On the Mixed Boundary Value Problem for n -Dimensional Non-Homogeneous Wave Equation in Trigonometric sincmap Coordinate Systems

This paper investigates the analytical dynamics and mathematical well-posedness of wave propagation in an [Formula: see text]-dimensional non-homogeneous space modeled by the non-linear spatial coordinate mapping [Formula: see text]. Under this transformation, the governing hyperbolic partial differential equation is formulated primarily with variable coefficients exhibiting strong spatial gradients. By conducting spectral and variational analysis within generalized weighted Sobolev spaces [Formula: see text] and [Formula: see text], we construct exact, closed-form multi-dimensional eigenfunctions within this specific coordinate framework and discrete eigenvalues, proving that the eigenbasis forms a complete orthonormal system under a non-trivial spatial weight. The core essence of this work consists of two novel lemmas proving the metric equivalence of the [Formula: see text] mapping and the weighted orthogonality of the eigenbasis, alongside the proof of two crucial theorems: the existence and uniqueness of the generalized solution in the Sobolev space [Formula: see text] is established via Theorem 4.1, and a new a priori stability estimate within a generalized energy norm is derived in Theorem 4.2.