Abstract
In this paper, we are concerned with the initial boundary value problem of arbitrarily high-dimensional Klein-Gordon equations, posed on a bounded domain Ω ⊂ ℝd for d ≥ 1 and equipped with the requirement of boundary conditions. We derive and analyze an integral formula which is proved to be adapted to different boundary conditions for general Klein-Gordon equations in arbitrarily high-dimensional spaces. The formula gives a closed-form solution to arbitrarily high-dimensional homogeneous linear Klein-Gordon equations, which is totally different from the well-known D’Alembert, Poisson, and Kirchhoff formulas. Some applications are included as well.
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