Polynomial quasisolutions of linear differential-difference equations

Valery B. Cherepennikov
Polina G. Ermolaeva

Abstract. The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment \(t=0\) the value of the desired function \(x(t)\) is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree \(N\) is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution) is understood in the sense of appearance of the residual \(\Delta (t)=O(t^N)\), when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.

Keywords: differential-difference equations, neutral type, initial value problem, polynomial quasisolution.

Mathematics Subject Classification: 34K15.

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