Especially, it is known that the degradation of pipe grade carbon steels under fatigue loading conditions is accelerated by hydrogen diffusion steels, but the classical diffusion equation cannot address this specific issue.
Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order (alphain 0,1)).
This new technique is based on the modification of the classical diffusion rule by using a nonlinear sigmoidal function.
The time fractional sub-diffusion equation (FSDE) is a kind of linear integro-differential equation which can be obtained from the classical diffusion equation by employing fractional derivatives of order α to describe the procedure of anomalous diffusion, where (alphain 0,1)).
In the following, we shall present the classical diffusion model followed by the new proposed anisotropic weld defect detection algorithm.
In this equation, the non-local diffusion term ∫ R k ( x, y ) u ( t, y ) d y replaces the classical diffusion behavior given by u x x ( t, x ).
This model was proposed by Shin-Min Chao and Tsai [14], it incorporated a sharpening strategy in the classical diffusion model in order to enhance the anomalies effectively in defected surfaces.
They are a significant improvement of the classical diffusion models.
The case (sigma =1) corresponds to the classical diffusion, and the transport phenomenon exhibits sub-diffusion for (sigma <1) while super-diffusion is associated with (sigma >1).
The classical diffusion equation in Euclidean space with d dimensions can be expressed by frac{partial}{partial t}u x,t)=frac{1}{x^{d-1}}frac{partial }{partial x} biggl(x^{d-1} frac{partial}{partial x}u x,t) biggr).
