A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations (original) (raw)
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Numerical Solution of Fredholm Integral Equations of Second Kind using Haar Wavelets
2016
Integral equations have been one of the most important tools in several areas of science and engineering. In this paper, we use Haar wavelet method for the numerical solution of one-dimensional and two-dimensional Fredholm integral equations of second kind. The basic idea of Haar wavelet collocation method is to convert the integral equation into a system of algebraic equations that involves a finite number of variables. The numerical results are compared with the exact solution to prove the accuracy of the Haar wavelet method.
In this work, we present a numerical solution of nonlinear fredholm integral equations using Leibnitz-Haar wavelet collocation method. Properties of haar wavelet and its operational matrix is utilized to convert into a system of algebraic equations, solving these equations using MATLAB to compute the required Haar coefficients. The numerical result of the proposed method is presented in comparison with the solutions given in the literature [3, 18 & 19] of the illustrative examples. Error analysis is worked out, which shows the efficiency of the method.
NUMERICAL SOLUTION OF COUPLED FREDHOLM INTEGRAL EQUATIONS OF SECOND KIND USING HAAR WAVELETS
Integral equation, one-dimensional coupled Fredholm integral equation of second kind, two-dimensional coupled Fredholm integral equation of second kind, Haar wavelets, collocation points. Integral equations provide an important tool for modeling numerous phenomena and processes. In this paper, we use Haar wavelets to solve one-dimensional and two-dimensional coupled Fredholm integral equations of second kind. This method converts the system of linear integral equations into a system of linear algebraic equations. The numerical results are compared with the exact solution to prove the accuracy of the Haar wavelet method.Fredholm integral equations are widely used in mechanics, geophysics, electricity and magnetism, kinetic theory of gases, hereditary phenomena in biology, quantum mechanics, mathematical economics, and queuing theory. Babolian and Mordad [1] used hat basis functions for solving systems of linear and nonlinear integral equations of the second kind. Khalil and Khan [2] used Legendre polynomials for solving coupled system of Fredholm integral equations. Rashidinia and Zarebnia [3] used Sinc collocation method to obtain an approximate solution of system of Fredholm integral equations and also established the exponential rate of convergence of the method. Maleknejad et. al. [4] used collocation method with Legendre polynomials for solving system of integral equations. Ibraheem [5] solved system of linear Fredholm integral equations of second kind using open Newton-Cotes formulas. Sahu and Ray [6] applied semi-orthogonal B-spline wavelet collocation method to obtain a numerical solutions for the system of Fredholm integral equations of second kind. Vahidi and Damercheli [7] solved systems of linear Fredholm integral equations of the second kind using modified Adomian decomposition method. Ebrahimi and Rashidinia [8] used spline collocation for solving system of Fredholm and Volterra integral equations. Numerical solutions for solving single two-dimensional Fredholm integral equation have been treated using different methods. Hanson and Phillips [9] presented a general procedure for numerically solving linear Fredholm integral equations of the first kind in two integration variables. The method involves collocation followed by the solution of an appropriately scaled stabilized matrix least squares problem. Carutasu [10] obtained the numerical solution of two-dimensional nonlinear Fredholm integral equations of the second kind by Galerkin and iterated Galerkin method using spline functions. Han and Wang [11] used discrete Galerkin and iterated discrete Galerkin method to study the numerical solution of two-dimensional Fredholm integral equation. Ismail [12] used mechanical quadrature method to solve two-dimensional nonlinear singular integral equation with Hilbert kernel. Saeed and
Haar Wavelet Collocation Method for Solving Linear Volterra and Fredholm Integral Equations
Zenodo (CERN European Organization for Nuclear Research), 2022
The main purpose of this paper is to obtain the numerical solution of linear Volterra and Fredholm integral equations by using Haar wavelet collocation method. Specifically, a numerical solution of the second kind of LinearVolterra and Fredholm integral equations has been discussed.This equation cannot be easily evaluated analytically. As a result,an efficient numerical technique has been applied to find the solution which is indeed an approximate solution. In this paper,The Haar wavelet collocation methodis used to transform linear Volterra and Fredholm integral equations in to a system of linear algebraic equations. The resulting systems of algebraic equations are solved by using Gaussian elimination with partial pivoting to compute the Haar coefficients. The presented method is verified by means of different problems, where theoretical results are numerically confirmed. The numerical results of six test problems, for which the exact solutionsare known,are considered to verify the accuracy and the efficiency of the proposed method. The numerical results are compared with the exact solutions and the performance of the Haar wavelet collocation methodis demonstrated by calculating the error norm and maximum absolute errors for different number collocation points. The computational cost of the proposed methods is analyzed by examples and the error analysis is done by Haar wavelet collocation method numerically. The convergence of the Haar wavelet collocation methodis ensured at higher level resolution (J).The numerical results show that the method is applicable, accurate and efficient. Most of computations are performed using MATLAB R2015asoftware.
Bernoulli Wavelet Based Numerical Method for Solving Fredholm Integral Equations of the Second Kind
In this paper, a Bernoulli wavelet based numerical method for the solution of Fredholm integral equations of the second kind is proposed. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet (BW) is first presented and the resulting Bernoulli wavelet matrices are utilized to reduce the Fredholm integral equations into algebraic equations. Solving these equations using MATLAB to obtain Bernoulli coefficients. The numerical results of the proposed method through the illustrative examples is presented in comparison with the exact and existing methods (Haar wavelet method (HWM) [13], Hermite cubic splines (HCS) [11]) of solution from the literature are shown in tables and figures, which show that the validity and applicability of the technique with higher accuracy even for the smaller values of N.
Mathematics and Computer Science, 2017
The Haar wavelet method applied to different kinds of integral equations (Fredholm integral equation, integro-differential equations and system of linear Fredholm integral equations) and boundary value problems (BVP) representation of integral equations. Three test problems whose exact solutions are known were considered to measure the performance of Haar wavelet. The calculations show that solving the problem as integral equation is more accurate than solving it as differential equation. Also the calculations show the efficiency of Haar wavelet in case of F. I. E. S and integro-differential equations comparing with other methods, especially when we increase the number of collocation points. All calculations are done by the Computer Algebra Facilities included in Mathematica 10.2.
Chebyshev Wavelet Method for Numerical Solution of Fredholm Integral Equations of the First Kind
Mathematical Problems in Engineering, 2010
A computational method for solving Fredholm integral equations of the first kind is presented. The method utilizes Chebyshev wavelets constructed on the unit interval as basis in Galerkin method and reduces solving the integral equation to solving a system of algebraic equations. The properties of Chebyshev wavelets are used to make the wavelet coefficient matrices sparse which eventually leads to the sparsity of the coefficients matrix of obtained system. Finally, numerical examples are presented to show the validity and efficiency of the technique.
Journal of Mathematics and Computer Science
The Fredholm and Volterra types of integral equations are appeared in many engineering fields. In this paper, we suggest a method for solving Fredholm and Volterra integral equations of the first kind based on the wavelet bases. The Haar, continuous Legendre, CAS, Chebyshev wavelets of the first kind (CFK) and of the second kind (CSK) are used on [0,1] and are utilized as a basis in Galerkin or collocation method to approximate the solution of the integral equations. In this case, the integral equation converts to the system of linear equations. Then, in some examples the mentioned wavelets are compared with each other.
In this paper, the continues Legendre wavelets constructed on the interval [0,1] are used in solving numerical problems involving Fredholm integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. We use quadrature formula for the calculation of inner products of any functions, which are required in the approximation for the integral equations. So, Legendre wavelet method required for our subsequent development are given and are utilized to reduce Fredholm integral equation to some algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique. Legendre wavelet based numerical solutions are compared with the spline wavelet based solutions [8] and Exact solutions. Legendre wavelet method based solutions are in good agreement with exact one. Finally, the convergence and efficiency of this method is discussed with some illustrative examples, which indicate the ability and accuracy of the Legendre wavelets based numerical method.
Haar Wavelet Method for the System of Integral Equations
Abstract and Applied Analysis, 2014
We employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.