Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

Jafari, Hossein; Nemati, Somayeh; Ganji, R.M.

doi:10.1186/s13662-021-03588-2

Cited by 10 publications

(7 citation statements)

References 39 publications

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“…The above literature showed that the exact solutions of FDEs are usually very hard to acquire or contain some special functions. Therefore, more and more researchers turn to numerical methods, for example finite-difference methods [10][11][12] and spectral methods [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Spectral methods are well-known high-accuracy methods [27][28][29][30][31][32][33][34][35], normally carried out by a Galerkin, Tau, or collocation approach in practical problems.…”

Section: Introductionmentioning

confidence: 99%

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Zhang

Wang

et al. 2023

Fractal Fract

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This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods.

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Section: Introductionmentioning

confidence: 99%

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Zhang

Wang

et al. 2023

Fractal Fract

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“…For example, the authors of [31] used shifted Chebyshev polynomials of the fifth kind in conjunction with the collocation method to obtain approximate solutions for time-fractional partial integro-differential equations. In [32], the authors proposed a numerical scheme based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations. The authors of [33] used a collocation method based on shifted fifth-kind Chebyshev polynomials to solve a nonlinear variable-order fractional reaction-diffusion equation with a Mittag-Leffler kernel.…”

Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.

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“…For example, the authors in [21] employed the Boubaker wavelets together with the operation matrix of derivative to solve singular initial value problem. The collocation method is presented in [22] based on the second kind Chebyshev wavelets for solving calculus of variation problems. The use of the operational matrices of derivatives and integrals has been highlighted in the field of numerical analysis [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Operational Matrix of New Shifted Wavelet Functions for Solving Optimal Control Problem

Abass,

Shihab

2023

Preprint

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The present work concerns with presenting an explicit formula for new shifted wavelet (NSW) functions. Wavelet functions have many applications and advantages in both applied and theoretical fields. They are formulated with different orthogonal polynomials to construct new techniques for treating some problems in sciences, and engineering. A new important differentiation property of NSW in terms of NSW themselves is obtained and proved in this paper. Then it is utilized together with the state parameterization technique to find solution of optimal control problem (OCP) approximately. The suggested method converts the OCP into a quadratic programming problem, which can be easily determined on computer. As a result, the approximate solution closes with the exact solution even with a small number of NSW utilized in estimation. The error bound estimation for the proposed method is also discussed. Some test numerical examples are solved to demonstrate the applicability of the suggested method. For comparison, the exact known solutions against the obtained approximate results are listed in Tables.

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Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

Cited by 10 publications

References 39 publications

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Novel spectral schemes to fractional problems with nonsmooth solutions

Operational Matrix of New Shifted Wavelet Functions for Solving Optimal Control Problem

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