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In this book, we make an extension to the unified method that unifies all the known methods in the literature for finding the exact solutions of scalar or vector nonlinear PDE's with constant coefficients in the nonlinear sciences. The extended unified method unable us to investigate the effects of the inhomogeneity of the diffusion, diffraction dispersion super-diffusion of the medium trough considering the coefficients space-dependent. On the other hand, some problems have been studied when these coefficients are taken as time-dependent. The main objectives of the extended unified method...