Geodesic
The ill-posed problem for the heat transfer equation with involution
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 48-59.

Voir la notice de l'article provenant de la source Math-Net.Ru

A mixed problem for an equation of heat transfer with involution is considered. The uniqueness of the problem's solution is proved. The ill-posedness of the mixed problem with Dirichlet-type boundary conditions for this equation is shown. By application of Fourier method, we obtain a spectral problem for a second-order differential operator with involution with an infinite number of positive and negative eigenvalues. The Green function of obtained second-order differential operator with involution is constructed. Uniform estimate of the Green's function is established for sufficiently large values of the spectral parameter. The existence of the Green's function of a second-order differential operator with involution and with variable coefficient is proved. By estimation of the Green's function completeness of the eigenfunctions's system for operator discussed is proved. In the class of polynomials the existence of a solution of this ill-posed problem is proved.
Keywords: differential equation with involution, Fourier method, Green's function, eigenfunctions, basis. 
@article{SVMO_2019_21_1_a3,
     author = {A. A. Sarsenbi},
     title = {The ill-posed problem for the heat transfer equation with involution},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {48--59},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a3/}
}
TY  - JOUR
AU  - A. A. Sarsenbi
TI  - The ill-posed problem for the heat transfer equation with involution
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 48
EP  - 59
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a3/
LA  - ru
ID  - SVMO_2019_21_1_a3
ER  - 
%0 Journal Article
%A A. A. Sarsenbi
%T The ill-posed problem for the heat transfer equation with involution
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 48-59
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a3/
%G ru
%F SVMO_2019_21_1_a3
A. A. Sarsenbi. The ill-posed problem for the heat transfer equation with involution. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 48-59. http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a3/

[1] D. Przeworska-Rolewicz, Equations with transformed argument:an algebraic approach, PWN Elsevier, Amsterdam–Warszawa, 1973, 354 pp. | MR | Zbl

[2] J. Wiener, Generalized solutions of functional differential equations, PWN Elsevier, Singapore–New Jersey–London–Hong Kong, 1993, 410 pp.

[3] A. Kopzhassarova, A. L. Lukashov, A. Sarsenbi, “Spectral properties of non-self-adjoint perturbations for a spectral problem with involution”, Abstract and Applied Analysis, 2012, no. 4, 590781 | DOI | MR

[4] M. Sh. Burlutskaya, “Mixed problem for a first order partial differential equations with involution and periodic boundary conditions”, Comput. Mathematics and Math. Physics, 54:1 (2014), 3–12 | DOI | MR | Zbl

[5] A. G. Baskakov, I. A. Krishtal, E. A. Romanova, “Spectral analysis of a differential operator with an involution”, Journal of Evolution Equations, 17:2 (2017), 669–684 | DOI | MR | Zbl

[6] A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator”, Differential Equations, 46:4 (2010), 509–511 | DOI | MR | Zbl

[7] L. V. Kritskov, A. M. Sarsenbi, “Spectral properties of a nonlocal problem for second order differential equation with an involution”, Differential Equations, 51:8 (2015), 990–996 | DOI | MR | Zbl

[8] L. V. Kritskov, A. M. Sarsenbi, “Basicity in Lp of root functions for differential equations with involution”, Electronic Journal of Differential Equations, 2015:278 (2015), 1–9 | MR

[9] L. V. Kritskov, A. M. Sarsenbi, “Riesz basis property of system of root functions of second-order differential operator with involution”, Differential Equations, 53:1 (2017), 33–46 | DOI | MR | Zbl

[10] L. V. Kritskov, A. M. Sarsenbi, “Equiconvergence property for spectral expansions related to perturbations of the operator $-{u}''\left( -x \right)$ with initial data”, Filomat, 32:3 (2018), 1069–1078 | DOI | MR

[11] A. A. Sarsenbi, “Completeness and basicity of the eigenfunctions of the spectral problem with involution”, Matematicheskiy zhurnal, 17:2(64) (2017), 175–183 (In Russ.) | MR

[12] A. Cabada, F. A. F. Tojo, “On linear differential equations and systems with reflection”, Appl. Math. Comp., 305 (2017), 84–102 | DOI | MR | Zbl

[13] M. Kirane, N. Al-Salti, “Inverse problems for a nonlocal wave equation with an involution perturbation”, J. Nonlinear Sci. Appl., 9 (2016), 1243–1251 | DOI | MR | Zbl

[14] A. A. Sarsenbi, “On a class of inverse problems for a parabolic equation with involution”, AIP Conference Proceedings, 1880 (2017), 040021 | DOI

[15] M. J. Ablowitz, H. Z. Musslimani, “Integrable nonlocal nonlinear Schredinger equation”, Physical Review Letters, 110 (2013), 064105 | DOI

[16] A. M. Akhtyamov, I. M. Utyashev, “Restoration of polynomial potential in the Sturm-Liouville problem”, Zhurnal srednevolzhskogo matematicheskogo obshchestva, 20:2 (2018), 148–158 (In Russ.) | MR

[17] A. Ashyralyev, A. Sarsenbi, “Well-posedness of a parabolic equation with involution”, Numerical Functional Analysis and Optimization, 38:10 (2017), 1295–1304 | DOI | MR | Zbl

[18] A N. Tikhonov, A. A. Samarskiy, Equation of mathematical physics, Nauka Publ., Moscow, 1972, 736 pp. (In Russ.) | MR

[19] M. A. Naymark, “On some signs of completeness of the system of eigenvectors and associated vectors of a linear operator in a Hilbert space”, Doklady Akademii nauk SSSR, 98:5 (1954), 727–730 (In Russ.) | Zbl

[20] E. A. Koddington, N. Levinson, Theory of ordinary differential equations, Inostrannaya literatura Publ., Moscow, 1958, 474 pp. (In Russ.)