On linear partial equations of first-order

Authors

  • T. М. Aldibekov al-Farabi Kazakh National University, Almaty, Republic of Kazakhstan
  • M. M. Aldazharova Scientific Research Institute of the al-Farabi Kazakh National University

DOI:

https://doi.org/10.26577/jmmcs-2018-2-495

Keywords:

equation, first order partial derivatives

Abstract

We study a linear differential equation with first-order partial derivatives, where the coefficients
of the equation are given on an unbounded set and have continuous first-order partial derivatives.
Each partial differential equation is closely related to a system of ordinary differential equations,
a system of so-called characteristic equations of a given first-order partial differential equation.
Each first-order partial differential equation under certain conditions has a fundamental system of
integrals or an integral basis. We note that for a general linear partial differential equation of the
first order there can be no nontrivial integral. For a linear first-order partial differential equation,
where the coefficients of the equation are given on an unbounded set and have continuous firstorder
partial derivatives, with the first coefficient equal to one, an integral basis exists. For a
linear first-order partial differential equation, where the coefficients of the equation are given on
an unbounded set and have continuous first-order partial derivatives, with the first coefficient
equal to one, an integral basis exists. For a linear first-order partial differential equation, we define
the asymptotic stability of a linear first-order partial differential equation. A sufficient condition
for the asymptotic stability of a linear partial differential equation of the first order is given. At
present, the theory of partial differential equations finds its application in various fields of natural
science.

References

[1] Aldibekov T.M. "Obobshennye pokazateli lyapunova"[Generalized Lyapunov exponents"] Differential equations V.44,
№11(2008): 1582.
[2] Bers L., John D. and Shechter M. "Uravneniya s chastnymi proizvodnymi"[Partial equations ](М.: Mir, 1966): 352
[3] Caratheodory C. Variationsrechnung und partielle Differentialgleichungen erster Ordnung VI 6 (Leipzig und Berlin:B. G.
Teubner, 1935): 7-9.
[4] Courant R. "Uravneniya s chastnymi proizvodnymi"[Partial equations] (Mir, 1964): 23-27
[5] Elsgoltc L."Differentsialnye uravneniya"[Differential equations] (М., 2013): 57-67
[6] Gelfand I."Nekotorye zadachi teorii kvazilineinyh uravnenii"[Some problems of quazilinear equations theory] 14(2) (UMN,
1959): 87-158.
[7] Gross W. Bemerkung zum Existenzbeweise bei den partiellen Differentialgleichungen erster Ordnung VI.7-9 (S.-B.K.
Akad. Wiss. Wien, KI. Math. Nat., 123 (Abt. IIa), 1914): 2233-2251 .
[8] Hartman F. "Obyknovennye differentsialnye uravneniya"[ODE] ( М.:Mir, 1970): 627-629
[9] Hormander L. On the uniqueness of the Cauchy problem I No. 6 ( Math. Scand., 1958): 213-225.
[10] Hormander L. On the uniqueness of the Cauchy problem II No. 7 (Math. Scand., 1959): 177-190.
[11] Kamke E."Spravochnik po differentsialnym uravneniyam v chastnyh proizvodnyh pervogo poryadka"[Referense book in
first-order partial differential equations] (М.: Nauka, 1966): 46-48
[12] Kovalevskaya S. Zusatze und Bemerkungen zu Laplace’s Untersuchung uber die Gestalt der Saturnsringe CXI (Astronomische
Nachrichten, 1885): 18-21
[13] Kruzhkov S.N. "Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi"[First order quaziliniear
equations with many independent variables] 81(2)(Mat. sbornik, 1970): 228-255.
[14] Massera H.L. "Lineinye differentsialnye i funktsionalnye prostranstva"[Linear differential and functional spaces] ( М.:Mir,
1970): 50-59
[15] Mizohata S.S. "Teoriya uravnenii s chastnymi proizvodnymi"[Partial equations theory](М.: Mir, 1977): 504
[16] Perron O.Ueber diejenigen Integrale linearer Differentialgleichungen, welche sich an einer Unbestimmtheitsstelle bestimmt
verhalten VI.13, No 70, ( Math. Ann., 1911): 1-32
[17] Petrovsky I.G. "Lektsii ob uravneniyah s chastnymi proizvodnymi"[Lectures in partial equations] 3rd ed. (М., 1961): 38-43
[18] Petrovsky I.G. "Lektsii po teorii ODU"[Lectures in ODE] 6th ed.( М., 1970): 114-116
[19] Smirnov V."Kurs vysshei matematiki"[Higher math course] V.4, 2nd part ( М.: Nauka, 1981): 551
[20] Stepanov V.V."Kurs differentsialnyh uravnenii"[Diffenential equations course] 6-th ed. (Nauka.Fizmatgiz, 1959): 338-343
[21] Tichonov A.N and Samarsky A.A."Uravneniya matematicheskoi fiziki"[Mathematically physics equations] 7-th ed. (М.:
Izd. MGU; Nauka, 2004): 798
[22] Trikomi F. "Lektsii po uravneniyam v chastnyh proizvodnyh"[Lectures in partial equations] (IL., 1957): 67
[23] Wazewski T. Sur l’appreciation du domain d’existence des integrals de l’equation aux derives partielles du premier ordre
VI.9, No.14, (Ann. Soc. Polon. Math., 1935): 149-177
[24] Wazewski T.Ueber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung VI.7-9
No.43 (Math. Zeit., 1938): 522-532
[25] Yanenko N.N. and Rojdestvensky B.L. "Sistemy kvazilineinyh uravneenii i ih prilozhenie k gazovoi dinamike"[Systems
of quaziliniear differential equations and their application to gas dynamics] VI.7-9 (М., 1978): 223-225
[26] Zubov V.I."Voprosy teorii vtorogo metoda Lyapunova postroeniya obsh’ego v oblasti asimptoticheskoi ustoichivocti"[
General asimptotically stable domain building problems of the second method in lyapunov theory] Vol. XIX, 2nd
edition(PMM., 1955): 25-31

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Published

2018-08-29