A Jacobi–spectral framework for the heat equation with Dirichlet boundary conditions

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dc.contributor.authorJuan Toribio Milane
dc.contributor.authorJosé A. Gómez Hernández
dc.contributor.authorJuan R. Holguín
dc.contributor.authorPedro N. Tifa de Jesús
dc.date.accessioned2026-03-10T18:56:47Z
dc.date.available2026-03-10T18:56:47Z
dc.date.issued2026-03-09
dc.description.abstractWe 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 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 space. The Legendre case 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.
dc.identifier.citationJuan 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.
dc.identifier.urihttps://repositoriovip.uasd.edu.do/handle/123456789/1624
dc.language.isoen
dc.titleA Jacobi–spectral framework for the heat equation with Dirichlet boundary conditions
dc.typeArticle
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