Fractional Differential Equations

Introductory real analysis , Introductory real analysis , کتابخانه دیجیتال جندی شاپور اهواز

Introductory Real Analysis. By A. N. Kolmogorov and S. V. Fomin, translated by Richard A. Silverman. Pp. xii, 403. (Prentice-Hall Inc.)

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm’s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

https://www.mdpi.com/1099-4300/22/11/1213/pdf

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations

A new computational approach for optimal control problems with multiple time-delay

A new approach for solving of optimal nonlinear control problems

In this paper a numerical method is proposed to solve intuitionistic fuzzy nonlinear equations. The method is based on the Midpoint Newton's method in which the intuitionistic fuzzy quantities are presented in parametric form. The efficiency of the proposed method is tested with an illustrative example.

Solving Intuitionistic Fuzzy Nonlinear Equations

In this paper, we present a method based upon the moments problem for solving a class of fractional optimal control problems (FOCPs) with time delay. The performance index of the FOCP is considered as a function of both the state and control variables and the dynamics of system is given as an ordinary fractional differential equation with time delay. The fractional derivative (FD) is described in the Riemann–Liouville sense in which the FD order is α ∈ (0, 1]. The main reason of using this technique is the convexification of a non-linear and non-convex FOCP with time delay in which the non-linearities in the control variable can be expressed as polynomials. The Grünwald–Letnikov formula is used as an approximation for FD in numerical computations. Some numerical examples are given to illustrate the effectiveness of our method.

A numerical approximation for delay fractional optimal control problems based on the method of moments

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.

A semidefinite programming approach for solving fractional optimal control problems

Determination of shape parameter has a major role in the accuracy of the radial basis functions method. In this paper, we present a new method called Leave-Two-Out Cross Validation to determine the best shape parameter. In the proposed method, by deleting two data from the data set, a new formula is derived for determining the error value and then the optimal shape parameter is determined. Num results show that the method will be more accurate in comparison with other methods.

Mohammad Keyanpour

Papers

A new approach for solving of optimal nonlinear control problems

Solving Intuitionistic Fuzzy Nonlinear Equations

A numerical approximation for delay fractional optimal control problems based on the method of moments

A semidefinite programming approach for solving fractional optimal control problems

Leave-Two-Out Cross Validation to optimal shape parameter in radial basis functions