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The aim of this paper is to use the second derivative of Chebyshev polynomials (SDCHPs) as basis functions for solving linear and nonlinear boundary value problems (BVPs). Then, the operational matrix for the derivative was established by using SDCHPs. The established matrix via mixing between two spectral methods, collocation and Galerkin, has been applied to solve BVPs. Consequently, an error analysis is investigated to ensure the convergence of the technique used. Finally, we solved some problems involving real-life applications and compared their solutions with exact and other solutions from different methods to verify the accuracy and efficiency of this method. [ABSTRACT FROM AUTHOR]

Shifted Legendre polynomials (SLPs) with the Riemann–Liouville fractional integral operator have been used to create a novel fractional integration tool. This tool will be called the fractional shifted Legendre integration matrix (FSL B-matrix). Two algorithms depending on this matrix are designed to solve two different types of integral equations. The first algorithm is to solve fractional Volterra integro-differential equations (VIDEs) with a non-singular kernel. The second algorithm is for Abel's integral equations. In addition, error analysis for the spectral expansion has been proven to ensure the expansion's convergence. Finally, several examples have been illustrated, including an application for the population model. [ABSTRACT FROM AUTHOR]

A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices. [ABSTRACT FROM AUTHOR]

Legendre polynomials' first derivatives have been used as the basis function via the pseudo-Galerkin spectral method. Operational matrices for derivatives have been used and extended to deal with the system of ordinary differential-algebraic equations. An algorithm via those matrices has been designed. The accuracy and efficiency of the proposed algorithm had been shown by two techniques, theoretically, via the boundedness of the approximated expansion and numerically through numerical examples. [ABSTRACT FROM AUTHOR]

This research aims to assemble two methodical spectral Legendre's derivative algorithms to numerically attack the Lane-Emden, Bratu's, and singularly perturbed type equations. We discretize the exact unknown solution as a truncated series of Legendre's derivative polynomials. Then, via tau and collocation methods for linear and nonlinear problems, respectively, we obtain linear/nonlinear systems of algebraic equations in the unknown expansion coefficients. Finally, with the aid of the Gaussian elimination technique in the linear case and Newton's iterative method for the non-linear case - with vanishing initial guess- we solve these systems to obtain the desired solutions. The stability and convergence analyses of the numerical schemes were studied in-depth. The schemes are convergent and accurate. Some numerical test problems are performed to verify the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes. [ABSTRACT FROM AUTHOR]

This paper established a functional design of the spectral Tau method (STM) upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to execute the Tau integration analytically. Moreover, explicit algebraic systems for solving the Lane–Emden and Riccati equations and the contamination of a system of three artificial lakes are introduced. The convergence and error analyses are explored in depth. Some enlightening boundary value problems (BVPs) for real-life and physical applications, as singularly perturbed equations, are provided to guarantee that this approach is authentic, reliable, and appropriate. Accurate results are obtained using only a few number of retained nodes. The different outcomes, patterns, and better results showed that the FDCHPs differ from Chebyshev polynomials of the second kind. [ABSTRACT FROM AUTHOR]