Перейти до основного вмісту
Existence of a solution to the Cauchy problem for a linear equation with first-order partial derivatives with variable coefficients in the space of rapidly decreasing functions
Samoilenko Yulia Ivanivna 1
1 Taras Shevchenko National University of Kyiv, Kyiv, 01033, Ukraine
Keywords: the Cauchy problem, linear equation with first-order partial derivatives
Abstract
We find sufficient conditions for the existence of a solution of a linear non-homogeneous partial differential equation of the first order with variable coefficients, and the Cauchy problem for the equation.
References

[1] Lighthill M.J., Whitham G.B. On kinematic waves. I. Flood movement in long rivers // Proc. Roy. Soc. (London). – 1965. – 229 A . – P. 281-316.

[2] Scott E. Waves in active and nonlinear media in application to electronics. - M.: Soviet Radio, 1977. - 368 p.

[3] Lighthill M.J. Group velocity // Journ. of the Institute of Mathematical Applications. – 1965. – 1 . – P. 1-28.

[4] Lighthill M.J., Whitham G.B. On kinematic waves. II. A theory of traffic flow on long crowded roads // Proc. Roy. Soc. (London). – 1965. – 229 A . – P. 317- 345.

[5] Gazis D.C. Mathematical theory of automobile traffic // Science. – 1967. – 157 . – P. 273-281.

[6] Maslov V.P., Omelyanov G.A. Asymptotic soliton-like solutions of equations with small dispersion // Uspekhi mat. nauk. - 1981. -36 (219),N 2. - P. 63-124.

[7] Oleinik O.A. On the Cauchy problem for nonlinear equations in the class of discontinuous functions // Dokl.AN USSR. - 1954. -95, N 3. - P. 451-454.

[8] Goritsky A.Y., Kruzhkov S.N., Chechkin G.A. Equations with partial derivatives of the first order. - M.: Izd. of the Center for Applied Research at the Mechanical Engineering Department of Moscow State University, 1999. - 96 p.

[9] Ablowitz M.J. Nonlinear dispersive waves. Asymptotic analysis and solitons. – Cambridge: Cambridge University Press, 2011. – 348 p.

[10] Hopf E. On the right weak solution of the Cauchy problem for a quasilinear equation of first order // Journ. of Mathematics and Mechanics. – 1969. – 19 , N 6. – P. 483-487.

[11] Samoilenko Y.I. Existence of the solution of the Cauchy problem for the Hopf equation with variable coefficients in the space of rapidly decreasing functions // Collection of scientific works of the Institute of Mathematics of the National Academy of Sciences of Ukraine. - 2012. - 9, N 1. - P. 7-15.

[12] Samoilenko Y.I. Conditions for the existence of discontinuous solutions of quasilinear equations with variable coefficients // Bulletin of Kyiv National University. Mathematics. Mechanics. - 2009. - 22. - P. 25-32.

[13] Samoilenko, V.G.; Samoilenko, Yu.I. Pogranloy method and conditions of Hugoniot type for the Korteweg-de Friese equation // Vestnik of Brest State University. Series 4. Physics. Mathematics. - 2010. - N 2. - P. 128-143.

[14] Samoilenko V.G., Samoilenko Y.I. Existence of a solution of a nonhomogeneous equation with a one-dimensional Schrödinger operator in the space of rapidly decreasing functions // Ukrainian Mathematical Bulletin. - 9 , N 2. - P. 38-45.

[15] Samoilenko V.G., Samoilenko Y.I. Asymptotic expansions for single-phase soliton-like solutions of the Korteweg-de Vries equation with variable coefficients // Ukrainian Mathematical Journal - 2005. - 58, N 1. - P. 111-124.

[16] Samoylenko Yul. Asymptotical expansions for one-phase soliton type solution to perturbed Kortewegde Vries equation // Proceedings of the Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”. – K.: Institute of Mathematics. – 2004. – 3. – P. 1435-1441.

[17] Samoilenko Y.I. Asymptotic solutions of the Cauchy problem for the Korteweg-de Vries equation with variable coefficients and a small parameter of even degree at the highest derivative // Scientific Bulletin of Chernivtsi University: Collection of scientific works. Mathematics. - 2009. - 485. - P. 102-107.

[18] Samoilenko V.G., Samoilenko Y.I. Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients // Ukrainian Mathematical Journal - 2007. - 59, N 1. - P. 122-132.

[19] Faminsky A.V. Cauchy problem for the Korteweg-de Fries equation and its generalizations // Proceedings of the I.G. Petrovskii Seminar. - 1988. -13. - P. 56-105.

Cite
ACS Style
Samoilenko , Y.I. Existence of a solution to the Cauchy problem for a linear equation with first-order partial derivatives with variable coefficients in the space of rapidly decreasing functions. Bukovinian Mathematical Journal. 2016, 1
AMA Style
Samoilenko YI. Existence of a solution to the Cauchy problem for a linear equation with first-order partial derivatives with variable coefficients in the space of rapidly decreasing functions. Bukovinian Mathematical Journal. 2016; 1(1-2).
Chicago/Turabian Style
Yulia Ivanivna Samoilenko . 2016. "Existence of a solution to the Cauchy problem for a linear equation with first-order partial derivatives with variable coefficients in the space of rapidly decreasing functions". Bukovinian Mathematical Journal. 1 no. 1-2.
Export
We use own, third-party cookies, and localStorage files to analyze web traffic and page activities. Privacy Policy Settings