A Radiation of wire antennas has been of considerable concern. The equations which describe the behavior of the wire antennas are given by Pocklington and Hallen integral equations. These equations are in from of the first kind Fredholm integral equations with singular kernel. In this paper, we use the HAAR wavelet for solving Hallen integral equations. The method is based on constructing the HAAR wavelet to approximate the required solution for the wire antennas. The integral expressions that arise in the Hallen equations are converted into some algebric equations, thus greatly simplify the problem. The method yields to accurate results. As compared to the classical techniques, our method is simple and consumes less computer time. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made by the classical method of moments.      

This article proposes a new numerical technique for pricing asset-or-nothing options using the Black-Scholes partial differential equation (PDE). We first use the θ−weighted method to discretize the time domain, and then use HAAR wavelets to approximate the functions and derivatives with respect to the asset price variable. By using some vector and matrix calculations, we reduce the PDE to a system of linear equations that can be solved at each time step for different asset prices. We perform an error analysis to show the convergence of our technique. We also provide some numerical examples to compare our technique with some existing methods and to demonstrate its efficiency and accuracy.

Amid the previous three decades, the topic of image processing has gained vital name and recognition among researchers because of their frequent look in varied and widespread applications within the field of various branches of science and engineering. As an example, image processing is helpful to issues in signature recognition, digital video processing, remote sensing and finance. Image processing models are used for detecting the face. The aim of this thesis is to solve the face-detection in the first attempt using the HAAR-cascade classifier from images containing simple and complex backgrounds. It is one of the preeminent detectors in terms of reliability and speed. We introduced a new method to deal with the frontal face images by using a modified HAAR cascade algorithm. By using this algorithm, we can detect the image as well as the coordinates. The main attraction of this paper is to solve different types of images having one object, two objects, and three objects which can’ t be solved by any of the existing methods but can be solved by our proposed method.

A numerical method based on the HAAR wavelet is introduced in this study for solving the partial differential equation which arises in the pricing of European options. In the first place, and due to the change of variables, the related partial differential equation (PDE) converts into a forward time problem with a spatial domain ranging from 0 to 1. In the following, the HAAR wavelet basis is used to approximate the highest derivative order in the equation concerning the spatial variable. Then the lower derivative orders are approximated using the HAAR wavelet basis. Finally, by substituting the obtained approximations in the main PDE and doing some computations using the finite differences approach, the problem reduces to a system of linear equations that can be solved to get an approximate solution. The provided examples demonstrate the effectiveness and precision of the method.

We present here the numerical solution of damped forced oscillator prob-lem using HAAR wavelet and compare the numerical results obtained withsome well-known numerical methods such as Runge-Kutta fourth order clas-sical and Taylor Series methods. Numerical results show that the presentHAAR wavelet method gives more accurate approximations than above saidnumerical methods.

The main objective of this article is to solve stochastic delay differential equations via HAAR wavelets‎. ‎We present fundamental concepts of stochastic process‎, ‎HAAR‎, ‎block pulse functions‎, ‎and their operational matrix relevant to time-delayed HAAR‎. ‎Analytic solutions of two examples are solved for the first time to approximate two kinds of single time-delayed stochastic differential equations with additive and multiplicative noise‎. ‎This orthogonal basis function not only simplifies the problem but also speeds up the computations and lessens the computational complexity of the stochastic delay differential equations to a lower triangular system of linear algebraic equations‎. ‎The equation can be solved via forward substitution‎, ‎such as lower-upper decomposition method‎. ‎Finally‎, ‎we examine the order of convergence and error analysis of two visual samples to validate the efficiency and effectiveness of the suggested procedure.

Introduction: The stochastic calculus plays an important role in the study of stochastic integral equations and stochastic differential equations. The fractional Brownian motion has many applications in different branches of sciences such as economics, physics and biology. In many situations, the exact solution of these equations are not available or finding their exact solution is a very difficult process. Thus, finding an accurate and efficient numerical method for solving stochastic differential equations, and stochastic integral equations is important. Researchers have applied various numerical methods such as Dirichlet forms, Euler approximation, Skorohod integral, etc. In this paper, we used HAAR wavelet functions for solving fractional Wiener integrals. Moreover, the error analysis of the proposed method is investigated. Material and methods: In this scheme, first we present the properties of the HAAR wavelet functions then an efficient method based on these functions is proposed to estimate the solution of fractional Wiener integral with Hurst parameter H (1/2, 1). Results and discussion: We solve two numerical examples by using present method to demonstrate the efficiency and simplicity of the present method. For different values of, mean of error and standard deviation of error are shown in the tables. The obtained results confirm that proposed method enables us to find reasonable approximate solutions. Conclusion: The HAAR wavelet is the simplest possible wavelet, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems.