Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation

Zhou, Zhaojie; Tan, Zhiyu

doi:10.1007/s10915-018-0829-0

Cited by 34 publications

(15 citation statements)

References 26 publications

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“…Combining Equations (8) and (12) together, the quadratic FV scheme for model (1) now can be expressed in the following (2N − 1)-by-(2N − 1) matrix form:…”

Section: Fv Scheme and Matrix Formmentioning

confidence: 99%

“…However, because of the nonlocal nature of fractional integral and differential operators, the mathematical models, which described by fractional calculus, has only recently come to researchers' attention. For example, it is pointed out that fractional integrals and derivatives possess better modeling capabilities for describing challenging phenomena in stochastic computation, finance, physics, material science, biology, control, etc (see previous studies [1][2][3][4][5][6][7][8] ). The readers can refer to a very recent review paper 9 for a comprehensive understanding of real-world applications of fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

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An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

2020

Math Methods in App Sciences

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A quadratic finite volume (FV) method for steady‐state Riesz space‐fractional diffusion equations (sFDEs) with variable diffusivity coefficient is developed using piecewise quadratic basis functions, and a resulting linear algebra system with two‐by‐two block‐type Toeplitz‐like coefficient matrix is formulated. It is proved that the method requires a minimum memory of order scriptOfalse(Nfalse), where N is the number of spatial partitions. Moreover, as two of the produced Toeplitz‐like submatrices are not square, a new fast nonsquare Toeplitz‐like matrix‐vector product is specially designed, which requires an almost linear computational complexity of order scriptOfalse(Nlog2Nfalse). Then, a fast version of biconjugate gradient stabilized (BiCGSTAB) solution algorithm, named FBiCGSTAB, is proposed for the FV scheme. The FV method combined with Crank‐Nicolson (CN) time discretization is applied to solve time‐dependent sFDEs, and an efficient CN‐FV scheme is developed and analyzed. Finally, numerical results are presented to show the utility of the fast FV and fast CN‐FV methods.

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“…Combining Equations (8) and (12) together, the quadratic FV scheme for model (1) now can be expressed in the following (2N − 1)-by-(2N − 1) matrix form:…”

Section: Fv Scheme and Matrix Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

2020

Math Methods in App Sciences

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“…In [15], a fast gradient projection method is investigated for a constrained fractional optimal control. Finite element approximation of space fractional optimal control problem is considered in [16]. In [17], a priori error estimates of time-stepping discontinuous Galerkin finite element approximation for time fractional optimal control problem is established.…”

Section: Introductionmentioning

confidence: 99%

Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems

Tang¹

2021

AJCM

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“…In the past decades lots of works [14][15][16][17][18][19] are devoted to develop numerical methods or algorithms for fractional differential equations. In recent years optimal control problems governed by different types of fractional differential equations have attracted increasing attentions [20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

Wang

Zhou

2021

Fractal Fract

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In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

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Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation

Cited by 34 publications

References 26 publications

An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems

Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

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