In the present paper, we propose a new method for solving large-scale generalized differential Sylvester equations, by projecting the initial problem onto the extended block Krylov subspace with an orthogonality Galerkin condition. This projection gives rise to a low-dimensional generalized differential Sylvester matrix equation. The low-dimensional equations is then solved by Rosenbrock or BDF method. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed method.

https://jmm.guilan.ac.ir/article_3969_444c1efce1282718615df9f44418d764.pdf

The extended block Arnoldi method for solving generalized differential Sylvester equations

In this paper, we propose two new methods to solve large-scale systems of differential equations, which are based on the Krylov method. In the first one, the exact solution with the exponential projection technique of the matrix. In the second, we get a new problem of small size, by dropping the initial problem, and then we solve it in ways, such as the Rosenbrock and the BDF. Some theoretical results are presented such as an accurate expression of the remaining criteria. We give an expression of error report and numerical values to compare the two methods in terms of how long each method takes, and we also compare the approaches.

Numerical methods for solving large-scale systems of differential equations

We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods.

/pdf/the-novel-mittag-leffler-galerkin-method-application-to-a-ubgy93wz.pdf

The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order

Fractional BDF Methods for Solving Fractional Differential Matrix Equations

Application of MGA and EGA algorithms on large-scale linear systems of ordinary differential equations

In this paper, we present a new numerical methods for solving large-scale differential Sylvester matrix equations with low rank right hand sides. These differential matrix equations appear in many applications such as robust control problems, model reduction problems and others. We present two approaches based on extended global Arnoldi process. The first one is based on approximating exponential matrix in the exact solution using the global extended Krylov method. The second one is based on a low-rank approximation of the solution of the corresponding Sylvester equation using the extended global Arnoldi algorithm. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed methods compared with the extended block Krylov method given in Hached and Jbilou (Numer Linear Algebra Appl 255:e2187, 2018).

Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Controllability, observability and fractional linear-quadratic problem for fractional linear systems with conformable fractional derivatives and some applications

Lakhlifa Sadek

Papers

Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations

The extended block Arnoldi method for solving generalized differential Sylvester equations

Numerical methods for solving large-scale systems of differential equations

Controllability, observability and fractional linear-quadratic problem for fractional linear systems with conformable fractional derivatives and some applications

The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order