Numerical solutions for solving time fractional Fokker–Planck equations based on spectral collocation methods

Wang

et al. 2019

Therm sci

Self Cite

Original scientific paper https://doi.org/10.2298/TSCI180824220YIn this paper, we consider the numerical solution of the time-fractional non-linear Klein-Gordon equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. A rigorous error analysis is provided for the spectral methods to show both the errors of approximate solutions and the errors of approximate derivatives of the solutions decaying exponentially in infinity-norm and weighted L 2 -norm. Numerical tests are carried out to confirm the theoretical results.

Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The numerical solution of the time-fractional non-linear Klein-Gordon equation via spectral collocation method

Wang

et al. 2019

Therm sci

Self Cite

“…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

Liu

Mathematical Problems in Engineering

et al. 2018

Self Cite

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.

“…Bu [17,18] considered finite element multigrid method for time fractional advection diffusion equations. Recently, we provided Jacobi spectral-collocation method [19,20] for time-fractional equations. Bhrawy and Zaky [21] reported a spectral collocation method based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for solving one and two-dimensional variable-order fractional nonlinear Cable equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of time fractional Cable equations and convergence analysis

Numerical Methods Partial

Huang

Zhou

2017

Self Cite

In this article, the numerical solution of time fractional Cable equation is considered. We convert the time fractional Cable equations into equivalent integral equations with singular kernel, then propose a spectral collection method in both time and space discretizations with a spectral expansion of Lagrange interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results. K E Y W O R D S convergence analysis, Jacobi collocation method, time fractional Cable equation Numer Methods Partial Differential Eq. 2017;1-21.wileyonlinelibrary.com/journal/num