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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.

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%.

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.

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.

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.

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.

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.

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 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, 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.