Theoretical aspects and analysis of wavelets have applications in mathematical modeling, artificial neural networks, digital signal processing, and image processing and numerical methods. The term orthogonal deals with the mathematical part which covers a wide area of digital signal processing and image processing. An orthogonal wavelet generates the wavelet whose nature is orthogonal. This means an inverse or transpose wavelet transform is nothing but the adjoint of a wavelet transform. If this condition fails by missing orthogonality it may result in biorthogonal wavelets. Single scaling functions and single wavelets are generated but orthogonal wavelet filter bank. A biorthogonal wavelet associated with the wavelet transformation is invertible. There is no need that if it is invertible so it should be orthogonal. The biorthogonal wavelet allows maximum freedom in the case of designing the orthogonal wavelet. It also supports the construction of symmetric wavelet functions. In biorthogonal wavelets, as the name indicates, two scaling factors or functions are responsible for the generation of the various multi-resolutions on the basis of different wavelets. For the image and signal reconstruction purpose we need that wavelet. We get a better result in the presence of biorthogonal wavelets. In the present work, we analyzed the performance of orthogonal and biorthogonal wavelet filters for image processing. We test the image and observed that the filter coefficient and image quality for the orthogonal and biorthogonal wavelet. On the basis of performance analysis it is concluded that biorthogonal wavelets are better than orthogonal wavelets.

A partial differential equation has been using the various boundary elements techniques for getting the solution to eigenvalue problem. A number of mathematical concepts were enlightened in this paper in relation with eigenvalue problem. Initially, we studied the basic approaches such as Dirichlet distribution, Dirichlet process and the Model of mixed Dirichlet. Four different eigenvalue problems were summarized, viz. Dirichlet eigenvalue problems, Neumann eigenvalue problems, Mixed Dirichlet-Neumann eigenvalue problem and periodic eigenvalue problem. Dirichlet eigenvalue problem was analyzed briefly for three different cases of value of λ. We put the result for multinomial as its prior is Dirichlet distribution. The result of eigenvalues for the ordinary differential equation was extrapolated. The Basic mathematics was also performed for λ calculations which follow iterative method.