This chapter considers two main analyses, namely fracture mechanics analysis using dual integral equations, and dynamic fracture analysis using wave equations. Many crack problems can be formulated as mixed boundary value problems and this in turn can be expressed in dual integral equations. The mixed boundary value problem can be formulated as dual integral equations. The chapter discusses the mixed boundary value problem is not an easy problem and a standard technique is the use of dual integral equations. For the case of a flat punch contact, the integral can be evaluated analytically. A special form of the solution is assumed such that the dual integral equations can be reduced to a single Abel integral equation. The solutions of hyperbolic-type equations are expressed in terms of arbitrary functions of characteristics, one characteristic is a left-going wave and the other is a right-going wave.