Wavelet analysis has been developed dramatically fast and applied in many research areas. The first work presents geometric transformation models based on wavelet analysis to handle the partial differential equation. Shape distortion mainly arises in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by certain geometric transformation model, the inverse transformation can be applied to remove the distortion for the shape restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. However, it can be represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. A novel wavelet-based method is presented, which consists of three steps: 1, the partial differential equation is converted into boundary integral equation and representation by an indirect methodpotential theory; 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula; 3, the plane integral equation and representation are then solved by a method called wavelet collocation. The performance of our method is evaluated by numerical experiments. The second work discusses the protein sequence similarity comparison based on wavelet analysis. Similarity comparison of protein sequences in the area of bioinformatics and molecular biology helps the prediction and classification of protein structure and function. The protein sequences are firstly represented into the 1-dimensional feature vectors by their biochemical quantities. The proposed hybrid iii method is then applied to generate a new encoding feature, which includes discrete wavelet transform, fractal dimension calculation and sliding window. At last, through the computation between the feature vectors, we can obtain the distance matrix, by which, the phylogenic tree can be constructed. The experimental results show that the proposed model is more accurate than the existing methods, such as [J. Su, Appl. Math, 7(2013), pp. 1103-1110], [Y. Zhang, Int. Conf. (BIC-TA), 2010, pp. 1255-1258] and [Y. H. Yao, Proteins: Structure, Function and Bioinformatics, 73(2008), pp. 864-871], and it is consistent with the result generated from MEGA software and some known facts.