Generalized Riemann formulas for the solution of the first mixed problem for the general telegraph equation with variable coefficients in the first quadrant

F. E. Lomovtsev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 50-69.

Résumé

Using the well-known Riemann method and a new method for compensating the boundary regime with the right-hand side of the equation, we obtain the Riemann formulas for the unique and stable classical solution of the first mixed problem for a linear general inhomogeneous telegraph equation with variable coefficients in the first quadrant. From the formulation of the mixed problem, the definition of classical solutions, and the established criterion for the smoothness of the right-hand side of the equation, we obtain a criterion of the well-posedness in the Hadamard sense. This criterion consists of smoothness requirements and three conditions for matching the right-hand side of the equation and the boundary and initial data. The validity of the Riemann formulas and the well-posedness criterion is confirmed by their coincidence with the well-known formulas of the classical solution and the well-posedness criterion for the model telegraph equation.

Détail

Keywords: first mixed problem, telegraph equation, implicit characteristic, global correctness theorem, smoothness condition, consistency condition

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