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Lung cancer is one of the most serious health problems in the world. Lung Computer-Aided Diagnosis (CAD) is a potential method to accomplish a range of quantitative tasks such as early cancer and disease detection, analysis of disease progression. Edge detection in Lung image is a fundamental step in CAD system and many algorithms have suggested for edge detection in image. In this paper a new operator has developed for edge detection based on finite difference method. Experimentation showed that the proposed operator gave good results in detection the edges in Lung images.

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Discovered experimentally by Russell and described theoretically by Korteweg and de Vries, KdV equation has been a nonlinear evolution equation describing the propagation of weakly dispersive and weakly nonlinear waves. This equation received a lot of attention from mathematical and physical communities as an integrable equation. The objectives of this paper are: first, providing a rigorous mathematical derivation of an extended KdV equations, one on the velocity, other on the surface elevation, next, solving explicitly the one on the velocity. In order to derive rigorously these equations, we will refer to the definition of consistency, and to find an explicit solution for this equation, we will use the sine-cosine method. As a result of this work, a rigorous justification of the extended Kdv equation of fifth order will be done, and an explicit solution of this equation will be derived.

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