On Explicit Formulas of Hyperbolic Matrix Functions

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∈Mr(C)) can be calculated using numerous different techniques. In this article we derive some explicit formulas of sh(tA) and ch(tA) (t∈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.