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MULTIPOINT PROBLEM WITH EQUIDISTANT
NODES FOR PARTIAL DIFFERENTIAL
EQUATIONS
Iryna Klyus, Inna Kudzinovs’Ka
To cite this version:
Iryna Klyus, Inna Kudzinovs’Ka. MULTIPOINT PROBLEM WITH EQUIDISTANT NODES FOR PARTIAL DIFFERENTIAL EQUATIONS. [Research Report] National aviation university. 2021. �hal-03228147�
MULTIPOINT PROBLEM WITH EQUIDISTANT NODES FOR PARTIAL DIFFERENTIAL EQUATIONS
IRYNA KLYUS AND INNA KUDZINOVS’KA
Abstract. Correctness of the problem with multipoint conditions in time variable and frequency of the spatial coordinates for partial differential equations with shifts is investi-gated. The conditions of existence and uniqueness of the problem solution, metric theorems on lower bounds of small denominators arising in the construction of the solution of the problem are proved.
Introduction. Problems with multipoint conditions for partial differential equations are, in general, conditionally correct [1-4], and their solvability in many cases is related to the problem of small denominators. This note examines the problem with multipoint in time variable conditions for partial differential equations with shifts.
Statement of the problem. Let S be the unit circle, identified as Rp mod 2πq, H : S Ñ S the shift mapping such that Hpxq “ x`hp mod 2πq@x P S, where h P p0, 2πq. The mapping H generates a shift operator TH of the form THϕpxq “ ϕpHpxqq in the space of functions
given on S.
Let consider the problem:
Bnupt, xq Btn ` n´1 ÿ j“0 an´jTHn´j Bnupt, xq BtjBxn´j “ 0, pt, xq P p0, T q ˆ S, (0.1) uppj ´ 1qt0, xq “ ϕjpxq, j “ 1, ¨ ¨ ¨ , n, x P S, t0 “ T pn ´ 1q, (0.2)
where aj, j “ 1, ¨ ¨ ¨ , n are any complex numbers. We denote by λ1, ¨ ¨ ¨ , λm, m ď n different
roots of the equation λn` ia1λn´1` ¨ ¨ ¨ ` inan “ 0 and by Wα, α ą 0 the space of functions
ϕpxq on S for some finite norm: k ϕpxq k“
d ÿ
kě0
| ϕk|2 expp2α | k |q,
where ϕk, k P Z are the Fourier coefficients of the function ϕpxq.
Key words and phrases. differential equations; multipoint conditions; small denominators, metric theorems.
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Theorem 1. For the uniqueness of the solution of the problem (0.1), (0.2) in the space Cnpr0, T s; Wαq it is necessary and sufficient that the condition
tipλj´ λqqkt0exppikhq | k P Zzt0u, m ě j ą q ě 1u
č
2πZ “ H (0.3)
is fulfilled.
Theorem 2. Assume the condition (0.3) is valid and there exists positive constants βjq, m ě
j ą q ě 1 such that for all (except a finite number) numbers k P Z, the inequality | sh`pλj ´ λqqkt0 ě exppikhq
2
˘
|ě expp´βjq | k |q, m ě j ą q ě 1. (0.4)
is satisfied. If ϕjpxq P Wα`β, j “ 1, ¨ ¨ ¨ , n, where β is represented in terms of constants
n, T, λj, j “ 1, ¨ ¨ ¨ , m, βjq, m ě j ą q ě 1, then there exists a solution of the problem
(0.1), (0.2) from the space Cnpr0, T s; W
αq that continuously depends on functions ϕjpxq, j “
1, ¨ ¨ ¨ , n,
Theorem 3. For almost all (concerning the Lebesgue measure in R ) numbers h P p0, 2πq inequality (4) holds for all (except for a finite number) numbers k P Z for
βjq ą| λj ´ λq | t0{2, m ě j ą q ě 1.
Conclusion. It was established conditions for the existence, uniqueness and continuous dependence from the right parts of the boundary conditions of the solution of the multipoint problem for partial differential equations with shifts. The results complement scientific works that have been studied in [2-4]. They can be used in the study of specific practice problems which are modeled by means considered in the problem, and in further theoretical studies of problems with multipoint conditions for partial differential equations.
References
[1] Ptashnyk B.I. Incorrect boundary value problems for differential equations. – K.: Scientific thought, 1984. – 264p.
[2] Klyus I.S., Ptashnyk B.I. Multipoint problem with complex coefficients for partial differential equations not solved as to the highest derivative // Math. methods and physic.- mechanic. fields. – 1998. – 41,4. – P. 83–88.
[3] Klyus I. S., Ptashnyk B.I. Multipoint problem for partial differential equations with constant coeffi-cients not solved as to the highest derivative // Bulletin of the state. Univ. ”Lviv Polytechnic”. Applied Mathematics.-1998. – 1,337. – P. 112 – 115.
[4] Klyus I.S., Ptashnyk B.I. Multipoint problem for partial differential equations not solved as to the highest derivative // Ukr. Math. Journ. – 1999. – 51,12. – P. 1604–1613.
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Higher Mathematics Department, National Aviation Uinversity, Kyiv, Ukraine Email address: isklyus@gmail.com
Higher Mathematics Department, National Aviation Uinversity, Kyiv, Ukraine Email address: kudzinovskaya@ukr.net
