Laboratory Staff

Axmedov Dilshod Mashrabovich

Yetakchi ilmiy xodim More details →

G`ulomov Otabek Xudoyberdiyevich

Katta ilmiy xodim More details →

Hayotov Abdullo Raxmonovich

Hisoblash matematikasi laboratoriyasi mudiri More details →

Nuraliyev Farxod Abdug`aniyevich

Yetakchi ilmiy xodim More details →

Shadimetov Xolmatvay Maxkambayevich

Professor, Bosh ilmiy xodim More details →

Scientific Activity

Currently, the laboratory "Computational Mathematics" is conducting research in the following directions:

• Theory of optimal lattice quadrature and cubature formulas. The theory of optimal lattice quadrature and cubature formulas is a method for creating "recipes" for computing multidimensional integrals with the least computational resources and highest accuracy. Its main applications cover a wide range from numerical solution of complex integral and differential equations to improving the quality of medical images in computer tomography.

• Theory of optimal interpolation formulas and splines. The main and sole purpose of the theory of optimal interpolation formulas and splines is to construct a function (curve) that passes through a given set of points (data) and is simultaneously "best" (optimal) and "smoothest" possible. The solution to the problem of finding an optimal interpolation formula is a spline. That is, in a certain class of functions (for example, in Sobolev space), the "most optimal" curve that minimizes the interpolation error turns out to be a specific type of interpolation spline (often a cubic spline). In other words, splines are not only practically convenient (smooth and easy to control), but they are also theoretically considered the "most accurate" and "most optimal" solution. While spline theory is a practical tool showing how to construct smooth and flexible curves, optimal interpolation theory is the theoretical foundation proving why precisely those splines are the "best" solution with the least error.

• Optimal difference methods for approximate solution of differential equations. There is no single, universal "best" method for solving differential equations. "Optimality" is not a static property, but rather a dynamic goal that is constantly reconsidered depending on the characteristics of the problem being solved, the required accuracy, and available technologies. Based on this, the main goal of optimal difference methods is to develop and select numerical schemes that provide the best possible balance (optimal compromise) between accuracy, stability, and computational efficiency for a specific class of problems and available computational resources. This comprehensive goal includes three main components: Reliability: Ensuring that the solution is stable and free from physically inconsistent numerical artifacts (such as excessive diffusion, parasitic oscillations, or incorrect dispersion). This determines the quality of the solution and confidence in it. Efficiency: Achieving the specified accuracy with minimal computational costs (time and memory). This determines the practical applicability of the method and the ability to solve large-scale problems. Accuracy: Guaranteeing that the numerical solution is sufficiently close to the true solution. This determines the quantitative correctness of results and predictive capability. The search for an optimal method is a complex, multi-criteria optimization process. This process enables reliable, fast, and cost-effective numerical modeling of scientific and engineering problems, which in turn serves as a fundamental basis for the development of modern science and technology.

• Construction of discrete analogs of differential operators. The main goal of constructing discrete analogs of differential operators is to transform complex differential equations, which often do not have analytical (exact) solutions, into simpler systems of algebraic equations that can be solved on a computer. This approach is a fundamental tool for numerical modeling of continuous physical processes in almost all fields of science and engineering. The method of replacing differential operators with discrete analogs serves as a universal "bridge" for translating mathematical models describing the continuous world into numerical and computable form. This, in turn, enables understanding, predicting, and optimizing real-world problems through computer simulations.

• Optimal methods for approximate solution of integral equations. Integral equations appear as fundamental mathematical models in many fields of science and engineering, including potential theory, acoustics, elasticity, fluid mechanics, radiative heat transfer, and many other fronts. They can arise directly from physical principles or as a result of reformulating differential equations, where boundary conditions are often incorporated into the equation itself in a more natural way. Despite the widespread occurrence of integral equations, for most of them, especially in cases with complex kernels or geometry, closed-form analytical solutions do not exist. The space of functions with primitives that can be expressed through elementary functions is much smaller than the space of functions whose derivatives we can take. The main source of complexity is the kernel of the integral operator, which can be "arbitrarily complex." While there are systematic approaches for differential operators such as Sturm-Liouville theory, there is no single general paradigm for finding analytical solutions for integral operators. For this reason, relying on high-accuracy numerical methods to solve practical problems becomes inevitable. Here, the word "optimal" does not mean a single best method, but rather achieving a balance across several interrelated criteria. The improvements we seek can be quantitatively assessed by the following measures: Accuracy: How close the approximate solution is to the unknown true solution. Efficiency: How much computational resources (time and memory) are required to achieve the desired accuracy. Stability/Robustness: How sensitive the method is to small changes in input data, rounding errors, and discretization choices (such as mesh quality, geometric angles). The goal is to move from methods that are computationally impossible or incorrect to methods that enable practical implementation of large-scale, high-accuracy simulations.

International Cooperation

The scientific staff of the "Computational Mathematics" laboratory of the V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan and the faculty members of the "Informatics and Computer Graphics" department of Tashkent State Transport University are conducting scientific collaborations with scientists from the world's leading scientific centers and universities to solve urgent problems of modern computational mathematics. In particular, scientific cooperation has been well established with the following scientists and their scientific schools:

1. Professor V.L. Vaskevich, S.L. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Russia.
2. Professor M.D. Ramazanov, Institute of Mathematics, Ufa, Russia.
3. Professor M.V. Noskov, Siberian Federal University, Krasnoyarsk, Russia.
4. Academician G.V. Milovanović, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia.
5. Professor E. Novak, Friedrich Schiller University of Jena, Germany.
6. Professor A. Cabada, University of Santiago de Compostela, Spain.
7. Professor Ch.-O. Lee, KAIST, South Korea.
8. Professor S.S. Shcherbakov, Belarusian State University, Republic of Belarus.
9. Professor Parovik R.I., Vitus Bering Kamchatka State University, Russia.

Seminars