Abstract
In this paper, we consider a scalar linear differential-difference equation (LDDE) of neutral type \( \overset{\cdot }{x} \)(t) + p(t)\( \overset{\cdot }{x} \)(t − 1) = a(t)x(t − 1) + f(t). We examine the initial-value problem with an initial function in the case where the initial condition is given on an initial set. We use the method of polynomial quasisolutions based on the representation of the unknown function x(t) in the form of a polynomial of degree N. Substituting this function in the original equation we obtain the discrepancy Δ(t) = O(tN), for which an exact analytic representation is obtained. We prove that if a polynomial quasisolution of degree N is taken as an initial function, then the smoothness of the solution generated by this initial functions at connection points is no less than N.
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References
A. D. Myshkis, Linear Differential Equations with Delayed Arguments [in Russian], Gostekhizdat, Moscow–Leningrad (1951).
V. Cherepennikov, Numerical and Analytical Methods of the Study of Linear Functional-Differential Equations [in Russian], Novosibirsk (2013).
V. Cherepennikov and P. Ermolaeva, “Polynomial quasisolutions of linear differential-difference equations,” Opuscula Math., 26, No. 3, 431–443 (2006).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 132, Proceedings of International Symposium “Differential Equations–2016,” Perm, 2016.
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Cherepennikov, V.B. Smooth Solutions to Some Differential-Difference Equations. J Math Sci 230, 786–789 (2018). https://doi.org/10.1007/s10958-018-3790-4
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DOI: https://doi.org/10.1007/s10958-018-3790-4
Keywords and phrases
- differential-difference equation
- initial-value problem with an initial function
- polynomial quasisolution
- smooth solution