Abstract: The Goursat partial differential equation arises in
linear and non linear partial differential equations with mixed
derivatives. This equation is a second order hyperbolic partial
differential equation which occurs in various fields of study such as
in engineering, physics, and applied mathematics. There are many
approaches that have been suggested to approximate the solution of
the Goursat partial differential equation. However, all of the
suggested methods traditionally focused on numerical differentiation
approaches including forward and central differences in deriving the
scheme. An innovation has been done in deriving the Goursat partial
differential equation scheme which involves numerical integration
techniques. In this paper we have developed a new scheme to solve
the Goursat partial differential equation based on the Adomian
decomposition (ADM) and associated with Boole-s integration rule to
approximate the integration terms. The new scheme can easily be
applied to many linear and non linear Goursat partial differential
equations and is capable to reduce the size of computational work.
The accuracy of the results reveals the advantage of this new scheme
over existing numerical method.
Abstract: A mammography image is composed of low contrast area where the breast tissues and the breast abnormalities such as microcalcification can hardly be differentiated by the medical practitioner. This paper presents the application of active contour models (Snakes) for the segmentation of microcalcification in mammography images. Comparison on the microcalcifiation areas segmented by the Balloon Snake, Gradient Vector Flow (GVF) Snake, and Distance Snake is done against the true value of the microcalcification area. The true area value is the average microcalcification area in the original mammography image traced by the expert radiologists. From fifty images tested, the result obtained shows that the accuracy of the Balloon Snake, GVF Snake, and Distance Snake in segmenting boundaries of microcalcification are 96.01%, 95.74%, and 95.70% accuracy respectively. This implies that the Balloon Snake is a better segmentation method to locate the exact boundary of a microcalcification region.