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The Haar wavelet collocation approach (HWCM) is an impressive numerical method for solving linear initial value problems when compared to the existing numerical methods (Adomian decomposition method (ADM) & Runge–Kutta method (RK4)). The objective of this study is to use the Haar-wavelet technique, Adomian decomposition technique (ADM) and Runge–Kutta (RK4) method to achieve the numerical solution of second-order ordinary differential equations. The proposed methods are applied to three different problems and the numerical results show that the HWCM has better agreement with analytic solutions than the other numerical methods.

This research work is related to establish a powerful algorithm for the computation of numerical solution to nonlinear variable order integro-differential equations (VO-IDEs). The adopted procedure is based on the Haar Wavelet Method (HWM) to compute the required numerical solution to the proposed problem. Further, in the considered problem, a proportional-type delay term is involved, which is also known as the pantograph equation. For a physical problem to investigate the computational purposes, we need to first ensure its existence. For this purpose, we utilize classical fixed results given by Banach and Schauder to establish the sufficient conditions for existence of at least one approximate solution to the proposed problem. Two pertinent examples are given, where the error analysis is also recorded.

The main goal of this paper is to present a novel numerical scheme based on the Fibonacci wavelets for solving the brain tumor growth model governed by the Burgess equation. At the first instance, the Fibonacci-wavelet-based operational matrices of integration are obtained by following the well-known Chen–Hsiao technique. These matrices play a vital role in converting the said model into an algebraic system, which could be handled with any standard numerical method. To access the effect of medical treatment over the brain tumor growth, we have investigated both the linear and nonlinear cases of Burgess equation. The nonlinearity arising in the Burgess equation is handled by invoking the quasilinearization technique. In order to compare the efficiency of the Fibonacci-wavelet-based numerical technique, we formulated an analogous numerical scheme based on the Haar wavelets. Subsequently, both the methods are testified on several test problems and it is demonstrated that the Fibonacci wavelet method yields a much more stable solution and a better approximation than the Haar wavelet method.

A numerical method for the solution of nonlinear variable-order (VO) fractional differential equations (FDEs) is proposed in this paper. To determine the numerical solution of nonlinear VO FDEs, we used the Haar wavelet collocation method (HWCM) with a combination of Caputo fractional derivatives. For checking the efficiency of the HWCM, some examples are given. The maximum absolute error and mean square root errors of each test problem are computed for a different number of collocation points (CPs) to check the validity and applicability of the presented technique. The comparison of the exact and approximate solution is shown in figures for various numbers of CPs.

This paper is related to some qualitative results about the existence and uniqueness of a solution to a third-order problem by using a fixed point approach. Haar technique is applied for numerical solution of a third-order linear integro-differential equation (IDE) with initial conditions. In IDE, the third-order derivative is computed by Haar functions, and the integration is used to get the expression of second- and first-order derivatives, as well as an approximate solution. Some examples from the literature are used to verify the validity of the proposed method. Error analysis is performed. Also, comparison between the exact and numerical solutions at different collocation points (CPs) is derived. The convergence rate is recorded taking different numbers of CPs, which is approximately equal to 2.

Wavelet thresholding is an effective method for noise reduction of a wide class of naturally occurring signals. However, bias near to a discontinuity and Gibbs phenomenon are a drawback in wavelet thresholding. The extent to which this is a problem is investigated. The Haar wavelet basis is good at approximating discontinuities, but is bad at approximating other signal artefacts. A method of detecting jumps in a signal is developed that uses non-decimated Haar wavelet coefficients. This is designed to be used in conjunction with most existing thresholding methods.

A detailed simulation study is carried out and results show that when discontinuities are present a substantial reduction in bias can be obtained, leading to a corresponding reduction in mean square error.

This paper describes a wavelet-based preconditioning technique for conjugate gradient method for linear systems derived from the Poisson equation. The linear systems solved with a conventional iterative matrix solver resulted in a marked increase in computing time with respect to an increase in grid points. Use of our wavelet-based technique leads to a matrix with a bounded condition number so that computing time is reduced significantly. In this study, one of the simplest wavelets, the Haar wavelet, is used for the purpose of developing a simple but efficient preconditioning algorithm. Simple wavelets having low data communication property such as the Haar wavelet are expected to be suitable for the purpose of improving computing performance. In this study, we also pay attention to the basic characteristics of the Haar-wavelet-based preconditioning method for a Poisson equation solver.

Estimations and optimal experimental designs for two-dimensional Haar-wavelet regression models are discussed. It is shown that the eigenvalues of the covariance matrix of the best linear unbiased estimator of the unknown parameters in a two-dimensional linear Haar-wavelet model can be represented in closed form. Some common discrete optimal designs for the model are constructed analytically from the eigenvalues. Some equivalences among these optimal designs are also given, and an example is demonstrated.

In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ₊. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared.

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products $U{P}_{\lambda }$ and $U{P}_G$ that are similar to the Heisenberg uncertainty product. $U{P}_{\lambda }$ and $U{P}_G$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $U{P}_{\lambda }$ and $U{P}_G$ in terms of Walsh and Haar basis are given.

Computing solutions of singular differential equations has always been a challenge as near the point of singularity it is extremely difficult to capture the solution. In this research paper, Haar wavelet coupled with quasilinearization approach (HWQA) is proposed for computing numerical solution of nonlinear SBVPs popularly also referred as Lane–Emden equations. This technique is the combination of quasilinearization and Haar wavelet collocation method. To show the accuracy of the HWQA, several examples are presented. Convergence of the proposed method is also established in this paper, which shows that proposed method converges very fast.

In this study, we develop a generalized wavelet-based collocation method to solve the fractional Pennes bioheat transfer model during hyperthermia treatment. Unlike the existing operational matrix methods based on orthogonal functions, we formulate the Haar wavelet operational matrices of general order integration without using the block pulse functions. Consequently, the governing problem is transformed into an equivalent system of algebraic equations, which can be tackled with any classical method. Some prime features of the proposed method include no requirement of the inverse of the Haar matrices, no need to convert the boundary value problem into the initial-value problem, which in turn eliminates the possibility of unstable solutions. The proposed technique is testified for different values of fractional parameter $\alpha$ and is observed that as the fractional parameter $\alpha$ increases, the tissue temperature at the target region also increases appreciably. Moreover, the obtained results also indicate that the overall time taken to attain the hyperthermia temperature for the fractional model is comparatively less than the classical bioheat model.

In this paper, two numerical methods are being considered for simulations of 1D elliptic type single and double interface models. The first proposed method is based on Haar wavelet collocation while the second method is based on meshless collocation which is realized on radial basis functions. Numerical experiments are carried out to check performance of both the methods. The accuracy of the methods is assessed in terms of $L_{\infty }$ error norm. Comparison of numerical results is shown to establish validity and superiority of the methods.

In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

In this study, Haar wavelet method is implemented for solving the nonlinear age-structured population model which is the nonclassic type of partial differential equation associated with boundary integral equation. This paper develops the flexibility of Haar wavelet method for reduction of the partial differential equation with nonlocal boundary conditions to an algebraic system. In fact, the simple structure of piecewise orthogonal Haar basis functions which leads to sparse matrices causes the convergence and computational efficiency. Some illustrative results show the reliability and accuracy of the presented method.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

In this work, an efficient numerical method is proposed for solving generalized Burger’s type equations. The generalized Burger’s type equations are first converted into a nonlinear ordinary differential equation by choosing some suitable wave variable transformation. Linearize such nonlinear differential equations by using quasilinearization technique. For solving algebraic system of linear equations Haar wavelet-based collocation method is used. A distinct feature of the proposed method is their simple applicability in a variety of two- and three- dimensional nonlinear partial differential equations. Numerical experiments are performed to illustrate the accuracy and efficiency of the proposed method.

Biomedical image compression plays an important role in the medical field. Mammograms are medical images used in the early detection of breast cancer. Mammogram image compression is a challenging task because these images contain information that occupies huge size for storage. The aim of image compression is to reduce the image size and the time taken for recovering the original image without any loss. In this paper, two different techniques of mammogram compression are introduced. The proposed algorithm includes two main steps. First, a preprocessing step is applied to enhance the image, and then a compression algorithm is applied to the enhanced image. The algorithm is tested using 322 mammogram images from the online MIAS database. Three parameters are used to evaluate the performance of the compression techniques; compression ratio (CR), Peak Signal to Noise Ratio (PSNR) and processing time.

According to the results, Haar wavelet-based compression for enhanced images is better in terms of CR of 26.25% and PSNR of 47.27dB.

The paper presents an efficient ear biometrics system for human recognition based on discrete Haar wavelet transform. In the proposed approach the ear is detected from a raw image using template matching technique. Haar wavelet transform is used to decompose the detected image and compute coefficient matrices of the wavelet which are clustered in its feature template. Decision is made by matching one test image with 'n' trained images using Hamming distance approach. It has been implemented and tested on two image databases pertaining to 600 individuals from IITK and 350 individuals from Saugor University, India. Accurcy of the system is more than 96%.