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Solution of the Cauchy Problem for One Degenerate Equation with the Dzhrbashyan–Nersesyan Fractional Derivative 用日尔巴先-涅尔塞先分式导数求解一个退化方程的考奇问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120170
B. Yu. Irgashev
{"title":"Solution of the Cauchy Problem for One Degenerate Equation with the Dzhrbashyan–Nersesyan Fractional Derivative","authors":"B. Yu. Irgashev","doi":"10.1134/s0012266123120170","DOIUrl":"https://doi.org/10.1134/s0012266123120170","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A solution of the Cauchy problem is obtained for one degenerate equation with the\u0000Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using\u0000the Kilbas–Saigo function.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"295 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold 论向量束上的线性微分算子的可解性
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120078
M. S. Smirnov
{"title":"On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold","authors":"M. S. Smirnov","doi":"10.1134/s0012266123120078","DOIUrl":"https://doi.org/10.1134/s0012266123120078","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A necessary and sufficient condition is established for the closedness of the range or\u0000surjectivity of a differential operator acting on smooth sections of vector bundles. For connected\u0000noncompact manifolds it is shown that these conditions are derived from the regularity conditions\u0000and the unique continuation property of solutions. An application of these results to elliptic\u0000operators (more precisely, to operators with a surjective principal symbol) with analytic\u0000coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the\u0000Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively,\u0000Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic)\u0000manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies\u0000solvability in generalized sections.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"39 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Analysis of a Multipoint Boundary Value Problem for a Nonlinear Matrix Differential Equation 非线性矩阵微分方程的多点边界值问题分析
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120017
A. N. Bondarev, V. N. Laptinskii
{"title":"Analysis of a Multipoint Boundary Value Problem for a Nonlinear Matrix Differential Equation","authors":"A. N. Bondarev, V. N. Laptinskii","doi":"10.1134/s0012266123120017","DOIUrl":"https://doi.org/10.1134/s0012266123120017","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a nonlinear differential matrix equation, we study a multipoint boundary value\u0000problem by a constructive method of regularization over the linear part of the equation using the\u0000corresponding fundamental matrices. Based on the initial data of the problem, sufficient\u0000conditions for its unique solvability are obtained. Iterative algorithms containing relatively simple\u0000computational procedures are proposed for constructing a solution. Effective estimates are given\u0000that characterize the rate of convergence of the iteration sequence to the solution, as well as\u0000estimates of the solution localization domain.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"2016 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Features of Numerical Solution of Coefficient Inverse Problems for Nonlinear Equations of the Reaction–Diffusion–Advection Type with Data of Various Types 论各类数据的反应-扩散-对流型非线性方程系数逆问题数值解法的特点
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120133
D. V. Lukyanenko, R. L. Argun, A. A. Borzunov, A. V. Gorbachev, V. D. Shinkarev, M. A. Shishlenin, A. G. Yagola
{"title":"On the Features of Numerical Solution of Coefficient Inverse Problems for Nonlinear Equations of the Reaction–Diffusion–Advection Type with Data of Various Types","authors":"D. V. Lukyanenko, R. L. Argun, A. A. Borzunov, A. V. Gorbachev, V. D. Shinkarev, M. A. Shishlenin, A. G. Yagola","doi":"10.1134/s0012266123120133","DOIUrl":"https://doi.org/10.1134/s0012266123120133","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these\u0000approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"3 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Initial–Boundary Value Problems for Homogeneous Parabolic Systems in a Semibounded Plane Domain and Complementarity Condition 半约束平面域中同质抛物系统的初始边界问题及互补条件
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120066
S. I. Sakharov
{"title":"Initial–Boundary Value Problems for Homogeneous Parabolic Systems in a Semibounded Plane Domain and Complementarity Condition","authors":"S. I. Sakharov","doi":"10.1134/s0012266123120066","DOIUrl":"https://doi.org/10.1134/s0012266123120066","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider initial–boundary value problems for homogeneous parabolic systems with\u0000coefficients satisfying the double Dini condition with zero initial conditions in a semibounded\u0000plane domain with nonsmooth lateral boundary. The method of boundary integral equations is\u0000used to prove a theorem on the unique classical solvability of such problems in the space of\u0000functions that are continuous together with their first spatial derivative in the closure of the\u0000domain. An integral representation of the obtained solutions is given. It is shown that the\u0000condition for the solvability of the posed problems considered in the paper is equivalent to the\u0000well-known complementarity condition.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"82 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Existence of Feedback Control for One Fractional Voigt Model 论一个分数沃伊特模型反馈控制的存在性
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120169
A. V. Zvyagin, E. I. Kostenko
{"title":"On the Existence of Feedback Control for One Fractional Voigt Model","authors":"A. V. Zvyagin, E. I. Kostenko","doi":"10.1134/s0012266123120169","DOIUrl":"https://doi.org/10.1134/s0012266123120169","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the feedback control problem for a mathematical model that describes the\u0000motion of a viscoelastic fluid with memory along the trajectories of the velocity field. We prove\u0000the existence of an optimal control that delivers a minimum to a given bounded and lower\u0000semicontinuous cost functional.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Singular Heat Equation 关于奇异热方程
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s001226612312011x
E. L. Shishkina, A. K. Yusupova
{"title":"On Singular Heat Equation","authors":"E. L. Shishkina, A. K. Yusupova","doi":"10.1134/s001226612312011x","DOIUrl":"https://doi.org/10.1134/s001226612312011x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In physics, the singular heat equation with the Bessel operator is used to explain the basic\u0000process of heat transport in a substance with spherical or cylinder symmetry. This paper examines\u0000the solution of the Cauchy problem for the heat equation with the Bessel operator acting in the\u0000space variable. We obtain some properties of the solution and consider the normalized modified\u0000Bessel function of the first kind.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"436 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algorithms for Robust Inversion of Dynamical Systems 动态系统鲁棒反演算法
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-01-25 DOI: 10.1134/s001226612314001x
E. I. Atamas’, A. V. Il’in, S. K. Korovin, V. V. Fomichev
{"title":"Algorithms for Robust Inversion of Dynamical Systems","authors":"E. I. Atamas’, A. V. Il’in, S. K. Korovin, V. V. Fomichev","doi":"10.1134/s001226612314001x","DOIUrl":"https://doi.org/10.1134/s001226612314001x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A new methodology for solving inverse dynamics problem is developed. The methodology\u0000is based on using a mathematical model of a dynamical system and robust stabilization methods\u0000for a system under uncertainty.\u0000</p><p>Most exhaustively the theory is described for linear finite-dimensional\u0000time-invariant scalar systems and multiple-input multiple-output systems.\u0000</p><p>The study shows that with this approach, the zero dynamics of the original system\u0000is of crucial significance. This dynamics, if exists, is assumed to be exponentially stable.\u0000</p><p>It is established that zero-dynamics, relative order, and the corresponding\u0000equations of motion cannot be defined correctly in multiple-input multiple-output systems. For\u0000correct inverse transformation of the solution of the problem, additional assumptions have to be\u0000introduced, which generally limits the inverse system category.\u0000</p><p>Special attention is given to the synthesis of elementary (minimal) inverters, i.e.,\u0000least-order dynamical systems that solve the transformation problem.\u0000</p><p>It is also established that the inversion methods sustain the efficiency with finite\u0000parameter variations in the initial system as well as with uncontrolled exogenous impulses having\u0000no impact on the system’s internal dynamics.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"69 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty 关于计算具有不确定性的线性系统的可解集问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2023-12-29 DOI: 10.1134/s00122661230110083
A. A. Melnikova, P. A. Tochilin
{"title":"On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty","authors":"A. A. Melnikova, P. A. Tochilin","doi":"10.1134/s00122661230110083","DOIUrl":"https://doi.org/10.1134/s00122661230110083","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a linear-convex control system defined by a set of differential equations with\u0000continuous matrix coefficients. The system may have control parameters, as well as uncertainties\u0000(interference) the possible values of which are subject to strict pointwise constraints. For this\u0000system, over a finite period of time, taking into account the constraints, we study the problem of\u0000guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The\u0000main stage of solving the problem is the construction of an alternating integral and a solvability\u0000set. To construct the latter, the greatest computational complexity is the calculation of the\u0000geometric difference between the target set and the set determined by the uncertainty. A\u0000two-dimensional example of this problem is considered for which a method is proposed for finding\u0000the solvability set without the need to calculate the convex hull of the difference of the support\u0000functions of the sets.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"118 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Variation of the Nonlinearity Parameter in the “Super-Twisting” Algorithm 论 "超级扭曲 "算法中非线性参数的变化
IF 0.6 4区 数学
Differential Equations Pub Date : 2023-12-29 DOI: 10.1134/s00122661230110137
V. V. Fomichev, A. O. Vysotskii
{"title":"On the Variation of the Nonlinearity Parameter in the “Super-Twisting” Algorithm","authors":"V. V. Fomichev, A. O. Vysotskii","doi":"10.1134/s00122661230110137","DOIUrl":"https://doi.org/10.1134/s00122661230110137","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the stability of a modified (with variation in the nonlinearity parameter)\u0000“super-twisting” algorithm. The analysis is based on majorizing the trajectories of the system with\u0000an arbitrary nonlinearity parameter by the trajectories of systems of the classical “super-twisting”\u0000algorithm. Stability conditions for the modified systems are obtained, as well as estimates for the\u0000size of the stability domain depending on system parameters.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"79 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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