Geodesic
Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 4, pp. 444-456 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity $u_t '(x, 0)=\psi (x)$. The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for $\psi (x)\in W_p^1$ ($1 ), and it is also shown that if $\psi(x)\in L_p[0,1]$ ($1\le p\le2$), the formal solution is a generalized solution of the mixed problem. 
@article{ISU_2020_20_4_a3,
     author = {V. P. Kurdyumov and A. P. Khromov and V. A. Khalova},
     title = {Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {444--456},
     year = {2020},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2020_20_4_a3/}
}
TY  - JOUR
AU  - V. P. Kurdyumov
AU  - A. P. Khromov
AU  - V. A. Khalova
TI  - Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2020
SP  - 444
EP  - 456
VL  - 20
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ISU_2020_20_4_a3/
LA  - ru
ID  - ISU_2020_20_4_a3
ER  - 
%0 Journal Article
%A V. P. Kurdyumov
%A A. P. Khromov
%A V. A. Khalova
%T Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2020
%P 444-456
%V 20
%N 4
%U http://geodesic.mathdoc.fr/item/ISU_2020_20_4_a3/
%G ru
%F ISU_2020_20_4_a3
V. P. Kurdyumov; A. P. Khromov; V. A. Khalova. Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 4, pp. 444-456. http://geodesic.mathdoc.fr/item/ISU_2020_20_4_a3/

[1] Krylov A. N., On Some Differential Equations of Mathematical Physics Having Applications in Engineering, GITTL, M.–L., 1950, 368 pp. (in Russian) | MR

[2] Chernyatin V. A., Justification of the Fourier Method in a Mixed Problem for Partial Differential Equations, Moscow Univ. Press, M., 1991, 112 pp. (in Russian)

[3] Burlutskaya M. S., Khromov A. P., “Resolvent approach in the Fourier method”, Dokl. Math., 90:2 (2014), 545–548 | DOI | DOI | MR | Zbl

[4] Khromov A. P., “Behavior of the formal solution to a mixed problem for the wave equation”, Comput. Math. and Math. Phys., 56:2 (2016), 243–255 | DOI | DOI | MR | Zbl

[5] Gurevich A. P., Kurdyumov V. P., Khromov A. P., “Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 16:1 (2016), 13–29 (in Russian) | DOI | MR | Zbl

[6] Kurdyumov V. P., Khromov A. P., Khalova V. A., “A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 18:2 (2018), 157–171 (in Russian) | DOI | MR | Zbl

[7] Khromov A. P., “Mixed problem for a homogeneous wave equation with a nonzero initial velocity”, Comput. Math. and Math. Phys., 58:9 (2018), 1531–1543 | DOI | DOI | MR | Zbl

[8] Khromov A. P., “On the convergence of the formal Fourier solution of the wave equation with a summable potential”, Comput. Math. and Math. Phys., 56:10 (2016), 1778–1792 | DOI | DOI | MR | Zbl

[9] Burlutskaya M. S., Khromov A. P., “Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders”, Diff. Equat., 53:4 (2017), 497–508 | DOI | MR | Zbl

[10] Naymark M. A., Linear Differential Operators, Nauka, M., 1969, 526 pp. (in Russian) | MR

[11] Rasulov M. L., The Method of Contour Integral, Nauka, M., 1964, 462 pp. (in Russian) | MR

[12] Vagabov A. I., Introduction to the Spectral Theory of Differential Operators, Izd-vo Rostovskogo universiteta, Rostov-na-Donu, 1994, 160 pp. (in Russian)

[13] Bari N. K., Trigonometric Series, Gos. izd-vo fiz.-mat. lit., M., 1961, 936 pp. (in Russian) | MR