In the field of mathematical modeling and numerical analysis, singular perturbation problems occupy an important place due to their wide range of applications in various scientific and engineering disciplines. These problems arise in systems with multiscale dynamics, where the complex interaction of fast and slow processes significantly affects the behavior of the solution. Sobolev-type equations are widely used in mathematical physics and fluid mechanics to describe phenomena of various nature, including the propagation of plasma and ion-acoustic waves, filtration processes in porous media, compaction of clay materials, second-order fluid models, thermodynamic processes, and the dynamics of long waves of small amplitude. This work focuses on the study of nonlinear singularly perturbed Sobolev initial-boundary value problems and the development of new difference schemes for their numerical solution. The paper explores the mathematical aspects of the singularly perturbed problem.

This study examines zero-shot cross-topology generalization for DC power flow under varying load and renewable generation. A leakage-safe primary protocol uses 3500 physically generated IEEE-30 scenarios for development and internal evaluation, and 1000 IEEE-57 scenarios for external evaluation and reserve. Each bus is represented by 16 operating and topological features. The proposed KENO-Krylov operator combines bounded neural initialization, positive local preconditioning, explicit Kirchhoff residuals, and a 20-step Krylov/CG-like recurrence with a hard slack-bus constraint, using 2523 trainable parameters. On the locked IEEE-30 test, it achieves R² = 0.999999 and angle MAE 0.00159°. On unseen IEEE-57, it retains R² = 0.9440 and angle MAE 0.783°, but the median relative KCL residual remains 0.1433. Supplementary paired experiments quantify three renewable-generation regimes, external component ablations, and larger and radial networks. At the highest prescribed renewable scale, mean generation covers 49.63% and 53.25% of demand on IEEE-30 and IEEE-57. On 500 new IEEE-118 scenarios, angle MAE increases to 7.719° and median KCL to 0.974; radial stress tests reveal further accuracy limitations. Learned corrections slightly improve angle MAE over PCG-20, whereas convergent numerical solvers are more accurate and reused Cholesky factorization is faster in the measured setting. Nominal 90% uncertainty calibration does not transfer externally. The evidence supports comparative point-prediction transfer within the stated synthetic DC experiments, not universal topology generalization, operational security, or validity for nonlinear AC systems.

Математикалық есептерді шешуге арналған арнайы программалық пакеттер көптеп табылады, олар MathCad, MathLab, Maple және т.б. Солардың ішіндегі ең соңғысы және де қазіргі кездегі ең кең таралғаны ол – GeoGebra. GeoGebra бұл білім берудің барлық деңгейіне арналған динамикалық математиканың программалық қамтамасы, геометрияны, алгебраны, электрондық кестені, графиктер салуды, статистика мен есептеулерді бір ортаға біріктіріп пайдалануға мүмкіндік беретін қарапайым пакет. GeoGebra әлемдегі динамикалы математиканы, ғылымды, технологияны, инженерия мен математиканы, білім мен инновацияны, оқыту мен білім беруді сүйемелдеуші программалық қамтамасыз етушілердің ең алдығы қатарлысы болып отыр. GeoGebra циркуль мен сызғыштың көмегімен салынатын сызбаларды салуға мүміндік беретін тегін таралатын динамикалық геометриялық орта. Сонымен қатар, бұл бағдарлама ондағы кіріктірілген програмалау тілінің командаларының (геометриялық салуларды басқаруға мүмкіндік беретін) есебінен функция графиктері мен жұмыс істеуде (графиктер салу, түбірлерді есептеу, экстремумдар, интегралдар және т.б.) бай мүмкіндіктерге ие. Мақалада Geogebra программалау ортасында айнымалысы ажыратылатын дифференциалдық теңдеулерді шешу, шешімдерді анимациялау мысалдары келтірілген

This article presents a comprehensive methodology for evaluating the effectiveness of the educational process in computer science within an adaptive learning environment. Adaptive learning is an approach in which the content, pace, and form of educational material presentation are adjusted to each student's characteristics, such as their level of knowledge, learning style, interests, and the pace of mastering the material. Unlike traditional methods, where all students receive the same amount and structure of knowledge, the adaptive system offers flexibility and individualization of learning. The developed methodology includes designing, implementing, and evaluating the effectiveness of an interactive digital platform that enables students to diagnose their initial educational level, identify their strengths and weaknesses, and automatically select educational materials and practical assignments aligned with their learning trajectory. An experimental study was conducted at Atyrau University named after Khalel Dosmukhamedov (Atyrau, Kazakhstan) involving 118 students studying in the field of Computer Science. The article describes the stages of integrating adaptive technologies into the educational process and presents comparative learning outcomes for students using the adaptive platform versus those studying with traditional methodologies. The results indicate that the use of an adaptive learning environment facilitates deeper assimilation of educational material, increases motivation, and improves students' academic results in computer science. The presented methodology can be utilized by educators and educational institutions to enhance teaching quality through modern digital and analytical tools.

The attitude towards bilingualism in education has completely changed in the contemporary period. The current study shows that bilingualism has a clear advantage, as cumulative research data conducted over the past three decades in various locations around the world shows that people who can develop their linguistic potential receive cognitive, social, personal and economic benefits. In addition to linguistic, academic and cognitive outcomes, bilingual education promotes the absence of prejudice and acceptance of cultural diversity. At the same time, the issue of the development of professional, communicative and intercultural competencies and autonomy in the context of learning in several languages remains a rather acute topic. However, contrary to the concerns expressed due to higher cognitive requirements in bilingual programs, these problems are solved by effective teaching methods and curricula. The quality of teaching is another objective that the educational reform program is aimed at, and quality teaching will be crucial for the success of bilingual education. In addition, the development of appropriate complex and contextualized curricula for both languages is also crucial. It is these two aspects that will be discussed in detail in this article.