"Wavelet methods in partial differential equations on spheres"Iglewska-Nowak, IlonaIn this talk I would like to present a method of solving partial differential equations on the sphere using methods based on the continuous wavelet transform derived from approximate identities. It is to be stressed that the solution is analytical. In the last thirty years, many authors developed wavelet methods for solving differential equations. Already in the 1990s numerical solutions of ODEs and PDEs on the Euclidean space were found. The methods were further developed in the present century. One paper is known to me that presents a numerical solution of PDEs on the sphere. However, to my best knowledge, wavelet analysis methods were not used to finding analytical solutions to PDEs. Theories of continuous spherical wavelets have been developed in the last decades, simultaneously to theories of wavelets over Euclidean space. I was able to show that there exist only two essentially different continuous wavelet transforms for spherical signals, namely that based on group theory (pioneering works of Antoine and Vandergheynst) and that derived from approximate identities (Freeden, Windheuser, Schreiner, Bernstein). The latter one can be efficiently applied to solving the Poisson equation and the Helmholtz on the sphere. In the talk, I will present an explicit solution of the Poisson equation, as well as an algorithm for solving the Helmholtz equation. |
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