Numerical solution of integral equations system of the second kind by block-pulse functions
06 Jul 2005-Applied Mathematics and Computation (APPLIED MATHEMATICS AND COMPUTATION)-Vol. 166, Iss: 1, pp 15-24
TL;DR: The characteristic of Block–Pulse functions is described and it is indicated that through this method a system of Fredholm integral equations can be reduced to an algebraic equation.
About: This article is published in Applied Mathematics and Computation.The article was published on 2005-07-06. It has received 99 citations till now. The article focuses on the topics: Fredholm integral equation & Fredholm theory.
Citations
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TL;DR: The Kronecker convolution product is introduced and expanded to the Riemann-Liouville fractional integral of matrices and several operational matrices for integration and differentiation are studied.
171 citations
TL;DR: A way to solve the fractional differential equations using the Riemann-Liouville fractional integral for repeated fractional integration and the generalized block pulse operational matrices of differentiation are proposed.
Abstract: The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.
152 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: By using block pulse functions and their stochastic operational matrix of integration, a stochastically Volterra integral equation can be reduced to a linear lower triangular system, which can be directly solved by forward substitution.
95 citations
TL;DR: An approach for obtaining the numerical solution of the nonlinear Volterra-Fredholm integro-differential (NVFID) equations using hybrid Legendre polynomials and Block-Pulse functions that reduces NVFID equations to a system of algebraic equations, which greatly simplifying the problem.
Abstract: This paper introduces an approach for obtaining the numerical solution of the nonlinear Volterra-Fredholm integro-differential (NVFID) equations using hybrid Legendre polynomials and Block-Pulse functions. These hybrid functions and their operational matrices are used for representing matrix form of these equations. The main characteristic of this approach is that it reduces NVFID equations to a system of algebraic equations, which greatly simplifying the problem. Numerical examples illustrate the validity and applicability of the proposed method.
84 citations
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560 citations
"Numerical solution of integral equa..." refers background in this paper
...Multiplication of two BPF There are two different cases of this multiplication (see [1,4])....
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TL;DR: In this paper, the Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems and a new set of orthogonal functions is derived from Walsh functions.
Abstract: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.
207 citations
"Numerical solution of integral equa..." refers background in this paper
...Then several researchers (Gopalsami and Deekshatulu, 1997 [8]; Chen and Tsay, 1977 [9]; Sannuti, 1977 [10]) discussed the Block–Pulse functions and their operational matrix [1,2]....
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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
"Numerical solution of integral equa..." refers background in this paper
...Multiplication of two BPF There are two different cases of this multiplication (see [1,4])....
[...]
...Then several researchers (Gopalsami and Deekshatulu, 1997 [8]; Chen and Tsay, 1977 [9]; Sannuti, 1977 [10]) discussed the Block–Pulse functions and their operational matrix [1,2]....
[...]
01 Jun 1977
TL;DR: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions that gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration.
Abstract: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions. Here, block-pulse functions are chosen as the orthogonal set. The method gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration. Design of piecewise constant controls or feedback gains for dynamic systems can be simplified following this approach.
160 citations
"Numerical solution of integral equa..." refers background in this paper
...Then several researchers (Gopalsami and Deekshatulu, 1997 [8]; Chen and Tsay, 1977 [9]; Sannuti, 1977 [10]) discussed the Block–Pulse functions and their operational matrix [1,2]....
[...]
TL;DR: This paper presents a technique for determinating time-varying feedback gains of linear systems with quadratic performance criteria by developing an operational matrix for solving state equations and solving the piecewise constant gains problem.
Abstract: This paper presents a technique for determinating time-varying feedback gains of linear systems with quadratic performance criteria. The gains are approximated by the piecewise constants which axe naturally determined by Walsh functions. After introducing Walsh functions in the beginning we develop an operational matrix for solving state equations. Then using the operational matrix we solve the piecewise constant gains problem.
147 citations