Method for Analytical Approximation of Fractional Delay Differential Equation." International Journal of Emerging Multidisciplinaries: Mathematics 1, no. 2 (May 14, 2022): 104–23. http://dx.doi.org/10.54938/ijemdm.2022.01.2.35.

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The study of an entirely new class of differential equations known as delay differential equations or difference differential equations has resulted from the development and application of automatic control systems (DDEs). Time delays are virtually always present in any system that uses feedback control. Because it takes a finite amount of time to sense information and then react to it, a time delay is required. This exploration was carried out for the solution of fractional delay differential equations by using the reduced differential transform method. The results are presented in a series of form that leads to an exact answer. The proposed technique is found to be accurate and convergent. MAPLE 17 is used to illustrate the results graphically.

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Cesarano, Clemente, Sandra Pinelas, Faisal Al-Showaikh, and Omar Bazighifan. "Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations." Symmetry 11, no. 5 (May 3, 2019): 628. http://dx.doi.org/10.3390/sym11050628.

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In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

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Alenazy, Aneefah H. S., Abdelhalim Ebaid, Ebrahem A. Algehyne, and Hind K. Al-Jeaid. "Advanced Study on the Delay Differential Equation y′(t) = ay(t) + by(ct)." Mathematics 10, no. 22 (November 17, 2022): 4302. http://dx.doi.org/10.3390/math10224302.

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Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation y′(t)=ay(t)+byct belongs to such a set of delay differential equations. To the authors’ knowledge, there are no standard methods to solve the delay differential equations, i.e., unlike the ordinary differential equations, for which numerous and standard methods are well-known. In this paper, the Adomian decomposition method is suggested to analyze the pantograph delay differential equation utilizing two different canonical forms. A power series solution is obtained through the first canonical form, while the second canonical form leads to the exponential function solution. The obtained power series solution coincides with the corresponding ones in the literature for special cases. Moreover, several exact solutions are derived from the present power series solution at a specific restriction of the proportional delay parameter c in terms of the parameters a and b. The exponential function solution is successfully obtained in a closed form and then compared with the available exact solutions (derived from the power series solution). The obtained results reveal that the present analysis is efficient and effective in dealing with pantograph delay differential equations.

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Cherevko, I. M., and A. B. Dorosh. "Boundary value problem solution existence for linear integro-differential equations with many delays." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 65–70. http://dx.doi.org/10.15330/cmp.10.1.65-70.

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For the study of boundary value problems for delay differential equations, the contraction mapping principle and topological methods are used to obtain sufficient conditions for the existence of a solution of differential equations with a constant delay. In this paper, the ideas of the contraction mapping principle are used to obtain sufficient conditions for the existence of a solution of linear boundary value problems for integro-differential equations with many variable delays. Smoothness properties of the solutions of such equations are studied and the definition of the boundary value problem solution is proposed. Properties of the variable delays are analyzed and functional space is obtained in which the boundary value problem is equivalent to a special integral equation. Sufficient, simple for practical verification coefficient conditions for the original equation are found under which there exists a unique solution of the boundary value problem.

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Hu, Rong, and De Jun Shao. "Stability of Neutral Stochastic Delay Differential Equation." Applied Mechanics and Materials 192 (July 2012): 346–50. http://dx.doi.org/10.4028/www.scientific.net/amm.192.346.

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The paper discusses the pth moment stability of neutral stochastic differential equation with multiple variable delays. The stability is more general and representative than the exponential stability. This investigation uses a specific Lyapunov function based on usual methods. And it discusses the neutral stochastic differential equations with multiple variable delays which is much more general form. Finally a two dimensional half-linear example is discussed to illustrate the theory.

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Yüzbaşi, Şuayip. "A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations." International Journal of Biomathematics 10, no. 07 (September 21, 2017): 1750091. http://dx.doi.org/10.1142/s1793524517500917.

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In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improvement of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.

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Almarri, Barakah, Fahd Masood, Osama Moaaz, and Ali Muhib. "Amended Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of Higher-Order Functional Differential Equations." Axioms 11, no. 12 (December 12, 2022): 718. http://dx.doi.org/10.3390/axioms11120718.

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Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them.

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Zhan, Rui, Weihong Chen, Xinji Chen, and Runjie Zhang. "Exponential Multistep Methods for Stiff Delay Differential Equations." Axioms 11, no. 5 (April 19, 2022): 185. http://dx.doi.org/10.3390/axioms11050185.

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Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.

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Ohira, K. "Resonating delay equation." Europhysics Letters 137, no. 2 (January 1, 2022): 23001. http://dx.doi.org/10.1209/0295-5075/ac4ba3.

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Abstract We propose here a delay differential equation that exhibits a new type of resonating oscillatory dynamics. The oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. The optimal height of the power spectrum of the dynamical trajectory is observed with the suitably tuned delay. This resonant behavior contrasts itself against the general behaviors where an increase of the delay parameter leads to the persistence of oscillations or more complex dynamics.

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ARINO, OVIDE, and EVA SÁNCHEZ. "AN ABSTRACT DIFFERENTIAL EQUATION ARISING FROM CELL POPULATION DYNAMICS." Journal of Biological Systems 03, no. 02 (June 1995): 469–81. http://dx.doi.org/10.1142/s0218339095000447.

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We provide an analysis of the stability and bifurcation properties of the solutions of an abstract differential nonlinear equation arising from cell population dynamics. The work surveyed here stems from a remark we made with respect to these equations: that it is possible to associate to any of them a delay differential equation on an infinite dimensional vector space. Perturbation theory for nonlinear equations similar to the one known for delay differential equations on finite dimensional spaces could possibly yield the same results as for those equations.

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Byszewski, Ludwik. "Nonlinear second-order delay differential equation." Czasopismo Techniczne 3 (2019): 141–48. http://dx.doi.org/10.4467/2353737xct.19.038.10212.

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Berezansky, L., E. Braverman, and L. Idels. "Delay differential logistic equation with harvesting." Mathematical and Computer Modelling 40, no. 13 (December 2004): 1509–25. http://dx.doi.org/10.1016/j.mcm.2005.01.008.

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Sprott, J. C. "A simple chaotic delay differential equation." Physics Letters A 366, no. 4-5 (July 2007): 397–402. http://dx.doi.org/10.1016/j.physleta.2007.01.083.

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Ren, Jingli, and Zhibo Cheng. "On high-order delay differential equation." Computers & Mathematics with Applications 57, no. 2 (January 2009): 324–31. http://dx.doi.org/10.1016/j.camwa.2008.10.071.

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Paul, Christopher A. H. "Developing a delay differential equation solver." Applied Numerical Mathematics 9, no. 3-5 (April 1992): 403–14. http://dx.doi.org/10.1016/0168-9274(92)90030-h.

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Zhang, Xiang Mei, Xian Zhou Guo, and Anping Xu. "Stability Analysis of Fractional Delay Differential Equations by Chebyshev Polynomial." Advanced Materials Research 500 (April 2012): 586–90. http://dx.doi.org/10.4028/www.scientific.net/amr.500.586.

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The paper is devoted to the numerical stability of fractional delay differential equations with non-smooth coefficients using the Chebyshev collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Chebyshev polynomial of the first kind. Then we solve the stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is examined for a set of case studies that contain the complexities of periodic coefficients, delays and discontinuities.

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Philos, Ch G. "Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 6 (1998): 1371–87. http://dx.doi.org/10.1017/s0308210500027372.

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First-order scalar linear delay differential equations with periodic coefficients and constant delays are considered, where the coefficients have a common period and the delays are multiples of this period. A basic asymptotic criterion is given. Moreover, some results on the nonoscillation and on the stability of the trivial solution are obtained. An equation, which is in a sense the characteristic equation, plays an important role in establishing the results of the paper.

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Kanth, A. S. V. Ravi, and P. Murali Mohan Kumar. "A Numerical Technique for Solving Nonlinear Singularly Perturbed Delay Differential Equations." Mathematical Modelling and Analysis 23, no. 1 (February 12, 2018): 64–78. http://dx.doi.org/10.3846/mma.2018.005.

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This paper presents a numerical technique for solving nonlinear singu- larly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a se- quence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.

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Bakke, V. L., and Z. Jackiewicz. "Stability analysis of linear multistep methods for delay differential equations." International Journal of Mathematics and Mathematical Sciences 9, no. 3 (1986): 447–58. http://dx.doi.org/10.1155/s0161171286000583.

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Stability properties of linear multistep methods for delay differential equations with respect to the test equationy′(t)=ay(λt)+by(t), t≥0,0<λ<1, are investigated. It is known that the solution of this equation is bounded if and only if|a|<−band we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.

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Journal articles on the topic 'Delay differential equation'

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