AUTHOR=Gefter Sergiy L. , Piven' Aleksey L. 

TITLE=Some class of nonlinear partial differential equations in the ring of copolynomials over a commutative ring

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 10 - 2024

YEAR=2024

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2024.1466569

DOI=10.3389/fams.2024.1466569

ISSN=2297-4687

ABSTRACT=We study the copolynomials, i.e., K-linear mappings from the ring of polynomials K[x] into the commutative ring K. With the help of the Cauchy–Stieltjes transform of a copolynomial, we introduce and examine a multiplication of copolynomials. We investigate the Cauchy problem related to the nonlinear partial differential equation ∂u∂t=aum0(∂u∂x)m1(∂2u∂x2)m2(∂3u∂x3)m3,   m0,m1,m2,m3∈ℕ0,   ∑j=03mj>0,   a∈K in the ring of copolynomials. To find a solution, we use the series of powers of the δ-function. As examples, we consider the Cauchy problem with the Euler–Hopf equation ∂u∂t+u∂u∂x=0, for a Hamilton–Jacobi type equation ∂u∂t=(∂u∂x)2, and for the Harry Dym equation ∂u∂t=u3∂3u∂x3.