Collocation Methods for General Caputo Two-Point Boundary Value Problems

Liang, Hui; Stynes, Martin

doi:10.1007/s10915-017-0622-5

Cited by 49 publications

(19 citation statements)

References 15 publications

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“…Then 1is the multiterm linear fractional differential equation of constant order, and z(t) has t 1−max{ i } singularity near t = 0 under some regularity conditions, where { i } denote the fractional part of i , see Liang and Stynes. 6 However, to the best of our knowledge, there are no results for the nonlinear 1with variable-order fractional derivatives. To this end, we assume that z(t) satisfies the structure and regularity results established by 6…”

Section: The Legendre Spectral Collocation Schemementioning

confidence: 91%

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

Yan

Ding

et al. 2020

Math Methods in App Sciences

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A general class of nonlinear fractional differential equations is considered. Some sufficient conditions for the existence and uniqueness of exact solution are established by using Weissinger's fixed point theorem and the Gronwall-Bellman lemma. A spectral collocation method based on the smoothing technique is presented to solve the problem numerically. Then the rigorous error estimates under the L 2 and L ∞ norms are derived. The most remarkable feature of the method is its capability to achieve spectral convergence for weakly singular solutions. Finally, numerical results are given to support the theoretical conclusions with smooth and weakly singular solutions.

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Section: The Legendre Spectral Collocation Schemementioning

confidence: 91%

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

Yan

Ding

et al. 2020

Math Methods in App Sciences

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“…For instance, Liang and Stynes studied the convergence behavior of a collocation method for general Riemann-Liouville two-point boundary value problems (Liang and Stynes 2018b). Liang and Stynes presented a piecewisepolynomial collocation method for a class of boundary value problems involving Caputo fractional-order derivatives (Liang and Stynes 2018a). Wang et al proposed a Legendre spectral collocation method for solving nonlinear fractional differential equations with Caputo derivatives (Wang et al 2018).…”

Section: Introductionmentioning

confidence: 99%

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

Yan

Han

et al. 2019

Comp. Appl. Math.

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In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach's fixed-point theorem and the Gronwall-Bellman lemma. Third, we employ the Legendre-Gauss and Jacobi-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces L 2 [0, 1] and L ∞ [0, 1]. Finally, numerical results are given to support the theoretical conclusions. Keywords Spectral collocation method • Variable fractional order • Initial value problem • Convergence analysis Mathematics Subject Classification 65L60 • 41A05 • 41A10 • 41A25 Communicated by José Tenreiro Machado.

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“…One of them is to use numerical methods based on basis functions which reflect the singular behavior of the exact solution (see, for example, te Riele 1982;Brunner 1983;Hu 1997;Gu et al 2016). The second way is based on the polynomial spline collocation on graded meshes (see Brunner and van der Houwen 1986;Brunner 1985Brunner , 2004Cao et al 2003;Liang and Stynes 2018;Ma and Huang 2013). The third way is to use some smoothing transformations to guarantee the transformed VIEs possess a smooth solution with respect to the new variable.…”

Section: Introductionmentioning

confidence: 99%

Barycentric rational collocation methods for Volterra integral equations with weakly singular kernels

Huang

Ming

2019

Comp. Appl. Math.

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In this paper, we investigate barycentric rational collocation methods for weakly singular Volterra integral equations. Since the exact solution usually has a weak singularity at the initial point, a smoothing transformation is first used to convert the original equation to a new equation whose exact solution is smooth enough. Then, by the collocation approach based on barycentric rational interpolation, a discrete numerical scheme is constructed, and a general convergence result is established. Finally, the theoretical results are illustrated by some numerical examples.

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Collocation Methods for General Caputo Two-Point Boundary Value Problems

Cited by 49 publications

References 15 publications

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

Barycentric rational collocation methods for Volterra integral equations with weakly singular kernels

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