Singular Hyperbolic Type Equations and Tsunami Propagation for Irregular Topographies
Altybay A. Ruzhansky M. Sebih M.E. Tokmagambetov N.
2025Springer
Journal of Mathematical Sciences (United States)
2025
We consider the tsunami wave equation with singular coefficients and prove that it has a very weak solution. We show the uniqueness and consistency of the very weak solution with the classical one in an appropriate sense. In one space dimension, we analyze the behavior of the waves in singular topographies. We observe the appearance of a substantial reflected wave, travelling in the opposite direction from the point of singularity. Its structure and strength are analyzed numerically. In particular, we illustrate the limiting behavior of the solution to the regularized problems when the regularizing parameter tends to zero. A surprising conclusion is that while, in general, the solution of the equation may not exist in the “classical” sense, the limit of the net of solutions of the regularized problems may exist, calling it the limiting very weak solution.
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Institute of Mathematics and Mathematical Modeling, 125, Pushkin St., Almaty, 050010, Kazakhstan
Ghent University, 281, Krijgslaan, Building S8, Ghent, B 9000, Belgium
Queen Mary University of London, 327, Mile End Road, London, E1 4NS, United Kingdom
Mustapha Stambouli University, G47H+QJ6, Cheikh El Khaldi Av., Mascara, 29000, Algeria
Al-Farabi Kazakh National University, 71, Al-Farabi Av., Almaty, 050040, Kazakhstan
Centre de Recerca Matemática, Universitat Autònoma de Barcelona, Campus UAB Edifici C, Bellaterra, Barcelona, 08193, Spain
Institute of Mathematics and Mathematical Modeling
Ghent University
Queen Mary University of London
Mustapha Stambouli University
Al-Farabi Kazakh National University
Centre de Recerca Matemática
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