Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method

Laib, Hafida; Boulmerka, Aissa; Bellour, Azzeddine; Birem, Fouzia

doi:10.1016/j.cam.2022.114537

Cited by 13 publications

(4 citation statements)

References 24 publications

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“…In [32], the authors investigated a space-time Sinc-collocation method for treating the fourth-order nonlocal heat model appearing in viscoelasticity. Laib et al [33], based on the using Taylor polynomials, suggested an algorithm to construct a collocation solution for approximating the solution of 2D-VIEs. Wang et al [34], by utilizing the zeros of Chebyshev polynomial as collocation points, proposed a new collocation method to solve the second kind VIE.…”

Section: Introductionmentioning

confidence: 99%

An Improved Bernoulli Collocation Method for Solving Volterra Integral Equations

Khansari,

Khezerloo,

Nouri

et al. 2024

Preprint

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In this work, an improved collocation method based on the Bernoulli polynomials is presented to solve the Volterra integral equation (VIE) of the second kind. The main idea of the proposed method is to improve the results of the classic Bernoulli collocation method (BCM) by dividing the interval into some sub-intervals and considering the collocation points on each of them. Here, the zeros of the shifted Chebyshev polynomials (SCPs) are considered as collocation points. Then, BCM is applied step by step from the first sub-interval to the last one. By this process, a system of algebraic equations is attained for each sub-interval that could be easily solved. Convergence of the scheme is analyzed. For the purpose of demonstrating the validity, applicability, and efficiency of the suggested scheme several numerical examples are provided. Numerical results illustrate that the accuracy of the improved Bernoulli collocation method (IBCM) is more than BCM.

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Section: Introductionmentioning

confidence: 99%

An Improved Bernoulli Collocation Method for Solving Volterra Integral Equations

Khansari,

Khezerloo,

Nouri

et al. 2024

Preprint

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“…Nonlinear integral equations are mathematical equations that involve both integrals and nonlinear functions [1,2]. These equations play a crucial role in various scientific and engineering fields, describing phenomena where the relationship between variables is not linear [3][4][5]. The general form of a nonlinear integral equation can be represented as:…”

Section: Introductionmentioning

confidence: 99%

An iterative method for robust solutions to nonlinear Volterra integral equations: Stability, convergence, and practical applications

Azure

2024

Math. Syst. Sci.

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<p>The paper introduces an iterative method for solving nonlinear Volterra integral equations and analyzes its convergence, stability, and application through examples. It expresses the general nonlinear Volterra integral equation as a series and decomposes the nonlinear operator to derive a recursive formula for the proposed iterative method. The method ensures absolute and uniform convergence, with stability analysis conducted to ensure bounded errors in the presence of perturbations. Convergence analysis utilizes the Lipschitz condition, demonstrating the uniform convergence of the solution series. Illustrative examples, including power nonlinearity and trigonometric functions, validate the stability and convergence of the method. Through graphical representations, convergence analyses for specific integral equations demonstrate the method’s effectiveness and applicability in solving diverse nonlinear integral equations. Overall, the paper contributes a robust iterative method with insights into its stability and convergence properties, supported by practical examples.</p>

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“…Taylor's theorem is one of the central elementary tools in mathematical analysis, e.g., in numerical methods, topology optimization and optimal control [5,11,13]. It provides simple arithmetic formulas, in polynomial terms, to accurately compute values of various transcendental functions, such as trigonometric and exponential ones.…”

Section: Introductionmentioning

confidence: 99%

Generalized Taylor’s formula for power fractional derivatives

Zitane,

Torres

2023

Bol. Soc. Mat. Mex.

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We establish a new generalized Taylor’s formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor’s formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor’s expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.

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Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method

Cited by 13 publications

References 24 publications

An Improved Bernoulli Collocation Method for Solving Volterra Integral Equations

An Improved Bernoulli Collocation Method for Solving Volterra Integral Equations

An iterative method for robust solutions to nonlinear Volterra integral equations: Stability, convergence, and practical applications

Generalized Taylor’s formula for power fractional derivatives

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