Explicit and approximate solutions of second‐order evolution differential equations in Hilbert space

The explicit closed‐form solutions for a second‐order differential equation with a constant self‐adjoint positive definite operator coefficient A (the hyperbolic case) and for the abstract Euler–Poisson–Darboux equation in a Hilbert space are presented. On the basis of these representations, we propose approximate solutions and give error estimates. The accuracy of the approximation automatically depends on the smoothness of the initial data. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 111–131, 1999