Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets
01 Mar 2009-Journal of Computational and Applied Mathematics (Elsevier Science Publishers B. V.)-Vol. 225, Iss: 1, pp 87-95
TL;DR: A computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets is presented, which shows efficiency of the method.
About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2009-03-01 and is currently open access. It has received 154 citations till now. The article focuses on the topics: Fredholm integral equation & Fredholm theory.
Citations
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TL;DR: The Haar wavelet operational matrix is derived and used to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations.
250 citations
TL;DR: In this paper, a numerical scheme based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented.
123 citations
TL;DR: The advantage of the proposed new algorithms based on Haar wavelets is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations.
117 citations
TL;DR: An efficient direct solver for solving numerically the high-order linear Fredholm integro-differential equations (FIDEs) with piecewise intervals under initial-boundary conditions is developed.
96 citations
TL;DR: A quadrature rule based on uniform Haar wavelets and hybrid functions is proposed to find approximate values of definite integrals for double, triple and improper integrals.
Abstract: A quadrature rule based on uniform Haar wavelets and hybrid functions is proposed to find approximate values of definite integrals. The wavelet-based algorithm can be easily extended to find numerical approximations for double, triple and improper integrals. The main advantage of this method is its efficiency and simple applicability. Error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method.
86 citations
Cites background from "Numerical solution of nonlinear Fre..."
...(60) Babolian and Shahsavaran [16] have shown that the square of the error norm for wavelet approximation is given by ‖f (x)− fM(x)‖(2) = K 3 3 · 1 (2M)2 ....
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...[16] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. Comput....
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...(60) Babolian and Shahsavaran [16] have shown that the square of the error norm for wavelet approximation is given by ‖f (x)− fM(x)‖2 = K 3 3 · 1 (2M)2 ....
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References
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Book•
28 Jun 1997TL;DR: In this paper, a brief discussion of integral equations is given, and the Nystrom method is used to solve multivariable integral equations on a piecewise smooth planar boundary.
Abstract: Preface 1. A brief discussion of integral equations 2. Degenerate kernel methods 3. Projection methods 4. The Nystrom method 5. Solving multivariable integral equations 6. Iteration methods 7. Boundary integral equations on a smooth planar boundary 8. Boundary integral equations on a piecewise smooth planar boundary 9. Boundary integral equations in three dimensions Discussion of the literature Appendix Bibliography Index.
1,719 citations
Book•
01 Feb 1997TL;DR: A mathematical introduction to the theory of orthogonal wavelets and their uses in analysing functions and function spaces, both in one and in several variables, can be found in this article.
Abstract: This book presents a mathematical introduction to the theory of orthogonal wavelets and their uses in analysing functions and function spaces, both in one and in several variables. Starting with a detailed and self contained discussion of the general construction of one dimensional wavelets from multiresolution analysis, the book presents in detail the most important wavelets: spline wavelets, Meyer's wavelets and wavelets with compact support. It then moves to the corresponding multivariable theory and gives genuine multivariable examples. Wavelet decompositions in Lp spaces, Hardy spaces and Besov spaces are discussed and wavelet characterisations of those spaces are provided. Also included are some additional topics like periodic wavelets or wavelets not associated with a multiresolution analysis. This will be an invaluable book for those wishing to learn about the mathematical foundations of wavelets.
542 citations
01 Jan 1997
TL;DR: In this article, an operational matrix of integration based on Haar wavelets is established, and a procedure for applying the matrix to analyse lumped and distributed-parameters dynamic systems is formulated.
Abstract: An operational matrix of integration based on Haar wavelets is established, and a procedure for applying the matrix to analyse lumped and distributed-parameters dynamic systems is formulated. The technique can be interpreted from the incremental and multiresolution viewpoint. Crude as well as accurate solutions can be obtained by changing the parameter m; in the mean time, the main features of the solution are preserved. Several nontrivial examples are included for demonstrating the fast, flexible and convenient capabilities of the new method.
516 citations
Book•
01 Jun 1983
TL;DR: In this article, the authors proposed piecewise constant orthogonal basis functions (PCF) for linear and non-linear linear systems, and the optimal control of linear lag-free and time-lag systems.
Abstract: I Piecewise constant orthogonal basis functions.- II Operations on square integrable functions in terms of PCBF spectra.- III Analysis of lumped continuous linear systems.- IV Analysis of time delay systems.- V Solution of functional differential equations.- VI Analysis of non-linear and time-varying systems.- VII Optimal control of linear lag-free systems.- VIII Optimal control of time-lag systems.- IX Solution of partial differential equations (PDE) [W55].- X Identification of continuous lumped parameter systems.- XI Parameter identification in distributed systems.
188 citations
TL;DR: A combination of Taylor and Block-Pulse functions on the interval [0,1], that is called Hybrid functions, is used to estimate the solution of a linear Fredholm integral equation of the second kind.
124 citations