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Item Open Access The Generalization of the Hurwitz Theorem(Suleyman Demirel University, 2010) Ender Dogan; Vilademir TenThis paper examines a generalized form of the classical Hurwitz theorem, originally established in 1895, which provides necessary and sufficient conditions for all roots of a real-coefficient polynomial to lie in the left half-plane of the complex plane. The study focuses on extending this theorem to broader regions, including semi-planes and bounded domains. The authors analyze polynomials with real coefficients and investigate how relationships among coefficients determine the localization of polynomial roots within specified regions of the complex plane. The project is divided into two main parts: a theoretical background covering complex numbers, polynomials, matrices, and determinants, and a second part presenting the proof of the generalized theorem along with its computational implementation. The results contribute to the stability analysis of differential equations and to broader applications in mathematical modeling and control theory.Item Open Access SYMBOLIC COMPUTATION TECHNIQUES(Suleyman Demirel University, 2012) Niyazi AriThis paper presents a brief overview of symbolic computation and introduces two widely used computer algebra systems: Maple and Maxima. The study outlines their basic characteristics, development background, programming languages, licensing models, and main functional capabilities. Several illustrative examples are provided to demonstrate symbolic integration, simplification of algebraic expressions, graphical visualization, differential equation solving, and contour plotting. The paper emphasizes that symbolic computation systems are able to produce exact and general solutions, which makes them especially valuable in mathematical analysis, education, and scientific research. At the same time, certain limitations of symbolic computation methods are discussed, including high computational cost and difficulties in solving complex problems such as higher-degree polynomials and complicated systems of differential equations. In such cases, numerical computation methods are suggested as an effective alternative. The paper concludes that Maple and Maxima remain powerful and complementary tools in the field of symbolic and mathematical computation.Item Open Access Development and optimization of physics-informed neural networks for solving partial differential equations(SDU University, 2025) Sharimbayev B.This thesis talks about using physics-informed neural networks (PINNs) to solve Poisson equations in both one-dimensional and two-dimensional areas. These equations are common in many physical problems, like heat transfer and electrostatics. The results from PINNs are compared to the finite difference method (FDM), which is a classical numerical method often used to solve these kinds of equations. The study shows that PINNs can give results that are close to those from FDM, with the added benefit of being more flexible for different types of problems. Another part of this work focuses on using multi-task learning with PINNs. In this part, the neural network does more than one job. It not only finds the solution of the differential equation, but it also learns unknown values or parameters that are part of the equation. For example, in one test problem, the equation had a source term and a coefficient that changes depending on the position. The PINN was able to learn both of them correctly while still solving the equation with a low training error. The results show that PINNs can work well even when the equation is more complex or has unknown parts. The model showed good performance on new or unseen data and was able to find the correct hidden values in the system. Because of this, PINNs may be very useful in future applications for solving advanced problems in science and engineering, especially where traditional methods might be harder to use.