Bernoulli Galerkin Matrix Method and Its Convergence Analysis for Solving System of Volterra–Fredholm Integro-Differential Equations

Hesameddini, Esmail; Riahi, Mohsen

doi:10.1007/s40995-018-0584-y

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…In [9] Attary introduced an approximation to a system of FVIDEs with variable coefficients through the Shannon technique. There are also many methods to approximate the solution of the proposed system, such as Bezier control method [10], Chebyshev polynomial method [11], Modified decomposition method [14], Shifted Chebyshev polynomial method [12], Lagrange method [15]. In [1], Holmaker proved the stability in solving systems of IDEs that describe the construction of liver zones.…”

Section: Introductionmentioning

confidence: 99%

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

2023

RNA

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In this paper, we have implemented Sinc collocation method (SCM) to solve linear systems of higher order Fredholm Volterra integro-differential equations (FVIDEs) with variable coefficients. This method transforms the system FVIDEs into algebraic equations. Two examples are included to illustrate the accuracy and success of the proposed method. We also point out that, as the number of dimensions increases we get satisfactory results, as explained in the figures and tables.

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Section: Introductionmentioning

confidence: 99%

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

2023

RNA

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“…Recently, Erfanian et al [49] developed a new method for solving two-dimensional nonlinear Volterra integral equations, based on the use of rationalized Haar functions in the complex plane. Bernoulli Galerkin matrix method has been also proposed by Hesameddini and Riahi [50] for solving the system of Volterra-Fredholm integro-differential equations. A new hybrid orthonormal Bernstein and improved block-pulse functions method has been studied by Ramadan and Osheba [51] for solving mathematical physics and engineering problems.…”

Section: Introduction and State-of-artmentioning

confidence: 99%

Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

Momeni

Beni²,

Bedon

et al. 2021

Applied Sciences

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The solution of the motion equation for a structural system under prescribed loading and the prediction of the induced accelerations, velocities, and displacements is of special importance in structural engineering applications. In most cases, however, it is impossible to propose an exact analytical solution, as in the case of systems subjected to stochastic input motions or forces. This is also the case of non-linear systems, where numerical approaches shall be taken into account to handle the governing differential equations. The Legendre–Galerkin matrix (LGM) method, in this regard, is one of the basic approaches to solving systems of differential equations. As a spectral method, it estimates the system response as a set of polynomials. Using Legendre’s orthogonal basis and considering Galerkin’s method, this approach transforms the governing differential equation of a system into algebraic polynomials and then solves the acquired equations which eventually yield the problem solution. In this paper, the LGM method is used to solve the motion equations of single-degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems. The obtained outputs are compared with methods of exact solution (when available), or with the numerical step-by-step linear Newmark-β method. The presented results show that the LGM method offers outstanding accuracy.

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“…Over the past few years, Bernoulli polynomials gains an increasing importance in numerical analysis due to many reasons. The efficiency of these polynomials has been formally investigated in [8,9,21,28,38,39,[51][52][53] and can be used to treat other problems as in [6,36,37]. For more details about see [17,24,42,51] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Bernoulli polynomial and the numerical solution of high-order boundary value problems

El-Gamel¹,

Adel²,

El–Azab³

2019

Math. Nat. Sci.

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In this work we present a fast and accurate numerical approach for the higher-order boundary value problems via Bernoulli collocation method. Properties of Bernoulli polynomial along with their operational matrices are presented which is used to reduce the problems to systems of either linear or nonlinear algebraic equations. Error analysis is included. Numerical examples illustrate the pertinent characteristic of the method and its applications to a wide variety of model problems. The results are compared to other methods.

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Bernoulli Galerkin Matrix Method and Its Convergence Analysis for Solving System of Volterra–Fredholm Integro-Differential Equations

Cited by 6 publications

References 24 publications

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

Bernoulli polynomial and the numerical solution of high-order boundary value problems

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