G. Exadaktylos, "A study of the transient fluid flow around a semi-infinite crack," Int. J. Solids Structur., vol. 49, no. 23-24, pp. 3323-3334, Nov. 2012. doi:10.1016/j.ijsolstr.2012.07.012
https://doi.org/10.1016/j.ijsolstr.2012.07.012

Applying the implicit finite difference approximation of the time derivative term, the diffusion equation governing fluid-flow around a crack in a fluid-infiltrated undeformable porous medium is transformed into a non-homogeneous modified Helmholtz’s equation. Then, Vekua’s theory regarding the solution of linear, second order, elliptical partial differential equations is employed for its solution and the corresponding Riemann function is found. Subsequently, the general solution of the Dirichlet initial-boundary value problem for a prescribed arbitrary distribution of pressure acting along a semi-infinite crack is found in the form of a Cauchy singular integral equation of the second kind. A numerical Gauss–Chebyshev quadrature scheme is proposed to solve this singular integral equation that is first applied to the steady-state problem and then to the transient problem. It is shown that the density of the Cauchy integral of the transient problem View the MathML source bears a simple similarity relationship with the steady-state problem View the MathML source of the form View the MathML source for View the MathML source, wherein View the MathML source, with D denoting the diffusivity coefficient and t the time. This solution is the first step towards the solution of transient fluid flow around multiple cracks and then of the coupled problem of a crack or cracks in deformable porous media and for the study of fluid-driven cracks in poroelastic media.