Metadata (abstracts and keywords) for the articles in the journal
Zakharov V.G. Matrix method of polynomial solutions to constant coefficient PDE’s // Vestnik I. Yakovlev Chuvach State Pedagogical University. Series: Mechanics of a limit state . 2024. № 3(61). p. 92-116
Author(s):
Zakharov V.G.
Index of UDK:
517.956
DOI:
10.37972/chgpu.2024.61.3.008
Title:
Matrix method of polynomial solutions to constant coefficient PDE’s
Keywords:
polynomial solution to linear constant coefficient PDE’s, exponential solution to non homogeneous PDE, null-space of matrix.
Abstracts:
In the paper, we introduce a matrix method to constructively determine polynomial (in general, multiplied by exponentials) solutions to the constant coefficient linear Partial Difference Equations (PDE’s). Note our method also is applicable if the polynomials that induce PDE’s have constant terms (similarly Helmholtz’ equation) and consequently such PDE’s cannot have pure polynomial solutions. In this case our matrix method supplies polynomial solutions multiplied by exponentials. Moreover, the method allows to find a polynomial (multiplied by an exponential) solution to PDE with polynomial (multiplied by the exponential) right-hand side. Our matrix method reduces the funding of polynomial (multiplied by an exponential) solution to PDE’s to determine the differential operator null-space of algebraic block matrix linear system (with numerical entries). Furthermore, using our matrix approach, we can investigate some linear algebra properties such as dimension and basis of the space of polynomial solution (generally, multiplied by an exponential). In particular, for a polynomial solution space we can decide the problem up to some arbitrarily large degree, In particular, we generalize problem about infinite degree of polynomial solutions of polynomial PDE to exponential case. Moreover, PDE can contain a nonzero polynomial (multiplied by an exponential) right-hand side. Some examples of polynomial solutions (multiplied by exponentials, in general) to the Laplace, Helmholtz, and Poisson equations are considered.
The contact details of authors:
Victor G. Zakharov, Cand. Phys.-Math. Sci., Scientific Researcher; e-mail: victor@icmm.ru; https://orcid.org/0000-0003-3179-6753; AuthorID: 6178
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