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The aim of this paper is to investigate, from the numerical point of view, the Jacobi polynomials to solve fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). A direct numerical method for solving a general class of FVPs and FOCPs is presented. The fractional derivative in FVPs is in the Caputo sense and in FOCPs is in the Riemann–Liouville sense. The Rayleigh–Ritz method is introduced for the numerical solution of FVPs containing left or right Caputo fractional derivatives. Rayleigh–Ritz method is one of the well-known direct methods used for the solution of variational problems. In this technique, at first, we expand the unknown function in terms of the modified Jacobi polynomials and then we derive a compact form of fractional derivative of the unknown function in terms of the Jacobi polynomials. Examples indicate that the new technique has high accuracy and is very efficient to implement. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce the operational matrices of integral and fractional integral using the flatlet oblique multiwavelets. The operational matrices of integral and fractional integrals for flatlet scaling functions and wavelets are presented and are utilized to reduce the solution of the Abel integral equations of the first and the second kinds to the solution of algebraic equations. The main characteristic behind our approach in using this technique is that only a small number of flatlet oblique multiwavelets is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and fractional derivatives of the dual flatlet multiwavelets by multiplying some matrices is suggested. The system is then used to solve a fractional convection–diffusion equation. The biorthogonality and high vanishing moments properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from the other bases. [ABSTRACT FROM PUBLISHER]

The system of generalized Sylvester matrix equations{A1XD1+E1XB1=C1A2XD2+E2XB2=C2:   :      :AmXDm+EmXBm=Cm(including Sylvester and Lyapunov matrix equations as special cases) has nice applications in various branches of control and system theory. In the present paper, we consider this system over the generalized centro-symmetric matrix X. By extending the Jacobi and the Gauss–Seidel iterations and by applying the hierarchical identification principle, we propose a gradient-based iterative algorithm for finding the generalized centro-symmetric solution of the system. It is shown that the iterative algorithm consistently converges to the generalized centro-symmetric solution for any initial generalized centro-symmetric matrix. A numerical experiment is also given to show the effectiveness of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Solving the linear system Ax = b is an active area of research in engineering. Many researchers have applied preconditioners to the linear system Ax = b. In this work, we introduce two new preconditioners for solving linear systems. The best property of the introduced preconditioners is that they can be used under weaker conditions than the previous preconditioners in the literature. We propose preconditioned associated accelerated overrelaxation (AOR) iterative methods with these two new preconditioners, and give the corresponding convergence results. Numerical examples show an improvement in the convergence rate of the AOR preconditioned iterative matrices. [ABSTRACT FROM PUBLISHER]

In this paper we first introduce fractional orthogonal Jacobi functions then we obtain a new fractional derivative operational matrix for these orthogonal functions. It is based on the relationship between the coefficients of the fractional Taylor series and fractional Jacobi function expansions. We also apply this new operational matrix to the collocation method for solving general multi-order fractional differential equations (FDEs) and nonlinear fractional integro–differential equations (FIDEs). We also present several test problems. The numerical results show that our new scheme is very effective and convenient for solving FDEs and FIDEs. [ABSTRACT FROM PUBLISHER]

Solving linear matrix equations has various applications in control theory, in engineering, in scientific computations and various other fields. By applying generalization of the Hermitian and skew-Hermitian splitting (GHSS) iteration and the hierarchical identification principle, we propose a gradient-based iterative method for finding the solution of the generalized coupled Sylvester matrix equations (including (coupled) Sylvester and Lyapunov matrix equations as special cases). We prove that the iterative solution consistently converges to the solution for any initial matrix. Some numerical examples and applications are provided to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

The conjugate gradient method is one of the most useful and the earliest-discovered techniques for solving large-scale nonlinear optimization problems. Many variants of this method have been proposed, and some are widely used in practice. In this article, we study the descent Dai–Yuan conjugate gradient method which guarantees the sufficient descent condition for any line search. With exact line search, the introduced conjugate gradient method reduces to the Dai–Yuan conjugate gradient method. Finally, a global convergence result is established when the line search fulfils the Goldstein conditions. [ABSTRACT FROM PUBLISHER]

In this work, the problem of parameter identification in a parabolic partial differential equation is discussed. The shifted Legendre-tau method is presented for finding the unknown function and also the unknown coefficient in the one-dimensional diffusion model. The method consists of expanding the required approximate solution as the elements of a shifted Legendre polynomial and by using operational matrices we reduce the problem to a set of algebraic equations. Several examples are given to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and yields very accurate results. [ABSTRACT FROM PUBLISHER]

In this work, multi-point boundary value problems are considered. These problems have important roles in the modelling of various problems in physics and engineering. Although numerous works have been carried out on the existence and uniqueness to the solution of these problems, the numerical or analytical methods are not established for solving them. In this paper the well-known He's variational iteration method is applied for solving the multi-point boundary value problems. The method is modified and the results are shown using some test problems. These results show the efficiency of the new approach. [ABSTRACT FROM PUBLISHER]

In this paper we present a computational technique for solving the second-order Fredholm integro-differential equations. The method is based on a non-classical pseudospectral method. The differential matrices are computed and are utilized to reduce the second-order Fredholm integro-differential equations to algebraic equations. Numerical examples are presented to demonstrate the validity and applicability of the new method. [ABSTRACT FROM PUBLISHER]

A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results. [ABSTRACT FROM PUBLISHER]

In this work, we present a third-order, semi-discrete, central-upwind scheme for computing approximate solutions of 1D systems of conservation laws. We combine the third-order CWENO reconstruction proposed in Levy et al., the semi-discrete central-upwind numerical flux proposed in Kurganov et al. and the third-order TVD Runge—Kutta method, proposed in Shu and Osher. We are interested in the behavior of the total variation of the approximate solution obtained with this scheme. Also we test our scheme on both scalar and gas dynamics problems. We observe that the total variation of computed solutions is close to the total variation of the exact solution or a reference solution. [ABSTRACT FROM PUBLISHER]

This note proposes an iterative method to solve the Sylvester matrix equation AY + YB = C over a reflexive matrix Y which has wide applications in many systems and control applications. Analysis of convergence shows that the proposed iterative method converges to the reflexive solution for any initial reflexive matrix. Finally, a numerical example confirms the theoretical results and shows that the method works reliably. [ABSTRACT FROM PUBLISHER]

This paper presents two numerical methods for solving the controlled Rayleigh nonlinear oscillator problem. The first method is based on construction of interpolation polynomials to approximate the states and subsequent control using the roots of Legendre polynomials as collocation nodal points. In the second method, we look for a piecewise continuous approximation polynomial for each state and a piecewise constant function for the control function. We show that this method leads to better results and enables us to accurately compute switching times. These methods are easy to implement, and yield very accurate results. [ABSTRACT FROM AUTHOR]