Nonlinear differential equation with first order partial derivatives
DOI:
https://doi.org/10.26577/JMMCS-2018-3-508Keywords:
equation, first order partial derivativesAbstract
The asymptotic behavior of solutions of a nonlinear differential equation with first-order partial
derivatives solved with respect to one of the derivatives is investigated. 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 homogeneous 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. In this paper,
a nonlinear partial differential equation of the first order, which is solved with respect to one
of the derivatives, is estimated from two sides by first-order partial differential equations. Using
differential inequalities it is proved that a nonlinear differential equation with first-order partial
derivatives solved with respect to one of the derivatives has a solution that tends to zero as one
tends to infinity to one of the independent variables. At present, the theory of partial differential
equations finds its application in various fields of natural science.
References
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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 für die reine und angewandte Mathematik, 1877): 230-315
[23] Perron O.Ueber diejenigen Integrale linearer Differentialgleichungen, welche sich an einer Unbestimmtheitsstelle bestimmt verhalten VI.13, No 70, ( Math. Ann., 1911): 1-32
[24] Hartman P. On Jacobi brackets (Amer. J. Math., 1957): 187-189
[25] Hormander L. On the uniqueness of the Cauchy problem I No. 6 ( Math. Scand., 1958): 213-225.
[26] Hormander L. On the uniqueness of the Cauchy problem II No. 7 (Math. Scand., 1959): 177-190.
[27] Nagumo M. Ueber die Differentialgleichung y00 = f(x; y; y0)( Proc. Phys.-Math. Soc. Japan, 19 (3), 1937): 861-866
[28] Nagumo M. Ueber die Ungleichung du=dy > f(x; y; u; du=dy), (Japan J. Math., 1939): 51-56
[29] Nagumo M. Ueber das Randwertproblem der nicht linearen gewohnlichen Differentialgleichungen zweiter Ordnung( Proc. Phys.-Math. Soc. Japan, 1942): 845-851
[30] Aldibekov T.M., Aldazharova M.M. Integral basis of the first-order partial differential equation (International Conference MADEA-8, Mathematical analysis, differential equations and applications. Abstracts. Issik-Kul, Kyrgyz Republic, -June 17-23, 2018): 26
[31] Aldibekov T.M. On the stability of solutions of system of nonlinear differential equations with the first approximation method/ Program of the 3rd Workshop on Dynamical Systems and Applications. CAMTP. University of Maribor.
[32] T. Aldibekov. 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
[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): 67
[6] Hartman P. "Obyknovennye differentsialnye uravneniya"[Ordinary differential equations] ( M.:Mir, 1970): 627-629
[7] Petrovsky I.G. "Lektsii ob uravneniyah s chastnymi proizvodnymi"[Lectures in partial equations] 3rd ed. (М., 1961): 38-43
[8] Petrovsky I.G. "Lektsii po teorii obyknovennyh differentsialnyh uravnenii"[Lectures in ordinary differential equations] 6th ed.( М., 1970): 114-116
[9] Elsgoltc L. "Differentsialnye uravneniya"[Differential equations] (М., 2013): 57-67
[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): 223-225
[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] Massera H.L. "Lineinye differentsialnye i funktsionalnye prostranstva"[Linear differential and functional spaces] ( М.:Mir, 1970): 50-59
[13] 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
[14] Wazewski T.Ueber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung VI.7-9 No.43 (Math. Zeit., 1938): 522-532
[15] Gelfand I."Nekotorye zadachi teorii kvazilineinyh uravnenii"[Some problems of quazilinear equations theory] 14(2) (UMN, 1959): 87-158.
[16] 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
[17] Digel E. Uber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung (Math Z, 1938): 445-451
[18] Caratheodory C. Variationsrechnung und partielle Differentialgleichungen erster Ordnung 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 CXI (Astronomische Nachrichten, 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 für die reine und angewandte Mathematik, 1877): 230-315
[23] Perron O.Ueber diejenigen Integrale linearer Differentialgleichungen, welche sich an einer Unbestimmtheitsstelle bestimmt verhalten VI.13, No 70, ( Math. Ann., 1911): 1-32
[24] Hartman P. On Jacobi brackets (Amer. J. Math., 1957): 187-189
[25] Hormander L. On the uniqueness of the Cauchy problem I No. 6 ( Math. Scand., 1958): 213-225.
[26] Hormander L. On the uniqueness of the Cauchy problem II No. 7 (Math. Scand., 1959): 177-190.
[27] Nagumo M. Ueber die Differentialgleichung y00 = f(x; y; y0)( Proc. Phys.-Math. Soc. Japan, 19 (3), 1937): 861-866
[28] Nagumo M. Ueber die Ungleichung du=dy > f(x; y; u; du=dy), (Japan J. Math., 1939): 51-56
[29] Nagumo M. Ueber das Randwertproblem der nicht linearen gewohnlichen Differentialgleichungen zweiter Ordnung( Proc. Phys.-Math. Soc. Japan, 1942): 845-851
[30] Aldibekov T.M., Aldazharova M.M. Integral basis of the first-order partial differential equation (International Conference MADEA-8, Mathematical analysis, differential equations and applications. Abstracts. Issik-Kul, Kyrgyz Republic, -June 17-23, 2018): 26
[31] Aldibekov T.M. On the stability of solutions of system of nonlinear differential equations with the first approximation method/ Program of the 3rd Workshop on Dynamical Systems and Applications. CAMTP. University of Maribor.
[32] T. Aldibekov. 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
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Published
2018-12-21
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Section
Mathematics
