@article{Milane2026, 
author = {Juan Toribio Milane and José A. Gómez Hernández and Juan R. Holguín and Pedro N. Tifa de Jesús},
title = {A Jacobi–spectral framework for the heat equation with Dirichlet boundary conditions},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {5776-5797},
keywords = {Jacobi transform, Jacobi spectral methods, heat equation, Dirichlet boundary conditions, Sturm–Liouville theory, spherical Bessel functions},
url = {https://www.sciopen.com/article/10.3934/math.2026238},
doi = {10.3934/math.2026238},
abstract = {We developed a Jacobi–spectral framework for the heat equation in a spherical domain under axial symmetry and Dirichlet boundary conditions. The angular part of the Laplacian was realized as a Jacobi Sturm–Liouville operator on a weighted        L          2       space, enabling the Jacobi transform to diagonalize the angular component and project the partial diferential equation (PDE) onto a sequence of decoupled radial problems. Each projected equation reduced to a Euler-type radial ordinary diferential equation (ODE) driven by the corresponding Jacobi coefficient of the source term. These modal equations were solved in terms of spherical-Bessel eigenfunctions and radial Green kernels, yielding explicit Duhamel-type formulas for the time-dependent coefficients and establishing convergence in the weighted        L          2       space. The Legendre case    (  α  ,  β  )  =  (  0  ,  0  ) recovered the classical axisymmetric model, while general Jacobi parameters provided a unified extension of this setting. A central result was the demonstration of a rigorous equivalence between the Jacobi–spectral representation and the classical separation-of-variables solution written in spherical harmonics and spherical-Bessel modes. The proposed framework clarified the angular–radial coupling in spherical geometries and connected naturally with modern Jacobi and ultraspherical spectral methods.}
}