Distributional solutions for damped wave equations
Date
2020-12-26
Authors
Nualart, Marc
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
This work presents results on solutions to the one-dimensional damped wave equation, also called telegrapher's equation, when the initial conditions are general distributions. We make a complete deduction of its fundamental solutions, both for positive and negative times. To obtain them we only use self-similarity arguments and distributional calculus, making no use of Fourier or Laplace transforms. We next use these fundamental solutions to prove both the existence and the uniqueness of solutions to the distributional initial value problem. As applications we recover the semi-group property for initial data in classical function spaces, and we find the probability distribution function for a recent financial model of evolution of prices.
Description
Keywords
Partial differential equations, Damped wave equation, Distributional solution, Initial value problem
Citation
Nualart, M. (2020). Distributional solutions for damped wave equations. <i>Electronic Journal of Differential Equations, 2020</i>(131), pp. 1-16.
Rights
Attribution 4.0 International