Talaat S. El-Danaf

Numerical treatment for the modified burgers equation

In this paper, we consider the solution of the modified Burgers equation by using the collocation method with quintic splines. Applying the Von-Neumann stability analysis method we show that the proposed method is unconditionally stable. By conducting a comparison between the absolute error for our numerical results and the analytic solution of the modified Burgers equation we will test the accuracy of the proposed method.

Iterative positive definite solutions of the two nonlinear matrix equations X±A T X -2 A=I

In this paper iterative methods for obtaining positive definite solutions of the two matrix equations X+/-A^T X^-^2A=I are proposed. We show that under some conditions on the real square matrix A the constructed iterative methods converge to positive definite solutions for the two equations. Numerical examples are given to test and illustrate the performance of the algorithms in terms of convergence, accuracy as well as the efficiency.

Two iterative algorithms for the reflexive and Hermitian reflexive solutions of the generalized Sylvester matrix equation

This paper is concerned with iterative solutions to the generalized Sylvester matrix equation A 1 V + A 2 V ¯ + B 1 W + B 2 W ¯ = E 1 VF 1 + E 2 V ¯ F 2 + C . Two iterative algorithms are presented to obtain the reflexive and Hermitian reflexive solutions. With these iterative algorithms, for any initial reflexive and Hermitian reflexive matrices the solutions can be obtained. Some needed lemmas are first stated, then two theorems are stated and proved where the iterative solutions are obtained. Finally, we report two numerical examples to verify the theoretical results.

A finite iterative algorithm for the solution of Sylvester-conjugate matrix equations AV+BW=EV¯F+C and AV+BW¯=EV¯F+C

In this paper, we consider two iterative algorithms for the Sylvester-conjugate matrix equation AV+BW=EV¯F+C and AV+BW¯=EV¯F+C. When these two matrix equations are consistent, for any initial matrices the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Two numerical examples are given to illustrate the effectiveness of the proposed method.

Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation

We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form:

Solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

In many areas of principal component analysis, biology, electricity, solid mechanics, automatics control theory and vibration theory, linear matrix equations can be encountered. The presented paper proposes first an iterative method for finding the generalized bisymmetric solution to the generalized coupled Sylvester matrix equations. Second, when the generalized coupled Sylvester matrix equations matrix are consistent, for any generalized bisymmetric initial iterative matrix pair, we can obtain the generalized bisymmetric solution within finite iterative steps in the absence of round-off errors. Furthermore, the optimal approximate generalized bisymmetric solution of the matrix equation for a given generalized bisymmetric matrix can be obtained by finding the least-norm generalized bisymmetric solution of new generalized coupled Sylvester matrix equations. Finally, numerical examples are presented to support the theoretical results of this paper.