On solutions of PDEs by using algebras
The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form Auxx+Bux +Cu = 0,...
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| Formato: | Artículo |
| Idioma: | English |
| Publicado: |
2022
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| Subjects: | |
| Acceso en liña: | https://doi.org/10.1002/mma.8073 https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.8073 |
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| Summary: | The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form Auxx+Bux +Cu = 0, and for each affine planar vector field , we give an algebra A with unit e=e1, with respect to which the components of all functions of the form L◦ are all the solutions for this PDE, where L is differentiable in the sense of Lorch with respect to A. Solutions are also constructed for the following equations: Auxx+Bux +Cu +Dux+Eu +Fu =
0, 3rd-order PDEs, and 4th-order PDEs; among these are the bi-harmonic, the bi-wave, and the bi-telegraph equations. |
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