Solving Volterra integral equations of the second kind by wavelet-Galerkin scheme

Saberi‐Nadjafi, Jafar; Mehrabinezhad, Mohammad; Akbari, Hani

doi:10.1016/j.camwa.2012.03.043

Cited by 10 publications

(4 citation statements)

References 17 publications

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“…The Taylor collocation method (Wang and Wang, 2014 ), Lagrange collocation method (Wang and Wang, 2013 ; Nemati, 2015 ), and Fibonacci collocation method (Mirzaee and Hoseini, 2016 ) were effective and convenient for solving integral equations. The Sinc-collocation method (Rashidinia and Zarebnia, 2007 ) and Galerkin method (Saberi-Nadjafi et al, 2012 ) also give good performance in solving Volterra integral equation problems. However, most of these traditional methods have the following disadvantage: they provide the solution, in the form of an array, at specific preassigned mesh points in the domain, and they need an additional interpolation procedure to yield the solution for the whole domain.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

Zhang²,

Weng³

et al. 2023

Front. Comput. Neurosci.

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In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra–Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.

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Section: Introductionmentioning

confidence: 99%

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

Zhang²,

Weng³

et al. 2023

Front. Comput. Neurosci.

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“…Meanwhile, the Haar wavelet [18] is used to solve one-dimensional nonlinear equations. Likewise, the Daubechie wavelet and Galerkin method [19] are used to solve the second kind of linear Volterra equation. On the other hand, Maleknejad put forward many methods for different types of one-dimensional integral equations, and the Sinc function collocation method [20,21] is applied for solving one-dimensional linear and nonlinear Fredholm equations of the first kind.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving

Sun

Gong

2023

Fractal Fract

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The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations.

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“…The specific conditions under which a solution exists for the nonlinear Volterra integral equation are considered in [1]- [4] [7]. Many analytical and numerical methods have been proposed for solving this type of equations, such as the linearization and collocation method [10]- [14], the trapezoidal numerical integration and implicit scheme method [15], the implicit multistep collocation methods [16], the reproducing kernel method [17], the wavelet method [18] [19], the Adomian decomposition method [6] [7] [20] and the methods by using function approximation [21]- [23].…”

Section: Introductionmentioning

confidence: 99%

Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind

Chen

Duan²

2015

APM

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Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.

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Solving Volterra integral equations of the second kind by wavelet-Galerkin scheme

Cited by 10 publications

References 17 publications

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving

Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind

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