Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

Yang, Yin; Chen, Yanping; Huang, Yunqing

doi:10.1016/s0252-9602(14)60039-4

Cited by 61 publications

(28 citation statements)

References 22 publications

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“…According to the results in [6,23], we have (4.19) where C is related to the continuous solution and their norms. Now, summing from n = 0 to n = NT on both sides of (4.19),…”

Section: Theorem 1 Suppose That the Initial Boundary Value Problem (mentioning

confidence: 99%

A predictor–corrector approach for pricing American options under the finite moment log-stable model

Chen

Zhu

2015

Applied Numerical Mathematics

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“…According to the results in [6,23], we have (4.19) where C is related to the continuous solution and their norms. Now, summing from n = 0 to n = NT on both sides of (4.19),…”

Section: Theorem 1 Suppose That the Initial Boundary Value Problem (mentioning

confidence: 99%

A predictor–corrector approach for pricing American options under the finite moment log-stable model

Chen

Zhu

2015

Applied Numerical Mathematics

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“…Fractional integro-differential equations with a weakly singular kernel are used to model a lot of different physical problems, such as heat conduction problem 1 and elasticity and fracture mechanics. 2 There are several numerical methods for the fractional integro-differential equations such as hybrid collocation method, 3 the Jacobi spectral-collocation method, 4 operational Jacobi Tau method, 5 and discontinuous Galerkin method. 6 However, the fractional integro-differential equations with a weakly singular kernel are solved by only a few methods such as Legendre wavelets method 7 and second Chebyshev wavelets methods.…”

Section: Introductionmentioning

confidence: 99%

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

Chen

Yang

2018

Math Methods in App Sciences

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In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of the reproducing kernel is obtained. Using this fractional reproducing kernel space, a class of fractional integro-differential equations with a weakly singular kernel is solved. The error estimation is given. The final numerical experiments demonstrate the correctness of the theory and the effectiveness of the method.KEYWORDS fractional integro-differential equation, fractional reproducing kernel space, weakly singular kernel 1 0 k(x, )u( )d + g(x), m − 1 < ≤ m, u (k) (0) = 0, k = 0, 1, · · · , m − 1,Math Meth Appl Sci. 2018;41:3841-3855.wileyonlinelibrary.com/journal/mma

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“…However, the aforementioned approaches are associated with the solution of a Hamilton-Jacobi-type equation, which may considerably slow down the speed of optimization convergence. New strategies have been implemented in the topology optimization method to solve the Hamilton-Jacobi-type equation [24][25][26][27], aiming at improving the computational efficiency and enhancing the numerical stability [28][29][30]. Recently, Guos group made great progress in parametric level set topology optimization methods, which significantly reduced the number of the design variables and in turn tremendously increase the computational efficiency [31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Topology Description Function Method Based on Modified Sigmoid Function

Yang

Liu

Yang

et al. 2018

Mathematical Problems in Engineering

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Optimal geometries extracted from traditional element-based topology optimization outcomes usually have zigzag boundaries, leading to being difficult to fabricate. In this study, a fairly accurate and efficient topology description function method (TDFM) for topology optimization of linear elastic structures is developed. By employing the modified sigmoid function, a simple yet efficient strategy is presented to tackle the computational difficulties because of the nonsmoothness of Heaviside function in topology optimization problem. The optimization problem is to minimize the structural compliance, with highest stiffness, while satisfying the volume constraint. The design problem is solved by a Sequential Linear Programming method. Convergent, crisp, and smooth final layouts are obtained, which can be fabricated without postprocessing, demonstrated by a series of numerical examples. Further, the proposed method has a rather higher accuracy and efficiency compared with traditional TDFM, when the classical topology optimization methods, such as bidirectional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP) method, are taken as benchmark.

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Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

Cited by 61 publications

References 22 publications

A predictor–corrector approach for pricing American options under the finite moment log-stable model

A predictor–corrector approach for pricing American options under the finite moment log-stable model

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

An Efficient Topology Description Function Method Based on Modified Sigmoid Function

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