Global correctness theorem to the first mixed problem for the general telegraph equation with variable coefficients on a segment

F. E. Lomovtsev

Problemy fiziki, matematiki i tehniki, no. 1 (2022), pp. 62-73.

Résumé

The global theorem to Hadamard correctness to the first mixed problem for inhomogeneous general telegraph equation with all variable coefficients in a half-strip of the plane is proved by a novel method of auxiliary mixed problems. Without explicit continuations of the mixed problem data outside set of mixed task assignments the recurrent Riemann-type formulas of a unique and stable classical solution for the first mixed problem on a segment are derived. This half-strip of the plane is divided by the curvilinear characteristics of a telegraph equation into rectangles of the same height, and each rectangle into three triangles. The correctness criterion consists of smoothness requirements and matching conditions on the right-hand side of the equation, initial and boundary conditions of the mixed problem. The smoothness requirements are necessary and sufficient for twice continuous differentiability of the solution in these triangles. The matching conditions together with these smoothness requirements are necessary and sufficient for twice continuous differentiability of solution on the implicit characteristics in these rectangles.

Détail

Keywords: general telegraph equation, implicit characteristics of equation, correctness criterion, smoothness requirement, matching condition.

Bibliographie

[1] F.E. Lomovtsev, “Method of auxiliary mixed problems for semi-bounded string”, Sixth Bogdanov readings on ordinary differential equations, mater. Int. mat. conf. Minsk, BSU, 7–10 Dec 2015 at 2 pm, v. 2, ed. S.G. Krasovsky, IM NAS Belarus, Minsk, 2015, 74–75

[2] F.E. Lomovtsev, “First Mixed Problem for the General Telegraph Equation with Variable Coefficients on the Half-Line”, Journal of the Belarusian State University. Mathemaatics and Informatics, 2021, no. 1, 18–38

[3] F.E. Lomovtsev, “Conclusion of the Smoothness Criterion for the Right-Hand Side of the Model Telegraph Equation with the Rate $a(x, t)$ by the Correction Method”, Modern methods of the theory for boundary value problems, XXXIV Voronezh Spring Mathematical School «Pontryagin Readings-XXXII», dedicated to the memory of A.D. Baeva, mater. Int. Conf. (Voronezh, Voronezh State University, May 3–9, 2021), Voronezh State University. Publishing House, Voronezh, 2021, 284–287

[4] F.E. Lomovtsev, “Global Correctness Theorem for the First Mixed Problem with the General Oscillation Equations of a Bounded String”, Analytic Methods of Analysis and Differential Equations, AMADE 2018, Cambridge Scientific Publishers Ltd., Cottenham, Cambridge CB24 8QY UK, 2020, 165–186

[5] F.E. Lomovtsev, “Global Correctness Theorem of the First Mixed Problem for a Wave Equation in a Half-Band of the Plane”, Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Computer Engineering and Management, 11:1 (2021), 68–82

[6] V.V. Khromov, “On the convergence of a formal solution by the Fourier method of a wave equation with a summable potential”, Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 56:10 (2016), 1795–1809 | MR | Zbl

[7] V.V. Kornev, A.P. Khromov, “The resolvent approach to the Fourier method in the mixed problem for an inhomogeneous wave equation”, Izvestiya Saratovskogo Universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika, 16:4 (2016), 403–413 | MR

[8] A.P. Khromov, “A mixed problem for a wave equation with a summable potential and a nonzero initial velocity”, Doklady Akademii nauk, 474:6 (2017), 668–670 | MR | Zbl

[9] A.P. Khromov, “On the classical solution of the mixed problem for a homogeneous wave equation with fixed ends and zero initial velocity”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika, 19:3 (2019), 280–288 | MR | Zbl

[10] V.V. Kornev, A.P. Khromov, “Classical and generalized solutions to the mixed problem for an inhomogeneous wave equation”, Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 59:2 (2019), 107–121 | MR

[11] A.P. Khromov, “Necessary and sufficient conditions for the existence of a classical solution of a mixed problem for a homogeneous wave equation in the case of a summable potential”, Differential Equations, 55:5 (2019), 717–731 | DOI | MR | Zbl

[12] A.P. Khromov, V.V. Kornev, “Classical and generalized solutions of the mixed problem for an inhomogeneous wave-wave equation”, Doklady Akademii nauk, 484:1 (2019), 18–20 | MR | Zbl

[13] V.V. Kornev, A.P. Khromov, “Divergent series and a generalized solution of one mixed problem for the wave equation”, Materials of the International Mathematical Conference, Voronezh Spring Mathematical School (2020 May 3–9, Voronezh, Russia), ANO Nauka-Unipress, Voronezh, 2020, 99–102

[14] I.S. Lomov, “A generalized d'Alembert formula for the telegraph equation in the case of a substantially non-self-adjoint operator”, Materials of the International Mathematical Conference, Voronezh Spring Mathematical School (2020 May 3–9, Voronezh, Russia), ANO Nauka-Unipress, Voronezh, 2020, 124–126

[15] F.E. Lomovtsev, “Riemann Formula of the Classical Solution to the First Mixed Problem for the General Telegraph Equation with Variable Coefficients on the Half-Line”, Tez. dokl. Seventh Bogdanov Readings on differential equations (Minsk, June 1–4, 2021), v. 2, Minsk, 2021, 201–203

[16] E.N. Novikov, Mixed problems for the equation of forced vibrations of a bounded string at unsteady boundary conditions with the first and second oblique derivatives, Abstract of the thesis. dis. ... can-that physical-mat. Sciences (01.01.02), IM NAS Belarus, Minsk, 2017