An Analytical Solution for the Uncertain Damped Wave Equation

Damped wave equations, as an important class of partial differential equations, have wide applications in fields such as acoustics, signal processing, and fluid mechanics. However, wave propagation is often affected by noise interference, including medium randomness, random external forces and so on. Therefore, an uncertain term needs to be added to the damped wave equation to enhance the model’s fidelity to real-world scenarios. Based on uncertainty theory, this paper introduces the Liu process to the damped wave equation to characterize uncertainties, thereby establishing a new type of equation—the uncertain damped wave equation. In order to solve this equation, the method of separation of variables is employed to derive the analytical solution. The uniqueness of the solution is proved under given initial and boundary conditions. Finally, several examples are provided to illustrate the analytical solutions.