http://localhost:8080/xmlui/handle/123456789/547 2026-07-06T01:20:10Z 2026-07-06T01:20:10Z مفتاح سالم ابحور, نعيمة http://localhost:8080/xmlui/handle/123456789/1307 2026-06-28T10:09:53Z 2025-01-01T00:00:00Z

حل بعض المعادلات الإهليجية باستعمال طريقة الفروق المنتهية وطريقة العنصر المنتهي ومقارنتهما مفتاح سالم ابحور, نعيمة Abstract The majority of physical and engineering phenomena are described by partial differential equations. Solving these equations is essential for understanding such problems; however, obtaining their exact analytical solution is frequently challenging. Consequently, it becomes necessary to utilize numerical methods for approximating these equations. In this study, we focused on linear second-order elliptic partial differential equations, specifically employing two-dimensional Poisson's and Laplace's equations as representative models for the aforementioned phenomena. We successfully derived the analytical solutions for the equations under investigation: using the separation of variables method for homogeneous elliptic PDEs and the eigenfunction method for inhomogeneous ones. For the inhomogeneous equations, a comparative analysis was conducted between the analytical solution and the numerical solutions generated by applying both the Finite Difference Method and the Finite Element Method. This comparison was presented through numerical tables and graphical plots. It is worth noting that although some of these equations were found in existing literature and solved using similar or alternative techniques, their presentation often lacked detail and clarity for the reader. Our approach therefore involved re-formulating and presenting these solutions in a detailed, simplified manner to clearly illustrate each method's mechanism and enhance reader comprehension. Our findings from the examined examples indicate that both the Finite Differences Method and the Finite Element Method exhibit convergence towards the analytical solution when applied to a two-dimensional homogeneous elliptic (Laplace) equation defined on a regular domain (such as a rectangle or triangle) and subject to homogeneous and inhomogeneous Dirichlet boundary conditions. Conversely, when addressing a two-dimensional inhomogeneous elliptic (Poisson) equation under homogeneous and inhomogeneous Dirichlet boundary conditions, the Finite Element Method generally yields a superior approximate solution compared to the Finite Difference Method, resulting in lower errors, particularly when the domain of the inhomogeneous elliptic (Poisson) equation possesses a regular geometric form (rectangle, triangle, etc.). Keywords: Finite Difference Method, Finite Element Method, Two-Dimensional Poisson's Equation, Two-Dimensional Laplace's Equation

2025-01-01T00:00:00Z عبدالله عبدالسلام أبوفارس, فريدة http://localhost:8080/xmlui/handle/123456789/1304 2026-06-09T10:59:21Z 2025-01-01T00:00:00Z

بعض المشاكل المتعلقة بفصول الدوال التحليلية وتطبيقاتها عبدالله عبدالسلام أبوفارس, فريدة

2025-01-01T00:00:00Z سعيد، مروة عبدالسلام http://localhost:8080/xmlui/handle/123456789/1142 2026-03-25T21:29:47Z 2026-01-01T00:00:00Z

استخدام الطرق التحليلية والعددية لحل معادلة شرودنجر غير الخطية العشوائية سعيد، مروة عبدالسلام

2026-01-01T00:00:00Z الفرجاني، ملاك منصور عمر http://localhost:8080/xmlui/handle/123456789/1092 2026-03-19T21:42:57Z 2025-01-01T00:00:00Z

الحلول الموجية لبعض معادلات شرودنجر الغير خطية العشوائية ذات الضوضاء البيضاء المضاعفة باستخدام حسبان آتو الفرجاني، ملاك منصور عمر In this thesis, some modern mathematical methods is applied to construct many types of solitons and other exact wave solutions equation in mathematical physics which have multiplicative white noise via Itȏ calculus with the aid of a computer algebraic system (CAS). The mathematical methods are the (G'/G, 1/G )-expansion method and the new Jacobi elliptic function expansion method for solving the nonlinear Biswas–Milovic equation (BME). The modified Kudryashov's-method and the addendum Kudryashov-method for solving the stochastic resonant nonlinear Schrodinger equation (stochastic resonant nonlinear-SE). The addendum Kudryashov-method for solving the nonlinear stochastic Radhakrishnan-Kundu Lakshmanan equation (RKL). Finally, a direct method based on the generalized Lienard equation is used for finding many other exact solutions of the stochastic resonant nonlinear-SE. Solitary wave solutions, hyperbolic functions solutions, periodic functions solutions, Jacobi elliptic functions solutions, rational functions solutions, dark soliton solutions, singular soliton solutions, bright soliton solutions are obtained. Based on reductive perturbation technique and a series of transformation, the nonlinear PDEs had been derived by many authors which can be reduced to a nonlinear ordinary differential equation (ODE) using the wave transformation. Furthermore, comparison between our new results and the well-known results is given. Finally, plotting 2D and 3D graphics of the exact solutions are shown.

2025-01-01T00:00:00Z