Numerical solution of nonlinear integral equations using alternative Legendre polynomials

Bazm, Sohrab; Hosseini, Alireza

doi:10.1007/s12190-016-1060-5

Cited by 22 publications

(7 citation statements)

References 34 publications

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“…But so far, the numerical solution of the time‐domain displacement of the nonlinear variable fractional viscoelastic arch has not been studied. Compared to other polynomial algorithms, shifted Legendre polynomials can accurately and efficiently solve nonlinear integral differential equations 33‐36 . Therefore, the shifted Legendre polynomials are proposed to perform the dynamic analysis of viscoelastic arch directly in the time domain.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

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Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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“…Here, we briefly give some notations and preliminaries of Chelyshkov wavelets and fractional order Chelyshkov wavelets. Chelyshkov polynomials, P N, N (t) can be shown through the following power relation [2]:…”

Section: Chelyshkov Wavelets and Fractional Order Chelyshkov Waveletsmentioning

confidence: 99%

A Composite Collocation Method Based on the Fractional Chelyshkov Wavelets for Distributed-Order Fractional Mobile-Immobile Advection-Dispersion Equation

Marasi

Derakhshan

2022

Mathematical Modelling and Analysis

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In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.

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“…Indeed, algorithms depending upon the orthogonal basis functions converts the nonlinear problems under study to a set of linear or nonlinear algebraic equations and the obtained system could be solved to approximate the given problem. Previously, various kinds of polynomials including Bernstein, Chebyshev, Chelyshkov, Gegenbauer and shifted Gegenbauer, Jacobi, Laguerre, Laurent, Legendre and a few others had been utilized by researchers to examine different complex nature physical problems [22][23][24][25][26][27][28][29][30][31]. Lately, alternative kinds of orthogonal polynomials have been familiarized and adopted to evaluate the dynamics of the several types of problems governed by differential/integral equations.…”

Section: Introductionmentioning

confidence: 99%

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

et al. 2020

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The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

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Numerical solution of nonlinear integral equations using alternative Legendre polynomials

Cited by 22 publications

References 34 publications

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

A Composite Collocation Method Based on the Fractional Chelyshkov Wavelets for Distributed-Order Fractional Mobile-Immobile Advection-Dispersion Equation

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

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