Integral Equations

 In this chapter, we will derive the mathematical model for a two-dimensional fluid-driven fracture propagating in a permeable rock and the Cauchy principal value 
 integral derived from linear elastic fracture mechanics will be used to model the elasticity of the rock. The presence of the Cauchy principal value integral in the 
 mathematical formulation poses a challenge in solving the mathematical problem. Thus, it is appropriate to introduce integral equations. Problems having Cauchy 
 principal value integrals have been solved using a number of methods such as linear spline method, Chebyshev method, finite difference method, Galerkin method, etc. 
 Several books on singular integral equations have been published. Worth mentioning is the book by Mushkelishvili \cite{musk}. We will begin this chapter by reviewing 
 integral equations, in particular, singular integral equations; and their method of solution.