A.M. Samoilenko’s numerical-analytic method is a well-known and effective research method of solvability and approximate construction of the solutions of various boundary value problems for systems of differential equations. The investigation of boundary value problems for new classes of systems of functional- differential equations by this method is still an actual problem. A boundary value problem for a system of differential equations with finite quantity of transformed arguments in the case of linear two-point boundary conditions is considered at this paper. In order to study the questions of the existence and approximate construction of a solution of this problem, we used a modification of A.M. Samoilenko’s numerical-analytic method without determining equation, i.e. the method has an analytical component only. Sufficient conditions for the existence of a unique solution of the considered boundary value problem and an error estimation of the constructed successive approximations are obtained. The use of the developed modification of the method is illustrated by concrete examples.
{"title":"ON A TWO-POINT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS WITH MANY TRANSFORMED ARGUMENTS","authors":"M. Filipchuk","doi":"10.31861/bmj2021.01.24","DOIUrl":"https://doi.org/10.31861/bmj2021.01.24","url":null,"abstract":"A.M. Samoilenko’s numerical-analytic method is a well-known and effective research method of solvability and approximate construction of the solutions of various boundary value problems for systems of differential equations.\u0000The investigation of boundary value problems for new classes of systems of functional- differential equations by this method is still an actual problem.\u0000A boundary value problem for a system of differential equations with finite quantity of transformed arguments in the case of linear two-point boundary conditions is considered at this paper.\u0000In order to study the questions of the existence and approximate construction of a solution of this problem, we used a modification of A.M. Samoilenko’s numerical-analytic method without determining equation, i.e. the method has an analytical component only. Sufficient conditions for the existence of a unique solution of the considered boundary value problem and an error estimation of the constructed successive approximations are obtained. The use of the developed modification of the method is illustrated by concrete examples.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121220755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the inverse boundary value problem on determining a space-dependent component in the right-hand side of semilinear time fractional diffusion-wave equation. We find sufficient conditions for a time-local uniqueness of the solution under the time-integral additional condition [frac{1}{T}int_{0}^{T}u(x,t)eta_1(t)dt=Phi_1(x), ;;;xin Omegasubset Bbb R^n] where $u$ is the unknown solution of the first boundary value problem for such equation, $eta_1$ and $Phi_1$ are the given functions. We use the method of the Green's function.
{"title":"INVERSE SOURCE PROBLEM FOR A SEMILINEAR FRACTIONAL DIFFUSION-WAVE EQUATION UNDER A TIME-INTEGRAL CONDITION","authors":"H. Lopushanska","doi":"10.31861/bmj2022.02.11","DOIUrl":"https://doi.org/10.31861/bmj2022.02.11","url":null,"abstract":"We study the inverse boundary value problem on determining a space-dependent component in the right-hand side of semilinear time fractional diffusion-wave equation. We find sufficient conditions for a time-local uniqueness of the solution under the time-integral additional condition\u0000[frac{1}{T}int_{0}^{T}u(x,t)eta_1(t)dt=Phi_1(x), ;;;xin Omegasubset Bbb R^n]\u0000where $u$ is the unknown solution of the first boundary value problem for such equation, $eta_1$ and $Phi_1$ are the given functions. We use the method of the Green's function.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122955917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.
{"title":"THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER","authors":"V. Litovchenko","doi":"10.31861/bmj2021.02.06","DOIUrl":"https://doi.org/10.31861/bmj2021.02.06","url":null,"abstract":"The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115830490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concept of continuity in a strong sense for the case of functions with values in metric spaces is studied. The separate and joint properties of this concept are investigated, and several results by Russell are generalized. A function $f:X times Y to Z$ is strongly continuous with respect to $x$ /$y$/ at a point ${(x_0, y_0)in X times Y}$ provided for an arbitrary $varepsilon> 0$ there are neighborhoods $U$ of $x_0$ in $X$ and $V$ of $y_0$ in $Y$ such that $d(f(x, y), f(x_0, y))
{"title":"STRONG CONTINUITY OF FUNCTIONS FROM TWO VARIABLES","authors":"V. Nesterenko, V. Lazurko","doi":"10.31861/bmj2021.01.19","DOIUrl":"https://doi.org/10.31861/bmj2021.01.19","url":null,"abstract":"The concept of continuity in a strong sense for the case of functions with values in metric spaces is studied. The separate and joint properties of this concept are investigated, and several results by Russell are generalized.\u0000\u0000A function $f:X times Y to Z$ is strongly continuous with respect to $x$ /$y$/ at a point ${(x_0, y_0)in X times Y}$ provided for an arbitrary $varepsilon> 0$ there are neighborhoods $U$ of $x_0$ in $X$ and $V$ of $y_0$ in $Y$ such that $d(f(x, y), f(x_0, y)) <varepsilon$ /$d((x, y), f (x, y_0))<varepsilon$/ for all $x in U$ and $y in V$. A function $f$ is said to be strongly continuous with respect to $x$ /$y$/ if it is so at every point $(x, y)in X times Y$.\u0000\u0000Note that, for a real function of two variables, the notion of continuity in the strong sense with respect to a given variable and the notion of strong continuity with respect to the same variable are equivalent.\u0000\u0000In 1998 Dzagnidze established that a real function of two variables is continuous over a set of variables if and only if it is continuous in the strong sense with respect to each of the variables.\u0000\u0000Here we transfer this result to the case of functions with values in a metric space: if $X$ and $Y$ are topological spaces, $Z$ a metric space and a function $f:X times Y to Z$ is strongly continuous with respect to $y$ at a point $(x_0, y_0) in X times Y$, then the function $f$ is jointly continuous if and only if $f_{y}$ is continuous for all $yin Y$.\u0000\u0000It is obvious that every continuous function $f:X times Y to Z$ is strongly continuous with respect to $x$ and $y$, but not vice versa. On the other hand, the strong continuity of the function $f$ with respect to $x$ or $y$ implies the continuity of $f$ with respect to $x$ or $y$, respectively. Thus, strongly separately continuous functions are separately continuous.\u0000\u0000Also, it is established that for topological spaces $X$ and $Y$ and a metric space $Z$ a function $f:X times Y to Z$ is jointly continuous if and only if the function $f$ is strongly continuous with respect to $x$ and $y$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123542516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Signi cantly nonlinear non-autonomous di erential equations have begun to appear in practice from the second half of the nineteenth century in the study of real physical processes in atomic and nuclear physics, and also in astrophysics. The di erential equation, that contains in its right part the product of regularly and rapidly varying nonlinearities of an unknown function and its rst-order derivative is considered in the paper. Partial cases of such equations arise, rst of all, in the theory of combustion and in the theory of plasma. The rst important results on the asymptotic behavior of solutions of such equations have been obtained for a second-order di erential equation, that contains the product of power and exponential nonlinearities in its right part. For, no such equations have been obtained before. According to this, the study of the asymptotic behavior of solutions of nonlinear di erential equations of the second order of general case, that contain the product of regularly and rapidly varying nonlinearities as the argument tends either to zero or to in nity, is actual not only from the theoretical but also from the practical point of view. The asymptotic representations, as well as the necessary and su cient conditions of the existence of Pω(Y0, Y1,±∞)-solutions of such equations are investigated in the paper. This class of solutions is the one of the most di cult of studying due to the fact that, by the a priori properties of the functions of the class, their second-order derivatives aren't explicitly expressed through the rst-order derivative. The results obtained in this article supplement the previously obtained results for Pω(Y0, Y1,±∞)-solutions of the investigated equation concerning the su cient conditions of their existence and quantity.
{"title":"ASYMPTOTIC REPRESENTATIONS OF SOLUTIONS WITH SLOWLY VARYING DERIVATIVES OF THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH THE PRODUCT OF DIFFERENT TYPES OF NONLINEARITIES","authors":"O. Chepok","doi":"10.31861/bmj2020.02.081","DOIUrl":"https://doi.org/10.31861/bmj2020.02.081","url":null,"abstract":"Signi cantly nonlinear non-autonomous di erential equations have begun to appear in practice from the second half of the nineteenth century in the study of real physical processes in atomic and nuclear physics, and also in astrophysics. The di erential equation, that contains in its right part the product of regularly and rapidly varying nonlinearities of an unknown function and its rst-order derivative is considered in the paper. Partial cases of such equations arise, rst of all, in the theory of combustion and in the theory of plasma. The rst important results on the asymptotic behavior of solutions of such equations have been obtained for a second-order di erential equation, that contains the product of power and exponential nonlinearities in its right part. For, no such equations have been obtained before. According to this, the study of the asymptotic behavior of solutions of nonlinear di erential equations of the second order of general case, that contain the product of regularly and rapidly varying nonlinearities as the argument tends either to zero or to in nity, is actual not only from the theoretical but also from the practical point of view. The asymptotic representations, as well as the necessary and su cient conditions of the existence of Pω(Y0, Y1,±∞)-solutions of such equations are investigated in the paper. This class of solutions is the one of the most di cult of studying due to the fact that, by the a priori properties of the functions of the class, their second-order derivatives aren't explicitly expressed through the rst-order derivative. The results obtained in this article supplement the previously obtained results for Pω(Y0, Y1,±∞)-solutions of the investigated equation concerning the su cient conditions of their existence and quantity.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130467588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On fawar problem and problem of kolmogorov-nikolsky solved by V.K. Dzyadyk","authors":"P. Zaderei, S. Ivasyshen, N. Zaderei, G. Nefodova","doi":"10.31861/bmj2019.01.048","DOIUrl":"https://doi.org/10.31861/bmj2019.01.048","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130790052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Systems of differential-difference equations are mathematical models of many applied problems of biology, ecology, medicine, economics. The variety of mathematical models of real dynamic processes is due to the fact that their evolution does not occur instantaneously, but with some delays that have different biological interpretations. The introduction of delay allows you to build adequate mathematical models and describe new effects and phenomena in physics, ecology, immunology and other sciences. The exact solution of differential-difference equations can be found only in the simplest cases, so algorithms for finding approximate solutions of such equations are important. In this paper, a family of difference schemes is constructed for the approximate finding of solutions to initial problems with delay. Special cases are generalized Euler difference schemes. The conditions for the convergence of the generalized explicit Euler difference scheme are established. To automate the numerical simulation of systems with delays, an application program has been developed, which is used to approximate the solutions of SIR models with two delays.
{"title":"DELAY MODELING OF MATHEMATICAL MODELS OF BIOLOGY AND IMMUNOLOGY","authors":"T. Lunyk, I. Cherevko","doi":"10.31861/bmj2021.02.07","DOIUrl":"https://doi.org/10.31861/bmj2021.02.07","url":null,"abstract":"Systems of differential-difference equations are mathematical models of many applied problems of biology, ecology, medicine, economics. The variety of mathematical models of real dynamic processes is due to the fact that their evolution does not occur instantaneously, but with some delays that have different biological interpretations. The introduction of delay allows you to build adequate mathematical models and describe new effects and phenomena in physics, ecology, immunology and other sciences.\u0000The exact solution of differential-difference equations can be found only in the simplest cases, so algorithms for finding approximate solutions of such equations are important. In this paper, a family of difference schemes is constructed for the approximate finding of solutions to initial problems with delay. Special cases are generalized Euler difference schemes. The conditions for the convergence of the generalized explicit Euler difference scheme are established.\u0000To automate the numerical simulation of systems with delays, an application program has been developed, which is used to approximate the solutions of SIR models with two delays.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125739841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In recent decades, special attention has been paid to problems with nonlocal conditions for partial differential equations. Such interest in such problems is due to both the needs of the general therapy of boundary value problems and their rich practical application (the process of diffusion, oscillations, salt and moisture transport in soils, plasma physics, mathematical biology, etc.). A multipoint in-time problem for a nonuniformly 2b-parabolic equation with degeneracy is studied. The coefficients of the parabolic equation of order 2b allow for power singularities of arbitrary order both in the time and spatial variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions of auxiliary problems, the limiting value of which will be the solution of the given problem. Estimates of the solution of the multipoint time problem for the 2b-parabolic equation are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of degeneracy of the coefficients of the groups of higher terms and the power features of the coefficients of the lower terms of the parabolic equation. With certain restrictions on the right-hand side of the equation, an integral image of the solution to the given problem is obtained.
{"title":"A MULTIPOINT IN-TIME PROBLEM FOR THE 2b-PARABOLIC EQUATION WITH DEGENERATION","authors":"I. Pukalskyy, B. Yashan","doi":"10.31861/bmj2022.02.18","DOIUrl":"https://doi.org/10.31861/bmj2022.02.18","url":null,"abstract":"In recent decades, special attention has been paid to problems with nonlocal conditions for partial differential equations. Such interest in such problems is due to both the needs of the general therapy of boundary value problems and their rich practical application (the process of diffusion, oscillations, salt and moisture transport in soils, plasma physics, mathematical\u0000biology, etc.).\u0000A multipoint in-time problem for a nonuniformly 2b-parabolic equation with degeneracy is studied. The coefficients of the parabolic equation of order 2b allow for power singularities of arbitrary order both in the time and spatial variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions of auxiliary problems, the limiting value of which will be the solution of the given problem. Estimates of the solution of the multipoint time problem for the 2b-parabolic equation are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of degeneracy of the coefficients of the groups of higher terms and the power features of the coefficients of the lower terms of the parabolic equation. With certain restrictions on the right-hand side of the equation, an integral image of the solution to the given problem is obtained.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115313220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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