Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

A. P. Khromov ; M. Sh. Burlutskaya

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 171-198.

Résumé

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series. The results for other similar mixed problems given without proof.

Bibliographie

[1] Steklov V. A., The main tasks of mathematical physics, Nauka, Moscow, 1983, 432 pp. | MR | Zbl

[2] Petrovsky I. G., Lectures on partial differential equations, Dover Publ. Inc., 1992, 245 pp. | MR

[3] Smirnov V. I., A Course of Higher Mathematics, in 4 vol., v. 4, Gostekhizdat, Moscow, 1953, 804 pp.

[4] Ladyzhenskaya O. A., Mixed problem for a hyperbolic equation, Gostekhizdat, Moscow, 1953, 282 pp.

[5] Il'in V. A., Selected works, in 2 vol., v. 1, OOO “Maks-press”, Moscow, 2008, 727 pp.

[6] Il'in V. A., “The solvability of mixed problems for hyperbolic and parabolic equations”, Rus. Math. Surv., 15:1 (1960), 85–142 | DOI | MR | Zbl

[7] Chernyatin V. A., Justification of the Fourier method in a mixed problem for partial differential equations, Moscow Univ. Press, Moscow, 1991, 112 pp. | MR | Zbl

[8] Krylov A. N., On some differential equations of mathematical physics with applications in technical matters, GITTL, Leningrad, 1950, 368 pp.

[9] Krylov A. N., Lectures on approximate calculations, GITTL, Moscow–Leningrad, 1950, 398 pp. | MR

[10] Lanczos C., Discourse of Fourier Series, Oliver and Boyd, Ltd., Edinburgh–London, 1966, 255 pp. | MR | Zbl

[11] Nersesyan A. B., “Acceleration of convergence of eigenfunction expansions”, Dokl. NAN Armenii, 107:2 (2007), 124–131 | MR

[12] Chernyatin V. A., “To clarify the theorem of existence of the classical solution of the mixed problem for one-dimensional wave equation”, Differential Equations, 21:9 (1985), 1569–1576 | MR | Zbl

[13] Chernyatin V. A., “To the decision of one of the mixed problem for an inhomogeneous equation with partial derivatives of fourth order”, Differential Equations, 21:2 (1985), 343–345 | MR | Zbl

[14] Chernyatin V. A., “On necessary and sufficient conditions for the existence of the classical solution of the mixed problem for one-dimensional wave equation”, Dokl. AN SSSR, 287:5 (1986), 1080–1083 | MR | Zbl

[15] Chernyatin V. A., “Classical solution of the mixed problem for the inhomogeneous hyperbolic equation”, Numerical methods for solving boundary value and initial problems for differential equations, Moscow Univ. Press, Moscow, 1986, 17–36

[16] Chernyatin V. A., “To clarify the existence theorem for solutions of the mixed problem for the inhomogeneous heat equation”, Numerical analysis: methods, algorithms, programs, Moscow Univ. Press, Moscow, 1988, 126–132 | MR

[17] Chernyatin V. A., “On the solvability of the mixed problem for the inhomogeneous hyperbolic equations”, Differential Equations, 24:4 (1988), 717–720 | MR | Zbl

[18] Andreev A. A., “About the correctness of boundary problems for some equations with calimanesti shift”, Differential equations and their applications, Proceedings of the 2nd international workshop, Samara, 1998, 5–18

[19] Dankl Ch. G., “Differential-Difference Operators Associated to Reflection Groups”, Trans. Amer. Math. Soc., 311:1 (1989), 167–183 | DOI | MR

[20] Platonov S. S., “The eigenfunction expansion for some functional-differential operators”, Proceedings of Petrozavodsk State University. Ser. Math., 11, 2004, 15–35

[21] Khromov A. P., “Inversion of integral operators with kernels discontinuous on the diagonal”, Math. Notes, 64:6 (1998), 804–813 | DOI | DOI | MR | Zbl

[22] Burlutskaya M. Sh., Kurdyumov V. P., Lukonina A. S., Khromov A. P., “A functional-differential operator with involution”, Doklady Math., 75:3 (2007), 399–402 | DOI | MR | Zbl

[23] Kornev V. V., Khromov A. P., “Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the diagonals”, Sbornik: Mathematics, 192:10 (2001), 1451–1469 | DOI | DOI | MR | Zbl

[24] Kurdyumov V. P., Khromov A. P., “Riesz bases of eigenfunctions of integral operators with kernels discontinuous on the diagonals”, Izvestiya: Mathematics, 76:6 (2012), 1175–1189 | DOI | DOI | MR | Zbl

[25] Kurdyumov V. P., Khromov A. P., “On Riesz basises of the eigen and associated functions of the functional-differential operator with a variable structure”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 7:2 (2007), 20–25

[26] Kornev V. V., Khromov A. P., “Operator integration with an involution having a power singularity”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 8:4 (2008), 18–33

[27] Burlutskaya M. Sh., Khromov A. P., “On the same theorem on a equiconvergence at the whole segment for the functional-differential operators”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 9:4, pt. 1 (2009), 3–10

[28] Burlutskaya M. Sh., Khromov A. P., “Substantiation of Fourier method in mixed problem with involution”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 11:4 (2011), 3–12

[29] Khalova V. A., Khromov A. P., “Integral Operators with Non-smooth Involution”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 13:1, pt. 1 (2013), 40–45

[30] Khromov A. P., “The mixed problem for the differential equation with involution and potential of the special kind”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 10:4 (2010), 17–22

[31] Burlutskaya M. Sh., Khromov A. P., “Classical solution of a mixed problem with involution”, Doklady Math., 82:3 (2010), 865–868 | DOI | MR | Zbl

[32] Burlutskaya M. Sh., Khromov A. P., “Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution”, Computational Mathematics and Mathematical Physics, 51:12 (2011), 2102–2114 | DOI | MR | Zbl

[33] Burlutskaya M. Sh., Khromov A. P., “Initial-boundary value problems for first-order hyperbolic equations with involution”, Doklady Math., 84:3 (2011), 783–786 | DOI | MR | Zbl

[34] Burlutskaya M. Sh., “A mixed problem with an involution on the graph of two edges with the cycle”, Doklady Math., 447:5 (2012), 479–482 | MR | Zbl

[35] Marchenko V. A., Sturm–Liouville Operators and Applications, Naukova Dumka, Kiev, 1977, 392 pp. | MR

[36] Naymark M. A., Linear differential operators, Nauka, Moscow, 1969, 528 pp. | MR