Malaysian Journal of Mathematical Sciences, June 2023, Vol. 17, No. 2
On Explicit Formulas of Hyperbolic Matrix Functions
Laarichi, Y., Elkettani, Y., Gretete, D., and Barmaki, M.
Corresponding Email: yassine.Laarichi@uit.ac.ma
Received date: 19 September 2022
Accepted date: 19 March 2023
Abstract:
Hyperbolic matrix functions are essential for solving hyperbolic coupled partial differential equations. In fact the best analytic-numerical approximations for resolving these equations come from the use of hyperbolic matrix functions. The hyperbolic matrix sine and cosine $sh(A)$, $ch(A)$ ($A\in M_{r}(\mathbb{C})$) can be calculated using numerous different techniques. In this article we derive some explicit formulas of $sh(tA)$ and $ch(tA)$ ($t\in \mathbb{R}$) using the Fibonacci-H\"{o}rner and the polynomial decomposition, these decompositions are calculated using the generalized Fibonacci sequences combinatorial properties in the algebra of square matrices. Finally we introduce a third approach based on the homogeneous linear differential equations. And we provide some examples to illustrate your methods.
Keywords: matrix functions; generalized Fibonacci sequence; hyperbolic matrix functions; Fibonacci-Horner decomposition
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