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On Error Estimates for Discretization Operators for the Solution of the Poisson Equation 论求解泊松方程的离散化算子的误差估计值
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010117
A. B. Utesov
{"title":"On Error Estimates for Discretization Operators for the Solution of the Poisson Equation","authors":"A. B. Utesov","doi":"10.1134/s0012266124010117","DOIUrl":"https://doi.org/10.1134/s0012266124010117","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A discretization operator for the solution of the Poisson equation with the right-hand side\u0000from the Korobov class is constructed and its error is estimated in the <span>(L^{p} )</span>-metric, <span>(2leq pleq infty )</span>. It is proved that for <span>(p=2 )</span> the resulting error estimate for the discretization\u0000operator is order sharp on the power scale. An error in calculating the trigonometric Fourier\u0000coefficients used when constructing the discretization operator is also found. It should be noted\u0000that the obtained estimate in one case is better than previously known estimates of the errors of\u0000discretization operators constructed from the values of the right-hand side of the equation at the\u0000nodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with\u0000them up to constants.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"27 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574501","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
Holomorphic Regularization of Singularly Perturbed Integro-Differential Equations 奇异扰动积分微分方程的全态正则化
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010014
V. S. Besov, V. I. Kachalov
{"title":"Holomorphic Regularization of Singularly Perturbed Integro-Differential Equations","authors":"V. S. Besov, V. I. Kachalov","doi":"10.1134/s0012266124010014","DOIUrl":"https://doi.org/10.1134/s0012266124010014","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> S.A. Lomov’s regularization method has long been used to solve integro-differential\u0000singularly perturbed equations, which are very important from the viewpoint of applications. In\u0000this method, the series in powers of a small parameter representing the solutions of these\u0000equations converge asymptotically. However, in accordance with the main concept of the method,\u0000to construct a general singular perturbation theory one must indicate conditions for the ordinary\u0000convergence of these series. This is the subject of the present paper.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"5 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574617","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
Discrete Equations, Discrete Transformations, and Discrete Boundary Value Problems 离散方程、离散变换和离散边界值问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120108
E. B. Afanas’eva, V. B. Vasil’ev, A. B. Kamanda Bongay
{"title":"Discrete Equations, Discrete Transformations, and Discrete Boundary Value Problems","authors":"E. B. Afanas’eva, V. B. Vasil’ev, A. B. Kamanda Bongay","doi":"10.1134/s0012266123120108","DOIUrl":"https://doi.org/10.1134/s0012266123120108","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the solvability of discrete elliptic pseudodifferential equations in a sector of the\u0000plane. Using special factorization of the symbol, the problem is reduced to a boundary value\u0000problem for analytic functions of two variables. A periodic analog of one integral transformation is\u0000obtained that was used to construct solutions of elliptic pseudodifferential equations in conical\u0000domains. The formula for the general solution of the discrete equation under consideration and\u0000some boundary value problems are described in terms of this transformation.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"36 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979174","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 One Cauchy Problem for a Hyperbolic Differential-Difference Equation 论双曲微分差分方程的一个考奇问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120182
N. V. Zaitseva
{"title":"On One Cauchy Problem for a Hyperbolic Differential-Difference Equation","authors":"N. V. Zaitseva","doi":"10.1134/s0012266123120182","DOIUrl":"https://doi.org/10.1134/s0012266123120182","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We provide a formulation of the Cauchy problem in a strip for a two-dimensional\u0000hyperbolic equation containing a superposition of a differential operator and a shift operator with\u0000respect to the spatial variable varying along the entire real axis. The solution of the problem using\u0000integral Fourier transforms is constructed in explicit form.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979289","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
Searching for Parameters of a Model with the Best Local Controllability 寻找具有最佳局部可控性的模型参数
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120145
M. A. Velishchanskiy, V. N. Chetverikov
{"title":"Searching for Parameters of a Model with the Best Local Controllability","authors":"M. A. Velishchanskiy, V. N. Chetverikov","doi":"10.1134/s0012266123120145","DOIUrl":"https://doi.org/10.1134/s0012266123120145","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the problem of optimal choice of model parameters with respect to any\u0000functional. Locally controllable affine systems and integral functionals depending on the program\u0000control are considered. Local controllability of affine systems with nonnegative inputs is proved in\u0000the case where the columns multiplying the controls form a positive basis. For such systems, we\u0000introduce the local controllability coefficient and pose the problem of its maximization depending\u0000on the choice of model parameters. As an example, we consider a very simplified model of an\u0000underwater vehicle and study the problem of finding an arrangement of its control propellers in\u0000which the energy consumption of the vehicle is minimal.\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":"139979294","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
Pseudodifferential Equations and Boundary Value Problems in a Multidimensional Cone 多维锥体中的伪微分方程和边值问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120091
V. B. Vasil’ev
{"title":"Pseudodifferential Equations and Boundary Value Problems in a Multidimensional Cone","authors":"V. B. Vasil’ev","doi":"10.1134/s0012266123120091","DOIUrl":"https://doi.org/10.1134/s0012266123120091","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a special boundary value problem in the Sobolev–Slobodetskii space for a\u0000model elliptic pseudodifferential equation in a multidimensional cone. Taking into account the\u0000special factorization of the elliptic symbol, we write the general solution of the pseudodifferential\u0000equation that contains an arbitrary function. To determine it unambiguously, some integral\u0000condition is added to the equation, which makes it possible to write the solution in Fourier\u0000transforms.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"14 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979316","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
Existence of Sub-Lorentzian Longest Curves 存在次洛伦兹最长曲线
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120157
Yu. L. Sachkov
{"title":"Existence of Sub-Lorentzian Longest Curves","authors":"Yu. L. Sachkov","doi":"10.1134/s0012266123120157","DOIUrl":"https://doi.org/10.1134/s0012266123120157","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Sufficient conditions for the existence of optimal trajectories in general optimal control\u0000problems with free terminal time as well as in sub-Lorentzian problems are obtained.\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":"139979786","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
Cauchy Problem for the Loaded Korteweg–de Vries Equation in the Class of Periodic Functions 周期函数类中负载科特韦格-德-弗里斯方程的考奇问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s001226612312008x
A. B. Khasanov, T. G. Khasanov
{"title":"Cauchy Problem for the Loaded Korteweg–de Vries Equation in the Class of Periodic Functions","authors":"A. B. Khasanov, T. G. Khasanov","doi":"10.1134/s001226612312008x","DOIUrl":"https://doi.org/10.1134/s001226612312008x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The inverse spectral problem method is applied to finding a solution of the Cauchy\u0000problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.\u0000A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms\u0000and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It is\u0000shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin\u0000system of equations and the first trace formula actually satisfies the loaded nonlinear\u0000Korteweg–de Vries equation. In addition, we prove that if the initial function is a real\u0000<span>(pi )</span>-periodic analytic function, then the solution of the\u0000Cauchy problem is a real analytic function in the variable <span>(x )</span> as well, and also that if the number\u0000<span>( {pi }/{n} )</span>, <span>(nin mathbb {N})</span>,\u0000<span>(nge 2 )</span>, is the period of the initial function, then the\u0000number <span>({pi }/{n} )</span> is the period for solving the Cauchy problem with\u0000respect to the variable <span>(x)</span>.\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":"139979286","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
Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness 线性近似系统的正指数在高阶小扰动下变为负指数的反珀伦效应的存在性
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120029
N. A. Izobov, A. V. Il’in
{"title":"Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness","authors":"N. A. Izobov, A. V. Il’in","doi":"10.1134/s0012266123120029","DOIUrl":"https://doi.org/10.1134/s0012266123120029","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove the existence of a two-dimensional linear system <span>(dot {x}=A(t)x )</span>, <span>(tgeq t_0)</span>, with\u0000bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an\u0000infinitely differentiable <span>(m)</span>-perturbation\u0000<span>(f(t,y) )</span> having an order <span>(m&gt;1 )</span> of smallness in a neighborhood of the origin\u0000<span>(y=0 )</span> and an order of growth not exceeding\u0000<span>(m )</span> outside it, such that the perturbed system\u0000<span>(dot {y}=A( t)y+thinspace f(t,y))</span>, <span>(yin mathbb {R}^2 )</span>, <span>(tgeq t_0)</span>, has a\u0000solution <span>(y(t) )</span> with a negative Lyapunov exponent.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"23 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979181","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 Smoothness of the Poisson Potential for Second-Order Parabolic Systems on the Plane 论平面上二阶抛物系统泊松势能的平滑性
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-02-26 DOI: 10.1134/s0012266123120042
E. A. Baderko, K. D. Fedorov
{"title":"On the Smoothness of the Poisson Potential for Second-Order Parabolic Systems on the Plane","authors":"E. A. Baderko, K. D. Fedorov","doi":"10.1134/s0012266123120042","DOIUrl":"https://doi.org/10.1134/s0012266123120042","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the solution of the Cauchy problem in a strip on the plane for a homogeneous\u0000second-order parabolic system. The coefficients of the system satisfy the double Dini condition.\u0000The initial function is continuous and bounded along with its first and second derivatives. Using\u0000the Poisson potential, the nature of the smoothness of this solution is studied and the\u0000corresponding estimates are proved.\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":"139979180","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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