A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

Math Methods in App Sciences

et al. 2020

This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.

Section: Space–time Fractional Chebyshev Spectral Collocation Methodsmentioning

confidence: 99%

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Zadeh

Math Methods in App Sciences

et al. 2020

“…Now, we implement a new fractional pseudospectral method based on the FL and SL polynomials. In this method, the state and control variables of FHOC problem () are approximated as follows

x (t) \approx \sum_{j = 1}^{N} {\overline{x}}_{j} L_{j}^{α} (t), u (t) \approx \sum_{j = 1}^{N} {\overline{u}}_{j} ψ_{j}^{α} (t), 0 \leq t < 1,

where

{\overline{x}}_{j}

and

{\overline{u}}_{j}

for j = 1, 2,…, N are unknown variables, and

ψ_{j}^{α} (.)

for k = 1, 2,…, N are arbitrary interpolating polynomials (they can be considered as classical Lagrange polynomials or other types of fractional Lagrange polynomials 12,13 ). By interpolation property, we get the following relations

x (t_{k}) \approx {\overline{x}}_{k}, u (t_{k}) \approx {\overline{u}}_{k}, k = 1, 2, \dots, N .

It is easy to show that

\frac{d}{d t} L_{j}^{α} (t) = (\frac{α}{t})

…”

Section: A New Shifted Legendre Pseudospectral Methodsmentioning

confidence: 99%

“…Lotfi et al 10 solved the fractional FHOC problems directly based on the Legendre polynomials. A method based on the Bernouli polynomials is given to solve fractional FHOC problems by Keshavarz et al 11 Moreover, Noori Skandari et al 12,13 suggested two different approaches for fractional FHOC problems based on the spectral‐collocation methods.…”

Section: Introductionmentioning

confidence: 99%

Pseudospectral method for fractional infinite horizon optimal control problems

Yang

Optim Control Appl Methods

2020

Summary Up to now, several numerical methods have been presented to solve finite horizon fractional optimal control problems by researchers, while solving fractional optimal control problems on infinite domain is a challenging work. Hence, in this article, a numerical method is proposed to solve fractional infinite horizon optimal control problems. At the first stage, a domain transformation technique is used to map the infinite domain to a finite horizon. Also, fractional derivative defined on an unbounded domain is converted into an equivalent derivative on a finite domain. Then, a new shifted Legendre pseudospectral method is utilized to solve the obtained finite problem and a nonlinear programming problem is suggested to approximate the optimal solutions. Finally, some numerical examples are given to show the efficiency of the method.

On the convergence of Jacobi‐Gauss collocation method for linear fractional delay differential equations

Peykrayegan

Ghovatmand

Math Methods in App Sciences

2020

In this paper, we propose a convergent numerical method for solving linear fractional differential equations. We first convert the equation into an equivalent time-dependent equation and then discretize it at the Jacobi-Gauss collocation points. Using this method, we achieve a system of algebraic equations to approximate the solution of the original equation. Here, we gain the solution and its fractional derivative, simultaneously. We fully present the convergence analysis for the suggested method. We finally illustrate the efficiency of our method by solving some numerical examples.