In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.

An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

We introduce new differentiation matrices based on the pseudospectral collocation method. Monic Chebyshev polynomials (MCPs) were used as trial functions in differentiation matrices (D-matrices). Those matrices have been used to approximate the solutions of higher-order ordinary differential equations (H-ODEs). Two techniques will be used in this work. The first technique is a direct approximation of the H-ODE. While the second technique depends on transforming the H-ODE into a system of lower order ODEs. We discuss the error analysis of these D-matrices in-depth. Also, the approximation and truncation error convergence have been presented to improve the error analysis. Some numerical test functions and examples are illustrated to show the constructed D-matrices’ efficiency and accuracy.

This paper presents two operational matrices. The first one represents integer-order derivatives of the modified shifted Chebyshev polynomials of the second kind. These polynomials serve as basis functions in two spectral methods, Galerkin and Petrov–Galerkin. These techniques are then applied to solve even-order initial boundary value problems (IBVPs). Additionally, convergence and error analysis is provided and demonstrated. On the other hand, the second operational matrix represents an integer integration of the shifted Chebyshev polynomials of the second kind. The proposed polynomials are used in the expansions of the spectral approach to solve partial boundary value problems (PBVPs). As an example, we consider the one-dimensional linear telegraph equation and solve it using the Galerkin and Petrov–Galerkin methods. The demand for more accurate and efficient spectral approaches to solving BVPs and IBVPs encouraged this study. By using modified shifted Chebyshev polynomials this work aims to improve the accuracy and convergence of even-order IBVPs and PBVPs.

Herein, new orthogonal polynomials have been generated from shifted Chebyshev polynomials that fulfill a given set of homogeneous boundary conditions and the necessary formulae have been established. Moreover, an integer order derivative operational matrix has been introduced. Then, the presented novel polynomials are used together with the two spectral methods, namely, the Galerkin and Tau methods, as the basis functions. The convergence and error analyses were introduced and proved. Finally, some even-order boundary value problems (BVPs) have been approximated using the presented method.