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 $ (\alpha, \beta) = (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.
Citation: Juan Toribio Milane, José A. Gómez Hernández, Juan R. Holguín, Pedro N. Tifa de Jesús. A Jacobi–spectral framework for the heat equation with Dirichlet boundary conditions[J]. AIMS Mathematics, 2026, 11(3): 5776-5797. doi: 10.3934/math.2026238
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 $ (\alpha, \beta) = (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.
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Juan Toribio Milane, José A. Gómez Hernández, Juan R. Holguín, Pedro N. Tifa de Jesús. A Jacobi–spectral framework for the heat equation with Dirichlet boundary conditions[J]. AIMS Mathematics, 2026, 11(3): 5776-5797. doi: 10.3934/math.2026238
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