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- Drin, Bohdan, et al.
(författare)
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НЕЛІНІЙНА МОДЕЛЬ ПОВЕДІНКИ ДВОХ КОНКУРЕНТНИХ ФІРМ : [The Nonlinear Model of Behavior of Two Competitive Firms]
- 2021
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Ingår i: Bulletin of Chernivtsi Institute of Trade and Economics. - : Chernivtsi Institute of Trade and Economics of Kyiv National University of Trade and Economics. - 2414-5831 .- 2310-8185. ; 81:1, s. 115-128
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Tidskriftsartikel (refereegranskat)abstract
- The practical task of economics lies in applying the methods of substantiating its decisions. For economics, the main method is the modeling of economic phenomena and processes and, above all, mathematical modeling, which has been stipulated by the presence of stable MATHEMATICAL METHODS, MODELS AND INFORMATION TECHNOLOGIES IN ECONOMY Issue I (81), 2021 117 quantitative patterns and the possibility of a formalized description of many economic processes. The economic-mathematical model contains a system of equations of linear and nonlinear units that promote a mathematical description of economic processes and phenomena, consists of a set of variables and parameters and serves to study these processes and control them. Dynamic models of the economy describe it in development, as well as provide a detailed description of technological methods of production. Mathematical description of dynamic models is carried out with the use of a system of differential equations (in models with continuous time), difference equations (in models with discrete time), as well as systems of algebraic equations. It is important that the investigation of various economic issues has led to the development of the mathematical apparatus. In linear algebra, productive matrices are caused by the studies of intersectoral balance, whereas mathematical programming arose in the course of researching the optimal plan for the distribution of limited resources. In a similar way, there emerged the theory of economic indices and econometrics, the theory of production functions and the theory of consumption, the theory of general economic balance and social welfare, the theory of optimal economic growth. The paper under studies deals with the dynamic economic behavior of two competing objects, whose mathematical model is a nonlinear nonlocal problem for a system of ordinary differential equations with variable coefficients and argument deviation. The dynamic mathematical model is based on the assumption that the volume of output of both firms is determined by such factors on which output depends linearly. The model under discussion includes nonlinear factors, which describe the level of distrust of the competitors and depend on the time of observations and production volumes in previous moments, because the latter significantly affect the production activities of the firm. Such mathematical models are called time-delayed models.
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| 2. |
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| 3. |
- Drin, I. I., et al.
(författare)
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Representation of solution for fully nonlocal diffusion equations with deviation time variable
- 2018
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Konferensbidrag (refereegranskat)abstract
- We prove the solvability of the Cauchy problem for a nonlocal heat equation which is of fractional order both in space and time. The representation formula for classical solutions for time- and space- fractional partial differential operator Dat + a2 (-Δ) γ/2 (0 ≤ α ≤ 1, γ ε (0, 2]) and deviation time variable is given in terms of the Fox H-function, using the step by step method.
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| 4. |
- Drin, Iryna, et al.
(författare)
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The boundary problem by variable t for equation of fractal diffusion with argument deviation
- 2017
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Ingår i: Наукові записки НаУКМА. Фізико-математичні науки.. ; :201, s. 5-7
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- For a quasilinear pseudodifferential equation with fractional derivative by time variable t with order a e (0,1), the second derivative by space variable x and the argument deviation with the help o f the stepmethod we prove the solvability o f the boundary problem with two unknown functions by variable t.
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| 5. |
- Drin, Yaroslav, et al.
(författare)
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About the Approximate Solutions to Linear and Non-Linear Pseudodifferential Reaction Diffusion Equations
- 2019
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Ingår i: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 2, s. 41-45
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Tidskriftsartikel (refereegranskat)abstract
- Background: The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function.Objective: To show that the homotopy perturbation transform method (HPTM) is easily applied tolinear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations.Methods: The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation.Results: The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series.Conclusions: In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.
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| 6. |
- Drin, Yaroslav M., et al.
(författare)
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The Analytical View of Solution of the First Boundary Value Problem for the Nonlinear Equation of Heat Conduction with Deviation of the Argument
- 2023
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Ingår i: Journal of Optimization, Differential Equations and Their Applications. - : Dnipro Oles Honchar Dnipro National University. - 2617-0108 .- 2663-6824. ; 31:2, s. 115-124
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Tidskriftsartikel (refereegranskat)abstract
- In this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established.
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| 7. |
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| 8. |
- Drin, Yaroslav, et al.
(författare)
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The first boundary value problem for the nonlinear equation of heat conduction with deviation of the argument
- 2022
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Ingår i: Proceedings of the 12th International Conference on “Electronics, Communications and Computing". - : Technical University of Moldova. ; , s. 209-213
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Konferensbidrag (refereegranskat)abstract
- The initial-boundary problem for the heat conduction equation with the inversion of the argument are considered. The Green’s function of considered problem are determined. The theorem about the Poisson integral limitation is proved. The theorem declared that the Poisson integral determine the solution of the first boundary problem considered and proved.
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| 9. |
- Drin, Yaroslav, et al.
(författare)
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The nonlocal problem for fractal diffusion equation
- 2022
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Ingår i: International Scientific Technical Journal "Problems of Control and Informatics". - : V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine. - 2786-6491 .- 2786-6505. ; 67:1, s. 47-55
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.
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| 10. |
- Drin, Iryna, et al.
(författare)
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МАТЕМАТИЧНА МОДЕЛЬ ГЛОБАЛЬНОГО ЕКОНОМІЧНОГО ПРОЦЕСУ З НЕЛОКАЛЬНИМИ УМОВАМИ : [Mathematical model of global economic process with non-local conditions]
- 2018
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Ingår i: Bulletin of the Chernivtsi Trade and Economic Institute. - Chernivtsi : Chtei Knteu. - 2310-8185 .- 2414-5831. ; :1-2, s. 152-158
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Tidskriftsartikel (refereegranskat)abstract
- The essence of dynamic economic behavior of competing economic entities, which simulated nonlocal problem for systems of differential equations with variable coefficients is revealed. Developed a dynamic mathematical model assuming that total production is determined by the following factors: the number of products each side, changing certain equipment, which leads to changes in output, the degree of distrust of competitors, and assuming that the rate of change in output proportional to these factors. A partial case of such a model is the model of the arms race. The economic interpretation of nonlocal conditions is that the production volumes at different times governed by predetermined condition. A mathematical model designed to study the dynamics of the economic process of adjusting production volumes at different times.
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