Generalization of the d'Alembert formula and the Fourier method

We prove the well-posedness of higher order abstract Cauchy problems and give a representation of the solution in its closed form. When it is not possible we give an expansion of the solution to the series. Many well-known formulas of mathematical physics can be obtained as particular cases of more general formulas. In the present paper we also show this for an example of the d'Alembert formula. Further we prove that the Fourier method can be applied to more general problems than those already known.