The paper presents single-term Haar wavelet series (STHWS) approach to the solution of nonlinear stiff differential equations arising in nonlinear dynamics. The properties of STHWS are given. The method of implementation is discussed. Numerical solutions of some model equations are investigated for their stiffness and stability and solutions are obtained to demonstrate the suitability and applicability of the method. The results in the form of block-pulse and discrete solutions are given for typical nonlinear stiff systems. As compared with the TR BDF2 method of Shampine and Gill's method, the STHWS turns out to be more effective in its ability to solve systems ranging from mildly to highly stiff equations and is free from stability constraints.

The paper presents a novel method for the computation of eigenvalues and solutions of Sturm-Liouville eigenvalue problems (SLEPs) using truncated Haar wavelet series. This is an extension of the technique proposed by Hsiao to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh-Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.

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a b s t r a c tIn this paper, a novel single-term Haar wavelet series (STHWS) method is implemented for the solution of the Duffing equation and Painleve's transcendents (PI and PII). The results, in the form of a block pulse and a discrete solution, are presented. Unlike classical numerical schemes, the STHWS method has no restrictions on the coefficients of the Duffing equation as regards its solution. PI and PII are analysed as regards their solutions, up to nearest singularities (poles), using the STHWS. Also, an efficient computational implementation shows the remarkable features of wavelet based techniques.

Multiple image compression using wavelet based methods including Discrete Wavelet Transform (DWT) through sub band coding (SBC) and decoding are reviewed for their comparative study. True color image compression measuring parameters like compression ratio (CR), peak to signal noise ratio (PSNR), mean square error (MSE), bits per pixel (BPP) are computed using MATLAB code for each algorithm employed. Gray scale image like Magnetic Resonance Imaging (MRI) is chosen for wavelet transform to achieve encoding and decoding using multiple wavelet families and resolutions to examine their relative merits and demerits. Our main objective is to establish advantages of multiple compression techniques (compressions using multiresolution) helpful in transmitting bulk of compressed medical images via different gadgets facilitating early detection and diagnosis followed by treatments or referrals to specialists residing in different parts of the world. Contemporary compression techniques based on wavelet transform can serve as revolutionary idea in medical field for the overall benefit of humanity.