Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets
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TLDR
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.read more
Citations
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Journal ArticleDOI
A Numerical Method for Fractional Pantograph Delay Integro-Differential Equations on Haar Wavelet
TL;DR: In this paper, the authors studied the solution of fractional pantograph delay integro-differential equations based on Haar wavelet collocation (HWC) technique and derived the necessary conditions for the existence and uniqueness of at most one solution of the considered problem.
Numerical Analysis of Nonlinear Wave Propagation
Abstract: Numerical analysis of nonlinear wave propagation Nonlinear partial differential equations (PDEs) arise in many areas of mathematical physics in order to model various physical phenomena. However, in the context of the thesis, nonlinear PDEs that describe wave propagation have been considered. For example, the Korteweg–de Vries and the Kadomtsev–Petviashvili equations can be used in order to model the propagation of long waves of small amplitude in shallow water, the Burgers and the 2D Burgers equations to model shock waves in fluids and traffic flow, the modified Korteweg–de Vries equation to model wave propagation in 1D anharmonic lattices, and the sine–Gordon equation to model one-dimensional crystal dislocations. Within the thesis, numerous wave propagation problems were numerically solved using the Pseudospectral method (PsM), the Haar wavelet method (HWM) as well as the higher order Haar wavelet method (HOHWM). 2D spectral analysis was used in order to gain extra insights into the behaviour of two dimensional problems. It gave the opportunity to find complex phenomena in the solution that would not have been obvious without its use. The HOHWM was used in order to gain improved accuracy compared to the HWM. The recently developed HOHWM was shown to provide more accurate results for all model equations. A nonuniform grid approach to the HOHWM was developed. While a nonuniform grid HWM had previously found use, the HOHWM had not been used alongside a nonuniform grid before. The nonuniform grid approach was shown to give significant improvements to accuracy when abrupt changes within the solution were expected and their location was known before computation. While adaptive grids have been used for a long time, they have not seen wide use in combination with the HWM. Furthermore, the HOHWM has never been used alongside an adaptive grid approach. The novel adaptive HOHWM was developed and shown to be a good candidate for solving nonlinear PDEs with abrupt changes in the solution. The method allowed obtaining results of a better level of accuracy than the uniform or static nonuniform grid HOHWM. This significantly decreases the computational complexity needed to obtain results of a certain accuracy. Thus, the adaptive HOHWM allows one to either obtain a) numerical results of the same accuracy faster while using fewer resources or b) numerical results of greater accuracy using the same amount of resources.
Journal ArticleDOI
A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind
S. Saha Ray,Sudeshna Behera +1 more
TL;DR: In this article, a wavelet technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind, which can give high-accurate solutions and good convergence rate.
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An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations
TL;DR: A truncated Legendre Wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations.
Journal ArticleDOI
A New Wavelet Method for Solving a Class of Nonlinear Volterra-Fredholm Integral Equations
TL;DR: In this paper, a Coiflet-type wavelet Galerkin method was proposed for numerically solving the Volterra-Fredholm integral equations, where the nonsingular property of the connection coefficients significantly reduces the computational complexity and achieves high precision.
References
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Book
The Numerical Solution of Integral Equations of the Second Kind
TL;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.
Book
A mathematical introduction to wavelets
TL;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.
Journal ArticleDOI
Haar wavelet method for solving lumped and distributed-parameter systems
C.F. Chen,C.H. Hsiao +1 more
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.
Book
Piecewise Constant Orthogonal Functions and Their Application to Systems and Control
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.
Journal ArticleDOI
Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions
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.
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