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Mathematical modeling of infectious diseases and impact of vaccination
(Faculty of Engineering and Natural Sciences, 2023) Bolatova D.
The world is changing and evolving these days, and the various germs and viruses that surround us are also changing and evolving. For this reason, it is essential to choose the best strategy for combating these infections while preserving the health as much as possible. This coursework uses a fundamental mathematical term in bio-mathematics which is a basic reproduction number R0. It is a significant epidemiological term illustrating the number of infected people by one sick individual, hence giving the idea of a possible epidemics. Another thing to investigate in this coursework is a numerical approximation of the epidemic model under various scenarios to examine the model from a mathematical perspective. The following project covers the following topics: vaccination strategy, stability theory, mathematical biology, modeling of epidemics, optimization problems, and Python simulation. According to the epidemic model, one of the best strategies for immunizing the populace is pulse vaccination. The Next Generation matrix approach and the Hartman-Grobman method are two linearization techniques.
A symmetric analysis in LP space in two dimensions
(Faculty of Engineering and Natural Sciences, 2023) Adilkhanova Zh.
The p-Laplacian equation is a nonlinear partial differential equation of third order that arises as Euler-Lagrange equation of the gradient of function in L p norm which was first studied by Gunnar Aronsson in the late 80s [1]. Since then many explicit classical solutions and their generalisations are found. In this paper we find only the classical solutions of p-Laplacian equation in spatial dimensions, i.e. the p-harmonic functions in two dimensions. The p-harmonic functions are found by the use of Lie symmetry analysis method which deals with invariant solutions under some transformations of the solution of the partial differential equations. We obtain Lie algebra generators of the p-Laplacian equation, and the corresponding symmetry reductions of the p-Laplacian equation to ordinary differential equations. Finally, we use the Lie symmetries to construct invariant solutions of p-Laplacian that already known and some new ones in explicit form. Moreover, by using the Lie symmetries we can construct new solutions from known solutions of the p-Laplacian equation. In this article we use Lie symmetry analysis to find two-dimensional a new solution to Laplace’s p-equation. The p-Laplace equation is nonlinear partial differential equation (PDE), arising as an equation Euler-Lagrange expression of the gradient of a function in the L p norm. Using symmetry Lie, we reduce PDEs to ordinary differential equations. We find already known and new solutions of p-Laplace. It turns out using symmetric Lie analysis, we find the symmetry of the given u. We have 8 cases where we ended up getting rid of x,y,u. And in the end what remains is g and s. We were able to find the symmetry for u. In this research article, we employ the Lie symmetry analysis technique to discover two-dimensional solutions of the p-Laplacian equation. The p-Laplacian equation is nonlinear partial differential equation (PDE) that arises as EulerLagrange equation of the gradient of function in L p norm. By using Lie symmetries we reduce PDEs to ordinary differential equations. We find solutions of p-Laplacian that already known and some new. Lie symmetry analysis is a powerful mathematical tool used to investigate symmetries and simplify the solutions of differential equations. In this paper, the authors apply Lie symmetry analysis to the p-Laplacian equation, which is a nonlinear PDE that arises in the context of gradient optimization problems. iv The p-Laplacian equation can be written as: ∇ · (|∇u| p−2∇u) = 0 where ∇ represents the gradient operator and u is the unknown function. The parameter p determines the nonlinearity of the equation. By applying Lie symmetry analysis, the authors are able to identify symmetries of the p-Laplacian equation, which correspond to transformations that leave the equation invariant. These symmetries allow the authors to reduce the PDE to a system of ordinary differential equations (ODEs), which are often easier to solve. Using this approach, the authors are able to find two-dimensional solutions of the p-Laplacian equation. They also compare their results with known solutions in the literature and find some new solutions. Overall, this paper demonstrates the effectiveness of Lie symmetry analysis in simplifying and solving nonlinear PDEs, specifically the p-Laplacian equation. The identified solutions can have various applications in fields such as physics, engineering, and mathematical modeling.
Variety of Bicommutative Algebra defined by identities
(Faculty of Engineering and Natural Sciences, 2023) Akhmetova Zh.
One of the key inquiries in modern algebra is the investigation of algebras that satisfy specific identities.Within the domain of polynomial identities we have 2 questions. The first is to describe an algebra by a defined identity. The second is to describe identities in algebra. The study of identities will help us in the construction of basis of free algebra, in the study of the Hilbert sequence, the Specht problem and problems of the finite basis. In this work, we used two differ- ent research methods. First, the theory of representations of symmetric groups. Second, the theory of representations of linear groups. In this research paper, we have completely described the subvariety of varirty of bicommutative alge- bras defined by the identities (ab)c + (ba)c + (ca)b + c(ba) + c(ab) + a(bc) = 0, 7[(ab)e — 2(ba)e + (ca)b] + d[e(ba) — 2c(ab) + a(bc)] = 0.
Development of mathematical models of system for determi ing the optimal ob jects with multi-criterial parameters
(Faculty of Engineering and Natural Sciences, 2019) Rayev Zh.
This thesis is based on the construction of a mathematical:model for determining unknown parameters using multivariate regression analysis. Structured data are given for the derivation and elimination of significant factors and coefficients. Also, machine learning simple regression models are used for modelling. The results have been evaluated and shown for comparative purposes.
Method of generalized Laguerre polynomials in problems with a moving boundary for the generalized heat equation
(Faculty of Engineering and Natural Sciences, 2019) Yryskeldi Zh.
This thesis is about solving Heat equation. exactly. generalized heat equation. which. solved using Laguerre polvnomials. The heat equation is an important partial differential equation (PDE) which describes the distribution of heat (or variation in temperature) in a given region over time It considers heat transferring in electrical contact from one side to another. Simple examples to the Generalized Heat equation are the melting of ice and the freezing of water. Solving the Heat equation, known problem in numerous industrial and technological applications. such as the manufacture of steel. ablation of heat shields. contact melting in thermal storage systems. ice accretion on aircraft. evaporation of water. Gencralized Heat equation with moving boundary have been solved by Laplace Transform. My aim is to solve the same equation by Laguerre polynomials obtained from heat polynomials.