A COMPARATIVE STUDY OF LEGENDRE WAVELET AND SPLINE WAVELET BASED NUMERICAL METHODS FOR THE SOLUTION OF FREDHOLM INTEGRAL EQUATIONS (original) (raw)
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In this paper, an efficient direct method based on Legendre wavelets is introduced to approximate the solution of Fredholm integral equations of the first kind. These basic functions are orthonormal and have compact support. The properties of the Legendre wavelets are utilized to convert the integral equations into a system of linear algebraic equations. The main characteristic of the method is low cost of setting up the equations without using any projection method. Furthermore an estimation of error bound for the present method is proved. Finally, some numerical examples are provided to demonstrate the applicability and accuracy of the proposed technique. Index Terms-Fredholm integral equation of the first kind, Legendre wavelets, Direct method, Error bound.
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.
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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.
Mohamadzadeh “Numerical Solution of System of Linear Integral Equations by using Legendre Wavelets
2010
In this paper, a direct method for numerical solution of linear Fredholm integral equations system by using Legendre wavelets is presented. Another method for solving Volterra type system of linear integral equations which uses zeros of Legendre wavelets for collocation points is introduced and used to reduce this type of system of integral equations to a system of algebraic equations.
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
Linear Legendre Multi-Wavelets Methods for Solving Systems of Fredholm Integral Equations
2016
In this paper, continuous Legendre multi-wavelets are utilized as a basis in a practical direct method to approximate the solutions of the Fredholm integral equations system. To begin with we describe the characteristic of Legendre multi-wavelets and will go on to indicate that through this method a system of Fredholm integral equations can be reduced to an algebraic equation. Finally, numerical results of some examples show that the method is practical and has high accuracy.
Numerical solution of system of linear integral equations by using Legendre wavelets
In this paper, a direct method for numerical solution of linear Fredholm integral equations system by using Legendre wavelets is presented. Another method for solving Volterra type system of linear integral equations which uses zeros of Legendre wavelets for collocation points is introduced and used to reduce this type of system of integral equations to a system of algebraic equations.