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SUMMARY:Representations of solutions for a first order quasilinear equatio
 n in a form of series in powers of the copolynomial δ-function
DTSTART;VALUE=DATE-TIME:20260922T143500Z
DTEND;VALUE=DATE-TIME:20260922T144000Z
DTSTAMP;VALUE=DATE-TIME:20260914T055214Z
UID:indico-contribution-516@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Roman Skurikhin (V. N. Karazin Kharkiv National Univ
 ersity)\nBy a *copolynomial* we mean here a linear functional on the space
  of polynomials $\\mathbb{R}[x]$. The space of copolynomials is denoted by
  $\\mathbb{R}[x]'$. If $T\\in\\mathbb{R}[x]'$ and $p\\in\\mathbb{R}[x]$\, 
 then the result of application of $T$ to $p$ is written as $(T\,p)$. The d
 erivative $T'$ of a copolynomial $T\\in\\mathbb{R}[x]'$ is defined in the 
 same way as in the classical theory of generalized functions: $(T'\,p)=-(T
 \,p')$\, $p\\in\\mathbb{R}[x]$. An important example of a copolynomial is 
 the $\\delta$-function\, which is defined by $(\\delta\,p)=p(0)$\, $p\\in\
 \mathbb{R}[x]$.\n\nThe *Cauchy-Stieltjes transform* of a copolynomial $T\\
 in\\mathbb{R}[x]'$ is defined as the following formal Laurent series in th
 e ring $s^{-1}\\mathbb{R}[[s^{-1}]]$: $C(T)(s)=\\sum_{k=0}^{\\infty}(T\,x^
 k)/s^{k+1}$. The mapping $C:\\mathbb{R}[x]'\\to s^{-1}\\mathbb{R}[[s^{-1}]
 ]$ is an isomorphism of these vector spaces. The multiplication of copolyn
 omials is defined through the multiplication of their Cauchy-Stieltjes tra
 nsforms. The theory of linear and nonlinear PDEs in the ring $\\mathbb{R}[
 x]'[[t]]$ was studied in [1\,2]\, respectively.\n\nLet $f(z)=\\sum_{j=1}^{
 \\ell}f_jz^j\\in\\mathbb{R}[z]$. We prove that the following Cauchy proble
 m for the first order quasilinear equation\n\n$$\n\\frac{\\partial u}{\\pa
 rtial t}+f(u)\\frac{\\partial u}{\\partial x}=0\,\n\\qquad u(0\,x)=\\delta
 (x)\,\n$$\n\nin $\\mathbb{R}[x]'[[t]]$ has a unique solution\n\n$$\nu(t\,x
 )=\\sum_{k=0}^{\\infty}\\sum_{|\\alpha|=k}\nf_1^{\\alpha_1}\\ldots f_{\\el
 l}^{\\alpha_{\\ell}}\n\\frac{\\left(\\sum_{j=1}^{\\ell}j\\alpha_j+k\\right
 )!}\n{\\alpha_1!\\ldots\\alpha_{\\ell}!\n \\left(\\sum_{j=1}^{\\ell}j\\alp
 ha_j+1\\right)!}\n\\delta^{\\sum_{j=1}^{\\ell}j\\alpha_j+k+1}(x)t^k\,\n$$\
 nwhere $\\alpha=(\\alpha\\_1\,\\ldots\,\\alpha_{\\ell})\\in\\mathbb{N}\\_0
 ^{\\ell}$ is a multi-index\, $|\\alpha|=\\sum\\_{j=1}^{\\ell}\\alpha\\_j$.
 \n\n*Acknowledgement.* Roman Skurikhin was supported by the National Resea
 rch Foundation of Ukraine\, project No. 2025.07/0369 “Qualitative method
 s of nonlinear analysis of heterogeneous structures”.\n\n[1] Gefter S. L
 .\, Piven' A. L.: Partial differential equations in module of copolynomial
 s over a commutative ring\, *J. Math. Phys. Anal. Geom.* **21** (2025)\, N
 o. 1\, 56–83.\n\n[2] Gefter S. L.\, Piven' A. L.: Nonlinear Partial Diff
 erential Equations in Module of Copolynomials over a Commutative Ring. *J.
  Math. Phys. Anal. Geom.* **21** (2025)\, No. 3\, 319–345.\n\nhttps://in
 dico.bitp.kiev.ua/event/18/contributions/516/
LOCATION:Bogolyubov Institute for Theoretical Physics Conference Hall
URL:https://indico.bitp.kiev.ua/event/18/contributions/516/
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