## Theses and Dissertations

### Wavelet methods for solving fractional-order dynamical systems

#### ORCID

ORCID:0000-0003-0174-7328

#### Issuing Body

Mississippi State University

Razzaghi, Mohsen

Dang, Hai

Qian, Chuanxi

Kim, Seongjai

5-13-2022

#### Document Type

Dissertation - Open Access

Mathematics

#### Degree Name

Doctor of Philosophy (Ph.D)

#### College

College of Agriculture and Life Sciences

#### Department

Department of Mathematics and Statistics

#### Abstract

In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

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