Nonlinear Partial Differential Equations in Module of Copolynomials over a Commutative Ring
Let $K$ be an arbitrary commutative integral domain with identity of characteristic 0. We study the copolynomials of $n$ variables, i.e., $K$-linear mappings from the ring of polynomials $K[x_1,\ldots,x_n]$ into $K$. We consider copolynomials as algebraic analogues of distributions. With the help of...
Збережено в:
| Дата: | 2025 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2025
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| Онлайн доступ: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1108 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
Репозитарії
Journal of Mathematical Physics, Analysis, Geometry| Резюме: | Let $K$ be an arbitrary commutative integral domain with identity of characteristic 0. We study the copolynomials of $n$ variables, i.e., $K$-linear mappings from the ring of polynomials $K[x_1,\ldots,x_n]$ into $K$. We consider copolynomials as algebraic analogues of distributions. With the help of the Cauchy-Stieltjes transform of a copolynomial, we introduce and study a multiplication of copolynomials. We prove the existence and uniqueness theorem of the Cauchy problem for some nonlinear partial differential equations in the ring of formal power series with copolynomial coefficients. We study a connection between some classical nonlinear partial differential equations and integer sequences. In particular, for the Cauchy problem for the Burgers equation, we obtain the representation of the unique solution to this problem in the form of the series in powers of $\delta$-function with integer coefficients.
Mathematical Subject Classification 2020: 35R50, 13B25, 35G20, 11Y55 |
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