The Non-Eigenvalue Form of Liouville’s Formula and α-Matrix Exponential Solutions for Combined Matrix Dynamic Equations on Time Scales

In this paper, the non-eigenvalue forms of Liouville’s formulas for delta, nabla and α -diamond matrix dynamic equations on time scales are given and proved. Meanwhile, a diamond matrix exponential function (or α -matrix exponential function) is introduced and some classes of homogenous linear diamond- α dynamic equations which possess the α -matrix exponential solutions is studied. The difference and relation of non-eigenvalue forms of Liouville’s formulas among these representative types of dynamic equations is investigated. Moreover, we establish some sufficient conditions to guarantee transformational relation of Liouville’s formulas and exponential solutions among these types of matrix dynamic equations. In addition, we provide several examples on various time scales to illustrate the effectiveness of our result.

2nd Order Parallel Splitting Methods for Heat Equation

In this paper, heat equation in two dimensions with non local boundary condition is solved numerically by 2nd order parallel splitting technique. This technique used to approximate spatial derivative and a matrix exponential function is replaced by a rational approximation. Simpson’s 1/3 rule is also used to approximate the non local boundary condition. The results of numerical experiments are checked and compared with the exact solution, as well as with the results already existed in the literature and found to be highly accurate.

Numerical Range for the Matrix Exponential Function

For a given square matrix A, the numerical range for the exponential function e^(At), t in C, is considered. Some geometrical and topological properties of the numerical range are presented.

The Method of Calculate Exponential Function

The state-transition matrix in the linear control system has very important application. The calculation methods of state transition matrix Based on the theorem and an example have been given. Such as the method of Matrix diagonalization, Laplace transform, the matrix exponential function limited direct expansion, the differential equation and the method of undetermined coefficients usually can be used. Under different characteristic value and through contrast analysis, the author discusses the some of the calculation method. But the author requires readers according to the specific issues to make the best choice to calculate.

LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.