On a linear system of differential equations
DOI:
https://doi.org/10.26577/JMMCS-2019-1-615Keywords:
equation, first order partial derivatives.Abstract
The linear systems of partial differential equations of the first order with the identical main parts
is considered. Аpplying the well-known relation between a normal system of ordinary differential
equations and a linear system of partial differential equations of the first order with the same
main parts, the existence of integral basis of a linear inhomogeneous system of partial differential
equations of the first order adjoining to some solution of the same linear inhomogeneous system
of differential equations with partial derivatives of the first order is proved. A sign at which the
nonlinear system of ordinary differential equations has a neighborhood such that any solution with
initial values from it tends to zero is found. Using the equivalence of a linear system of partial
differential equations of the first order with identical main parts to a linear differential equation
with partial derivatives of the first order, the existence of integral basis of the adjoining to zero
linear homogeneous system of partial differential equations of first order with nonlinear coefficients
is shown.
References
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[28] Aldibekov T.M., Aldazharova M.M., ”On a first-order partial differential equation”, Veszprem Conference on Differential and Difference Equations and Applications. Program and Abstracts. Faculty of Information Technology University of Pannonia Veszprem, Hungary. July 2 - July 5, (2018): 34
[29] Demidovich B.P., Lektsii po matematicheskoi teorii ustoichivosti [Lectures on math. stability theory]. (M: 1967): 472 [30] Aldibekov T.M., Aldazharova M.M., ”On the Stability by the First Approximation of Lyapunov Characteristic Exponents in Critical Cases”, Differential Equations, Vol. 50, No. 10. (2014): 1-5
[2] Mizohata S.S., Teoriya uravnenii s chastnymi proizvodnymi. [Partial equations theory]. (М.: Mir, 1977): 504
[3] Smirnov V.,Kurs vysshei matematiki. [Higher math course]. V.4, 2nd part, (М.: Nauka, 1981): 551
[4] Bers L., John D. and Shechter M., Uravneniya s chastnymi proizvodnymi. [Partial equations ]. (М.: Mir, 1966): 352
[5] Trikomi F., Lektsii po uravneniyam v chastnyh proizvodnyh. [Lectures in partial equations]. (IL., 1957): 443
[6] Hartman P., Obyknovennye differentsialnye uravneniya. [Ordinary differential equations]. (M.:Mir, 1970): 719
[7] Petrovsky I.G., Lektsii ob uravneniyah s chastnymi proizvodnymi. [Lectures in partial equations]. 3rd ed. (М., 1961): 400
[8] Petrovsky I.G., Lektsii po teorii obyknovennyh differentsialnyh uravnenii. [Lectures in ordinary differential equations]. 6th ed., (М., 1970): 473
[9] Elsgoltc L., Differentsialnye uravneniya. [Differential equations]. (М., 2013): 312
[10] 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): 676
[11] Kamke E., Spravochnik po differentsialnym uravneniyam v chastnyh proizvodnyh pervogo poryadka. [Referense book in first-order partial differential equations]. (М.: Nauka, 1966): 260
[12] Massera H.L., Lineinye differentsialnye i funktsionalnye prostranstva. [Linear differential and functional spaces] ( М.:Mir, 1970): 456
[13] Wazewski T., ”Sur l’appreciation du domain d’existence des integrals de l’equation aux derives partielles du premier ordre” [On the appreciation of the domain of existence of the integrals of the equation with partial derivatives of the first order], Ann. Soc. Polon. Math. VI.9, No.14, (1935): 149-177
[14] Wazewski T., ”Ueber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung” [On the conditions of existence of the integrals of partial differential equations of the first order], Math. Zeit., VI.7-9 No.43. (1938): 522-532
[15] Gelfand I., Nekotorye zadachi teorii kvazilineinyh uravnenii [Some problems of quazilinear equations theory], No. 14(2). (UMN, 1959): 87-158.
[16] Gross W., "Bemerkung zum Existenzbeweise bei den partiellen Differentialgleichungen erster Ordnung"[Remark on proof of existence in the first order partial differential equations], S.-B.K. Akad. Wiss. Wien, KI. Math. Nat., VI.7-9, (1914): 2233-2251
[17] Digel E., ”Uber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung” [On the Conditions of Existence of the Integrals of Partial Differential Equations of the First Order], Math Z, (1938): 445-451
[18] Caratheodory C., ”Variationsrechnung und partielle Differentialgleichungen erster Ordnung” [Variational calculus and partial differential equations of first order], VI 6, Leipzig und Berlin:B. G. Teubner, (1935): 7-9
[19] 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.
[20] Kovalevskaya S., "Zusatze und Bemerkungen zu Laplace’s Untersuchung uber die Gestalt der Saturnsringe"[Additions and Remarks on Laplace’s Investigation of the Shape of Saturn’s Rings], Astronomische Nachrichten, CXI. (1885): 18-21
[21] 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
[22] Frobenius G., ”Ueber das Pfaffsche Problem”, Journal for die reine und angewandte Mathematik, (1877): 230-315
[23] Perron O., "Ueber diejenigen Integrale linearer Differentialgleichungen, welche sich an einer Unbestimmtheitsstelle bestimmt verhalten"[On the integrals of linear differential equations which are determined at an uncertainty point], Math. Ann., VI.13, No 70. (1911): 1-32
[24] Plis A.,”Characteristics of nonlinear partial differential equation”, Bull.Acad. Polon. Sci., No 2. (1954): 419-422
[25] Hartman P.,”On Jacobi bracket”, Amer. J. Math, (1957): 187-189
[26] Hormander L., ”On the uniqueness of the Cauchy problem I”, Math. Scand., No. 6. ( 1958): 213-225.
[27] Hormander L., ”On the uniqueness of the Cauchy problem II”, Math. Scand., No. 7. (1959): 177-190.
[28] Aldibekov T.M., Aldazharova M.M., ”On a first-order partial differential equation”, Veszprem Conference on Differential and Difference Equations and Applications. Program and Abstracts. Faculty of Information Technology University of Pannonia Veszprem, Hungary. July 2 - July 5, (2018): 34
[29] Demidovich B.P., Lektsii po matematicheskoi teorii ustoichivosti [Lectures on math. stability theory]. (M: 1967): 472 [30] Aldibekov T.M., Aldazharova M.M., ”On the Stability by the First Approximation of Lyapunov Characteristic Exponents in Critical Cases”, Differential Equations, Vol. 50, No. 10. (2014): 1-5
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Published
2019-04-23
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Section
Mathematics
