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Abstract
Wavelets are functions that satisfy certain requirements and used as a basis function to represent other functions . the name wavelet comes from the requirement that they should integrate to zero ,”waving” above and below the x-axis . The diminutive connotation of wavelet suggests the function has to be well localized . other requirements are technical and needed mostly to insure quick and easy calculation of the direct and inverse wavelet transform . There are many kinds of wavelets .One can choose between smooth wavelets ,compactly supported wavelets , wavelets with simple mathematical expressions ,wavelets with simple wavelets with simple associated filters ,etc . br The primary goal of the research presented in this thesis is to introduce the theory of the wavelet analysis limiting ourselves to one particular wavelet family ,namely the compactly supported orthogonal wavelet ,studying its application in solving differential equation for booth kind’s ordinary and partial differential equation by using the wavelet –Galerkin method then discuss the improvement by using functional with the wavelet –Galerkin method . br We have tried our best to make this exposition as self-contained and accessible as possible ,and it is our sincere hope that reader will find it a help for understanding the underlying ideas and principles of wavelets as well as useful recipe for applied wavelet analysis to the differential equations.