Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 7 3 2026 07 01 Construction of an iterative method for solving a class of complex symmetric generalized Lyapunov matrix equation and application to Helmholtz equation 361 376 5916 10.22060/ajmc.2025.24112.1366 EN Akbar Shirilord Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran polytechnic), No. 424, Hafez Ave., 15914, Tehran, Iran Mehdi Dehghan Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran polytechnic), No. 424, Hafez Ave., 15914, Tehran, Iran 0000-0002-2573-9755 Journal Article 2025 04 29 The Lyapunov matrix equations occur in many branches of control theory, such as stability analysis and optimal control. In this work, we introduce a novel iterative approach to address the generalized Lyapunov matrix equation within the framework of complex matrices. At each iteration, the procedure involves solving two conventional Lyapunov equations with real-valued coefficient matrices. The scheme incorporates two positive parameters, for which we establish sufficient conditions to guarantee the convergence of the method under certain assumptions. Then we solve the Lyapunov equation arising by applying a finite difference procedure to Helmholtz equation by proposed method. https://ajmc.aut.ac.ir/article_5916_c5f441cd5f43eb2f2c024e1f8b5d00cd.pdf