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In this paper, the piecewise Chebyshev cardinal functions (PCCFs) are handled to design a highly accurate direct method to solve a category of fractional optimal control problems (FOCPs) involving the Caputo–Katugampola fractional derivative (as a generalisation of the Caputo derivative) in a large domain. These cardinal functions have many useful features, such as cardinality, exponential accuracy and orthogonality. Notice that the exponential accuracy makes it possible to obtain an approximate solution with very high accuracy for the problem under consideration by applying only a small number of basis functions. In this regard, first, the ordinary and fractional integral matrices of the PCCFs are obtained to construct a direct method using them. Then, the constructed approach turns to solve the FOCP under consideration into deriving the solution of an algebraic system (by the aid of approximating the state and control variables based on the PCCFs). To show the validity of the established technique some illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss–Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples. [ABSTRACT FROM AUTHOR]

In this study, two categories of fractional optimal control problems are investigated. One category is optimal control problems, which includes a fractional dynamic system, and the other category, in addition to the fractional dynamics system, includes inequality constraints. The Atangana–Baleanu fractional derivative in the Caputo sense is used to define these problems. The extended Chebyshev cardinal wavelets as an appropriate class of basis functions are introduced to construct two numerical methods for these problems. To solve these problems, at first, an operational matrix for the Atangana–Baleanu fractional integral of these wavelets is derived. Then, by approximating the fractional derivative of the state variables and control variables in terms of the extended Chebyshev cardinal wavelets, and employing the fractional integral operational matrix of these wavelets and the Lagrange multipliers method, the problems under consideration are converted into systems of algebraic equations, which can be easily solved. To examine the accuracy of the established methods, some numerical examples are solved. [ABSTRACT FROM AUTHOR]

In this study, a new family of cardinal wavelets called the extended Chebyshev cardinal wavelets is introduced to investigate problems on arbitrary finite domains. These wavelets possess many beneficial properties, such as spectral (exponential) accuracy, cardinality and orthogonality. The classical and fractional derivative matrices together with the fractional integral matrix of these wavelets are obtained. Two formulations of delay optimal control problems with a dynamical system involved with fractional derivatives are introduced. Two direct approaches based on these wavelets, together with their fractional derivative and integral matrices, are adopted for solving such problems. The presented methods transform solving the problems under study into solving constrained minimisation problems by approximating the state and control variables via the expressed wavelets. Some numerical examples are provided to show the efficiency of the expressed wavelet methods. [ABSTRACT FROM AUTHOR]

This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is governed by a nonlinear fractional differential equation. A new fractional integral matrix associated with these basis functions is derived. This matrix significantly reduces the computations in solving such problems. The proposed approach transforms the original problem into a nonlinear programming one by expanding the control and state variables in terms of the orthonormal piecewise Chelyshkov functions and employing the derived fractional integral matrix. Some numerical problems are examined for verification of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, an accurate numerical approach based on the Wilson wavelets and collocation method together with Kumar and Sloan scheme is proposed to numerical solution of a nonlinear fractional Fredholm–Hammerstein integro-differential equations. The presented method transforms solving such problems into solving systems of nonlinear algebraic equations. Error analysis of the established method is investigated theoretically and numerically. The accuracy of the proposed method is investigated on some numerical examples. The obtained results confirms that the presented method is satisfactory for such problems. [ABSTRACT FROM AUTHOR]

In this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method. [ABSTRACT FROM AUTHOR]