Exact solution to a multidimensional wave equation with delay. (English) Zbl 07425035

Summary: This paper deals with a mixed problem for the wave equation with discrete delay \(\tau > 0\), \[u_{t t} (t, \boldsymbol{x}) = a_1^2 \Delta_{\boldsymbol{x}} u (t,\boldsymbol{x}) + a_2^2 \Delta_{\boldsymbol{x}} u(t-\tau, \boldsymbol{x}) + b_1 u ( t , \boldsymbol{x}) + b_2 u (t-\tau,\boldsymbol{x}), \quad t > \tau ,\, 0 \leq \boldsymbol{x} \leq \boldsymbol{l} ,\] with Dirichlet boundary conditions. The exact infinite series solution is constructed by the method of separation of variables, where the time-dependent functions of the decomposition satisfy second-order delay differential equations. Our approach is based on and extends the work by F. Rodríguez et al. [ibid. 219, No. 6, 3178–3186 (2012; Zbl 1309.35045)].


35R10 Partial functional-differential equations
35C05 Solutions to PDEs in closed form
35C10 Series solutions to PDEs


Zbl 1309.35045
Full Text: DOI


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