Non-homogeneous wave equation on a cone. (English) Zbl 1484.35141
Summary: The wave equation \((\partial_{tt} - c^2 \Delta_x)u(x,t)=e^{-t}f(x,t)\) in the cone \(\{(x,t): \|x\| \leq t, x \in \mathbb{R}^d, t \in \mathbb{R}_+\}\) is shown to have a unique solution if \(u\) and its partial derivatives in \(x\) are in \(L^2(e^{-t})\) on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.
MSC:
| 35C10 | Series solutions to PDEs |
| 33C50 | Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable |
| 35L05 | Wave equation |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
Software:
DLMFReferences:
| [1] | Xu, Y., Fourier series in orthogonal polynomials on a cone of revolution, J Fourier Anal Appl, 26 (2020) · Zbl 1530.42051 · doi:10.1007/s00041-020-09741-x |
| [2] | Xu, Y., Sobolev orthogonal polynomials defined via gradient on the unit ball, J Approx Theory, 152, 52-65 (2008) · Zbl 1197.42016 · doi:10.1016/j.jat.2007.11.001 |
| [3] | Atkinson, K.; Chien, D.; Hansen, O., A spectral method for elliptic equations: the Dirichlet problem, Adv Comput Math, 33, 169-189 (2010) · Zbl 1196.65177 · doi:10.1007/s10444-009-9125-8 |
| [4] | Atkinson, K.; Chien, D.; Hansen, O., A spectral method for elliptic equations: the Neumann problem, Adv Comput Math, 34, 295-317 (2011) · Zbl 1211.65158 · doi:10.1007/s10444-010-9154-3 |
| [5] | Boyd, JP; Yu, F., Comparing seven spectral methods for interpolation and for solving the poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions, J Comput Phys, 230, 1408-1438 (2011) · Zbl 1210.65192 · doi:10.1016/j.jcp.2010.11.011 |
| [6] | Li, H.; Xu, Y., Spectral approximation on the unit ball, SIAM J Numer Anal, 52, 2647-2675 (2014) · Zbl 1315.41002 · doi:10.1137/130940591 |
| [7] | Vasil, GM; Burns, KJ; Lecoanet, D., Tensor calculus in polar coordinates using Jacobi polynomials, J Comput Phys, 325, 53-73 (2016) · Zbl 1380.65392 · doi:10.1016/j.jcp.2016.08.013 |
| [8] | Bérenger, J-P., A perfectly matched layer for the absorption of electromagnetic waves, J Comput Phys, 114, 185-200 (1994) · Zbl 0814.65129 · doi:10.1006/jcph.1994.1159 |
| [9] | Zenginolu, A., Hyperboloidal layers for hyperbolic equations on unbounded geometries, J Comput Phys, 230, 2286-2302 (2011) · Zbl 1210.65178 · doi:10.1016/j.jcp.2010.12.016 |
| [10] | Givoli, D., High-order local non-reflecting boundary conditions: a review, Wave Motion, 39, 319-326 (2004) · Zbl 1163.74356 · doi:10.1016/j.wavemoti.2003.12.004 |
| [11] | Hagstrom, T.; Lau, S., Radiation boundary conditions for Maxwell’s equations: a review of accurate time-domain formulations, J Comput Math, 25, 305-336 (2007) |
| [12] | Szeftel, J., A nonlinear approach to absorbing boundary conditions for the semilinear wave equation, Math Comput, 75, 565-594 (2006) · Zbl 1092.35065 · doi:10.1090/S0025-5718-06-01820-5 |
| [13] | Slevinsky, RM., On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev-Jacobi transform, IMA J Numer Anal, 38, 102-124 (2018) · Zbl 1477.65275 · doi:10.1093/imanum/drw070 |
| [14] | Szegö, G., Orthogonal polynomials (1975), Amer. Math. Soc. · JFM 61.0386.03 |
| [15] | Townsend, A.; Webb, M.; Olver, S., Fast polynomial transforms based on Toeplitz and Hankel matrices, Maths Comp, 87, 1913-1934 (2018) · Zbl 1478.65147 · doi:10.1090/mcom/3277 |
| [16] | Deconinck, B.; Trogdon, T.; Vasan, V., The method of Fokas for solving linear partial differential equations, SIAM Rev, 56, 159-186 (2014) · Zbl 1295.35002 · doi:10.1137/110821871 |
| [17] | Fokas, AS, Pelloni, B, editors. Unified transform for boundary value problems: applications and advances. Society for Industrial and Applied Mathematics; 2014. · Zbl 1311.65001 |
| [18] | Deconinck, B.; Guo, Q.; Shlizerman, E., Fokas’s unified transform method for linear systems, Quart Appl Math, 76, 463-488 (2018) · Zbl 1407.35004 · doi:10.1090/qam/1484 |
| [19] | Dunkl, CF, Xu, Y.Orthogonal polynomials of several variables. Cambridge: Cambridge University Press; 2014. (Encyclopedia of Mathematics and its Applications, Vol. 155). · Zbl 1317.33001 |
| [20] | Olver, FWJ, Olde Daalhuis, AB, Lozier, DW, et al., editors. NIST digital library of mathematical functions. Available from: http://dlmf.nist.gov/, Release 1.0.22 of 2019-03-15. |
| [21] | Slevinsky, RM.Conquering the pre-computation in two-dimensional harmonic polynomial transforms, arXiv:1711.07866. |
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