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Journal ArticleDOI

Using triangular orthogonal functions for solving Fredholm integral equations of the second kind

15 Jul 2008-Applied Mathematics and Computation (Elsevier)-Vol. 201, Iss: 1, pp 452-464
TL;DR: The present work proposes a method for solving Fredholm integral equations using a complementary pair of orthogonal triangular functions set derived from the well-known block pulse functions set, and demonstrates validity and applicability of the method.
About: This article is published in Applied Mathematics and Computation.The article was published on 2008-07-15. It has received 42 citations till now. The article focuses on the topics: Fredholm integral equation & Fredholm theory.
Citations
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Journal ArticleDOI
TL;DR: Two-dimensional orthogonal triangular functions are presented as a new set of basis functions for expanding 2D functions and used to approximate solutions of nonlinear two-dimensional integral equations by a direct method.
Abstract: Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.

58 citations

Journal ArticleDOI
TL;DR: An original approach to the solution of Fredholm equations of the second kind is proposed, which interprets the standard Von Neumann expansion of the solution as an expectation with respect to a probability distribution defined on a union of subspaces of variable dimension.

44 citations


Additional excerpts

  • ...), a.m.johansen@warwick.ac.uk (A.M. Johansen), v.b.tadic@bristol.ac.uk (V.B. Tadić). and, by iterating, one obtains f ðx0Þ ¼ gðx0Þ þ X1 n¼1 Z En Yn k¼1 Kðxk 1; xkÞ ! gðxnÞdx1:n; ð3Þ where xi:j , (xi,. ....

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Journal ArticleDOI
TL;DR: An efficient numerical method is proposed for solving nonlinear mixed type Volterra–Fredholm integral equations, using two-dimensional orthogonal triangular functions (2D-TFs) in a direct approach that has several advantages in reducing computational burden.

35 citations

Journal ArticleDOI
TL;DR: In this paper, a numerical method for solving weakly singular Fredholm integral equations of the second kind is presented, which utilizes Legendre wavelets constructed on the unit interval as a basis in the Galerkin method.
Abstract: In this paper, we present a numerical method for solving, linear and nonlinear, weakly singular Fredholm integral equations of the second kind. The method utilizes Legendre wavelets constructed on the unit interval as a basis in the Galerkin method and reduces the solution of the Fredholm integral equation to the solution of a system of algebraic equations. The features of the wavelet coefficient matrices of weakly singular kernels are studied. Finally, numerical examples are presented to show the validity and efficiency of the technique.

29 citations


Cites methods from "Using triangular orthogonal functio..."

  • ...In recent years, several simple and accurate methods based on orthogonal basic functions, including wavelets, have been used to approximate the solution of integral equations (Akyz-Daciolu, 2004; Blyth et al., 2004; Alipanah and Dehghan, 2007; Babolian et al., 2008; Hsiao, 2009; Liu, 2009)....

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Journal ArticleDOI
TL;DR: A numerical method for solving nonlinear Stochastic Ito-Volterra equations is proposed based on delta function approximations and the properties of DFs and their operational matrix of integration together with the Newton-Cotes nodes are presented.

28 citations

References
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Book
01 Jan 1992
TL;DR: This research presents a new generation of block pulse operational matrices for integrations that combine nonparametric representations of dynamic systems with state space representations ofynamic systems.
Abstract: Operations of block pulse series.- Block pulse operators.- Block pulse transforms.- Block pulse operational matrices for integrations.- Nonparametric representations of dynamic systems.- Input-output representations of dynamic systems.- State space representations of dynamic systems.- Practical aspects in using block pulse functions.

150 citations


"Using triangular orthogonal functio..." refers background or methods in this paper

  • ...The operational matrix for integration of BPFs has been derived as the following upper triangular matrix [8]:...

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  • ...As proposed by many authors, specially in [8], a set of block pulse functions (BPFs) are usually defined in the interval 1⁄20; 1Þ as /iðtÞ 1⁄4 1 i m 6 t < iþ1 m ; 0 otherwise; ð1Þ where i 1⁄4 0; 1; ....

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Journal ArticleDOI
TL;DR: It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution.
Abstract: The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.

90 citations


"Using triangular orthogonal functio..." refers background in this paper

  • ...For more details about triangular orthogonal functions, see [1]....

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  • ...The main body of Section 2 is based on [1]....

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  • ...The authors of [1] called T1ðtÞ the left-handed triangular functions (LHTF) and T2ðtÞ the right-handed triangular functions (RHTF)....

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Journal ArticleDOI
TL;DR: The main purpose of this paper is to demonstrate that using rationalized Haar wavelet for solving linear Fredholm integral equation of the second kind has a good degree of accuracy.

86 citations


"Using triangular orthogonal functio..." refers methods or result in this paper

  • ...Spline functions and Haar wavelets are also applied for solving these problems in [7,2]....

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  • ...1 t BPF method Method of [2] TF method Exact solution 0 0....

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  • ...This example is also provided in [2], and we compare the results....

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Journal ArticleDOI
TL;DR: A combination of Legendre and Block-Pulse functions on the interval [0,1], to solve the linear integral equation of the second kind, and converts the integral equation, to a system of linear equations.

77 citations

Journal ArticleDOI
TL;DR: A combination of Chebyshev and Block-Pulse functions on the interval [0,1], to solve the linear integral equation of the second kind and converts the integral equation, to a system of linear equations.

49 citations