Classification and reduction to canonical form of linear differential equations partial of the sixth-order with non-multiple characteristics

• Published in: Қарағанды университетінің хабаршысы. Математика сериясы Vol. 119; no. 3; pp. 4 - 21
• Main Authors: Abdukodirov, A.T., Tulkinboev, T.A.
• Format: Journal Article
• Language: English
• Published: Academician Ye.A. Buketov Karaganda University 30.09.2025
• Subjects: a sixth-order partial differential equation; canonical form of differential equations; classification of differential equations; elliptic differential operator; hyperbolic differential operator; non-multiple characteristics
• ISSN: 2518-7929; 2663-5011; 2663-5011
• DOI: 10.31489/2025m3/4-21

This paper studies the problems of classification and reduction to canonical form of linear partial differential equations of the sixth-order with non-multiple characteristics and constant coefficients. Considering that with the growth of the order of the equation or the increase in the number of independent variables, the problems of classification and reduction to canonical form become more complicated. The article first provides a general formula for the coefficients of the new equation obtained after the transformation of variables, and then formulates and proves three lemmas that play an important role in finding the canonical form of the equation. The classification problems are considered and the corresponding canonical types of equations are found by a new method in four cases in which the equation with partial derivatives of the sixth-order has: 1) six different real characteristics; 2) four different real roots and two complexconjugate characteristics; 3) two real roots and four different complex-conjugate characteristics; 4) six different complex-conjugate characteristics and, consequently, the corresponding theorem is proved.