Global Correctness Theorem to the Mixed Problem for a Homogeneous Two-Velocity Model Telegraph Equation with Non-Characteristic Time-Dependent Oblique Derivatives at the Ends of a String

8eb97d8d-9e6a-4629-94eb-401f4ce3ec4320251107111939584wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON COMMUNICATIONS2224-28641109-274210.37394/23204http://wseas.org/wseas/cms.action?id=4021318202531820252410.37394/23204.2025.24https://wseas.com/journals/communications/2025.phpGlobal Correctness Theorem to the Mixed Problem for a Homogeneous Two-Velocity Model Telegraph Equation with Non-Characteristic Time-Dependent Oblique Derivatives at the Ends of a StringFedorLomovtsevBelarusian State University Mechanics and Mathematics Faculty pr. Nezavisimosty, 220030 Minsk BELARUSWe have proved the global correctness theorem to new mixed problem for two-velocity homogeneous model telegraph equation with non-characteristic time-dependent oblique derivatives at the ends of string. An explicit recurrent formula for a unique and stable classic solutions with a correctness criterion for mixed problem in the set of twice continuously differentiable functions are derived. Its correctness criterion consists of necessary and sufficient four matching conditions and three smoothness requirements on the initial and boundary data of non-characteristic mixed problem. These results are obtained by author’s ”method of auxiliary mixed problems for wave equations on a half-line” and ”implicit characteristic method”.1172025117202531436https://wseas.com/journals/communications/2025/a125104-005(2025).pdf10.37394/23204.2025.24.6https://wseas.com/journals/communications/2025/a125104-005(2025).pdf10.3934/math.2023438Svetlin G. Georgiev, Khaled Zennir, Keltoum Bouhali, Rabab Alharbi, Yousif Altayeb, Mohamed Biomy. Existence of solutions for impulsive wave equations, AIMS Mathematics, 8(4), 2023, pp. 8731–8755. doi: 10.3934/math.2023438 10.1134/s0012266122050020O. Jokhadze, Mixed Problem with Nonliniear Boundary Condition for Semilinear Wave Equa tion, Differential equations, Vol. 58, 30 August 2022, pp. 593–609. 10.18500/1816-9791-2023-23-3-311-319V. Kurdyumov, Classical and generalized solutions of a mixed problem for a homogeneous wave equation with summable potential. Part I. Classical solution of a mixed problem, Bulletin of Saratov University. New series: Mathematics. Mechanics. Computer science. Vol. 23, issue 3, 2023, pp. 311– 319. https://doi.org/10.18500/1816-9791-2023- 23-3-311-319. 10.37394/232021.2023.3.5F. Lomovtsev, Correction Method of Trial Solutions into Classical Ones to a Model Wave Equation with Variable Velocities a1(x, t) and a2(x, t) in the First Quarter of the Plane, Bulletin of the Foundation for Basic Research No. 4, 2023, pp. 53–83. 10.37394/23206.2023.22.103F. Lomovtsev, A. Kukharev, The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane, WSEAS Transaction on Mathematics, Vol. 22, December 20 2023, Art. 103, pp. 936–949. ISSN/EISSN:1109-2769/2224- 2880, DOI: 10.37394/23206.2023.22.103 https://wseas.com/journals/articles.php?id=8733 10.30546/1683-6154.24.2.2025.185F. Lomovtsev, Z. Liu, E. Cheb, A global Correctness Theorem for the Second Mixed Problem for a Model Wave Equation in a Half-Strip of the Plane, Appl. Comput. Math., Vol. 24, No. 2, 2025, pp. 185–202. DOI: 10.30546/1683- 6154.24.2.2025.185 10.1007/s41980-025-00968-2F. Lomovtsev, Z. Liu, Riemann Formulas of the Classical Solution and a Correctness Criterion to the Second Mixed Problem for the General Telegraph Equation with Variable Coefficients on a Segment, Bulletin of the Iranian Mathematical Society, 2025, 51:45. pp. 1–32. https://doi.org/10.1007/s41980-025-00968-2 F. Lomovtsev, Method of auxiliary mixed problems for a semi-bounded string, Proc. int. mat. conf., Minsk, December 710, 2015, IM NAS of Belarus, Part 2, 2015, pp. 74–75. F. Lomovtsev, Mixed Problem for a Model Telegraph Equation with Two-Velocities a1(x, t) and a2(x, t) with a Non-Characteristic Oblique Derivative at the End of a Semi-Bounded String. I, Vesn. Vitsebsk. dzyarzh. un-ta 3(128), 2025, pp. 3–17. F. Lomovtsev, The First Mixed Problem for a General Telegraph Equation with Variable Coefficients on a Half-Line, Journal of the BSU. Mathematics and Informatics No. 1, 2021, pp. 18–38. doi=10.33581/2520-6508-2021-1-18-38 10.22363/2413-3639-2024-70-3-451-486V. Rykhlov, Classical solution of the initialboundary value problem for the wave equation with mixed derivative, Sovrem. mat. Fundam. napr. , Vol. 70, No. 3, 2024, pp. 451–486. http://doi.org/10.22363/2413-3639-2024-70-3- 451-486 10.3934/math.2023778Hatice Taskesen, Qualitative results for a relativistic wave equation with multiplicative noise and damping term, AIMS Mathematics, 8(7), 2023, pp. 15232-15254. doi: 10.3934/math.2023778.