On a diagonal system of the first-order partial differential equations from two independent variables

Authors

DOI:

https://doi.org/10.26577/JMMCS.2020.v105.i1.01
        96 82

Keywords:

differential equations, diagonal system, first order partial derivatives, asymptotic behavior

Abstract

A diagonal system of three first-order partial differential equations in two indep endent variables
is cons idered. The equations entering into the diagonal system are indep endent from each other,
therefore, the compatibility condition of the system do es not arise. We consider the asymptotic b ehavior of solutions at an infinitely distant p oint, with resp ect to some parameter. The main place
in the system is o ccup ied by a nonlinear first-order partial differential equation, the remaining
equations are adjoining equations, the solutions of which contain the initial value of one indep endent variable as a parameter. The attached equations are chosen appropriately, and the solution
to the system is already studied, w hich already has an internal connection. The adjoint equations
are linear first-orde r partial differential equations. Using the fact that the zero solutions of the
characteristic equations are asymptotically stable on Lyapunov, the condi tion s when the set of
three differential equations, c on sidered as a diagonal system of partial differential equations of the
first order, has a solution w ith certain initial values and is an infinitesimal function in the vicinity
of an infini te ly remote p oint are de scrib ed . Metho ds of the theory of functions and differential
inequalities in the theory of first-order differential equations are used.

References

[1] 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
[2] Frobenius G., ”Ueber das Pfaffsche Problem”, Journal for die reine und angewandte Mathematik, (1877): 230-315
[3] Bendixcon I., ” Demostration de l’existence de l’integrale d’une equation aux derives partielles lineaire”, [Demonstration of the existence of the integral of a linear partial differential equation], Bull. Soc. Math. France, 24 (1896): 220-225.
[4] Picard E., Traite d’analyse [Treatise on analysis] (Paris: Gauthier-Villars. , 1896)
[5] Goursat E., Lesons sur l’integration des equations aux derives partielles du premier ordre. 2ed. [Lessons on integrating first order partial differential equations.] (Paris, 1921), 77-81
[6] Caratheodory C., Variationsrechnung und partielle Differentialgleichungen erster Ordnung [Variational calculus and partial differential equations of first order] (Leipzig und Berlin:B. G. Teubner, 1935), 7-9
[7] Gunter N.M., Integrirovaniye uravneniy pervogo poryadka v chastnykh proizvodnykh [Integration of first order partial differential equations] (L.-M., 1934)
[8] Kamke E., Spravochnik po differentsialnym uravneniyam v chastnyh proizvodnyh pervogo poryadka [Referense book in first-order partial differential equations]. (М.: Nauka, 1966), 260
[9] Wazewski T., ”Sur l’unicite et la limitation des integrals des equations aux deriveespartielles du premier ordre”, [On the uniqueness and the limitation of integrals of first order partial differential equations], Atti R. Accad. Naz. Lincei Rend. Cl. Sci.Fis. Mat. Nat., 18 (6) (1933): 372-376
[10] 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
[11] 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
[12] 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
[13] Plis A., ”Characteristics of nonlinear partial differential equation”, Bull.Acad. Polon. Sci., No 2. (1954): 419-422
[14] Digel E., ”Uber die Bedingungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnung” [On the conditions of existence of the integrals of the first order partial differential equations], Math Z, (1938): 445-451
[15] Haar A., ”Zur Characteristikentheorie”, [Characteristic Theory], Acta Sci. Math. Szeged, V.4. №2 (1928): 103-114
[16] Turski S., ”Sur l’unicite et la limitation des integrals des equations aux derives partielles du premier ordre”, [On the uniqueness and the limitation of integrals of first order partial derivative equations], Ann. Soc. Polon. Math., 120 (1933): 81-86;
[17] Petrovsky I.G., ”O probleme Koshi dlya sistem uravneniy s chastnymi proizvodnymi”, [On the Cauchy problem for systems of partial differential equations], Mat. Sbornik, V.2. No. 5. (1937)
[18] Nagumo M., ”Ueber die Ungleichung du/dy > f ( x, y , u, du/dy )” [About the inequality du/dy > f (x, y, u, du/dy )], Japan J. Math., 15 (1939): 51-56;
[19] Nagumo M., ”Ueber die Differentialgleichung y ′′ = f(x, y, y ′)”[About the differential equation y ′′ = f(x, y, y ′)], Proc. Phys.-Math. Soc. Japan, 19 (3) (1937): 861-866;
[20] Nagumo M., ”Ueber das Randwertproblem der nicht linearen gewohnlichen Differentialgleichungen zweiter Ordnung”[On the boundary value problem of the non-linear ordinary differential equations of the second order], Proc. Phys.-Math. Soc. Japan, 24 (1942): 845-851;
[21] Hartman P., ”On exterior derivatives and solutions of ordinary differential equations”, Trans. Amer. Math., Soc., 91 (1959): 277-292.
[22] Kamke E., Differentialgleichungen, Losungsmethoden und Losungen[Differential Equations, Solution Methods and Solutions] (Leipzig: II. Akademische Verlagsgesellschaft, 1959)
[23] Hartman P., Obyknovennye differentsialnye uravneniya [Ordinary differential equations] (M.:Mir, 1970): 719
[24] Courant R., Uravneniya s chastnymi proizvodnymi [Partial equations] (M: Mir, 1964), 845
[25] Petrovsky I.G., Lektsii ob uravneniyah s chastnymi proizvodnymi [Lectures in partial equations] 3rd ed. (М., 1961), 400 [26] Rashevsky P.K., Geometricheskaya teoriya s chastnymi proizvodnymi [Partial geometric theory] (M.-L., 1947)

Downloads

How to Cite

Aldibekov T. М., & Aldazharova, M. M. (2020). On a diagonal system of the first-order partial differential equations from two independent variables. Journal of Mathematics, Mechanics and Computer Science, 105(1), 3–9. https://doi.org/10.26577/JMMCS.2020.v105.i1.01