Algebrization of nonautonomous differential equations. (English) Zbl 1382.12002

Alcorta-García, María Aracelia; Frías-Armenta, Martín Eduardo; Grimaldo-Reyna, María Esther; López-González, Elifalet

J. Appl. Math. 2015, Article ID 632150, 10 p. (2015).

Summary: Given a planar system of nonautonomous ordinary differential equations, \(dw/dt = F(t, w)\), conditions are given for the existence of an associative commutative unital algebra \(\mathbb{A}\) with unit \(e\) and a function \(H: \Omega \subset \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^2\) on an open set \(\Omega\) such that \(F(t, w) = H(t e, w)\) and the maps \(H_1(\tau) = H(\tau, \xi)\) and \(H_2(\xi) = H(\tau, \xi)\) are Lorch differentiable with respect to \(\mathbb{A}\) for all \((\tau, \xi) \in \Omega\), where \(\tau\) and \(\xi\) represent variables in \(\mathbb{A}\). Under these conditions the solutions \(\xi(\tau)\) of the differential equation \(d \xi / d \tau = H(\tau, \xi)\) over \(\mathbb{A}\) define solutions \((x(t), y(t)) = \xi(t e)\) of the planar system.

MSC:

12H20	Abstract differential equations
34M15	Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain

Keywords:

algebraization; nonautonomous differential equations

References:

[1]	Blum, E. K., A theory of analytic functions in Banach algebras, Transactions of the American Mathematical Society, 78, 2, 343-370, (1955) · Zbl 0064.36401
[2]	Ketchum, P. W., Analytic functions of hypercomplex variables, Transactions of the American Mathematical Society, 30, 4, 641-641, (1928) · JFM 55.0787.02
[3]	Lorch, E. R., The theory of analytic functions in normed abelian vector rings, Transactions of the American Mathematical Society, 54, 3, 414-425, (1943) · Zbl 0060.27202
[4]	Ward, J. A., A theory of analytic functions in linear associative algebras, Duke Mathematical Journal, 7, 1, 233-248, (1940) · JFM 66.0110.04
[5]	Ward, J. A., From generalized Cauchy-Riemann equations to linear algebra, Proceedings of the American Mathematical Society, 4, 3, 456-461, (1953) · Zbl 0050.08604
[6]	López-González, E., Differential equations over algebras, Advances and Applications in Mathematical Sciences, 8, 2, 189-214, (2011) · Zbl 1251.12005
[7]	Alvarez-Parrilla, A.; Frías-Armenta, M. E.; López-González, E.; Yee-Romero, C., On solving systems of autonomous ordinary differential equations by reduction to a variable of an algebra, International Journal of Mathematics and Mathematical Sciences, 2012, (2012) · Zbl 1254.34053
[8]	Wilczynski, E. J., Review: Abraham Cohen, an introduction to the lie theory of one-parameter groups with applications to the solution of differential equations, Bulletin of the American Mathematical Society, 18, 10, 514-515, (1912)
[9]	Page, J. M., Ordinary Differential Equations with an Introduction to Lie’s Theory or the Group of One Parameter, (1897), London, UK: Macmillan Publishers, London, UK · JFM 28.0266.01
[10]	Steinhour, R. A., The truth about lie symmetries: solving differential equations with symmetry methods [Senior Independent Study Theses], (2013)
[11]	Pierce, R., Associative Algebras, (1982), New York, NY, USA: Springer, New York, NY, USA · Zbl 0497.16001
[12]	Al-Dosary, K. I. T., Inverse integrating factor for classes of planar differential systems, International Journal of Mathematical Analysis, 4, 29–32, 1433-1446, (2010) · Zbl 1223.34004
[13]	García, I. A.; Grau, M., A Survey on the inverse integrating factor, Qualitative Theory of Dynamical Systems, 9, 1-2, 115-166, (2010) · Zbl 1364.34005
[14]	Brown, W., Matrices over Commutative Rings. Matrices over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, (1992), Marcel Dekker