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Journal articles on the topic 'Neumann boundary control problems'

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Published: 6 September 2023

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1

López, Ginés, and Juan-Aurelio Montero-Sánchez. "Neumann boundary value problems across resonance." ESAIM: Control, Optimisation and Calculus of Variations 12, no. 3 (June 20, 2006): 398–408. http://dx.doi.org/10.1051/cocv:2006009.

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2

Kowalewski, Adam, and Anna Krakowiak. "Optimal boundary control problems of retarded parabolic systems." Archives of Control Sciences 23, no. 3 (September 1, 2013): 261–79. http://dx.doi.org/10.2478/acsc-2013-0016.

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Abstract Optimal boundary control problems of retarded parabolic systems are presented. Necessary and sufficient conditions of optimality are derived for the Neumann problem. A simple example of application is also presented.

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3

Bollo, Carolina M., Claudia M. Gariboldi, and Domingo A. Tarzia. "Neumann boundary optimal control problems governed by parabolic variational equalities." Control and Cybernetics 50, no. 2 (June 1, 2021): 227–52. http://dx.doi.org/10.2478/candc-2021-0012.

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Abstract We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα . We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

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4

Hamamuki, Nao, and Qing Liu. "A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 13. http://dx.doi.org/10.1051/cocv/2019076.

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This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.

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5

Gunzburger, Max D., Hyung-Chun Lee, and Jangwoon Lee. "Error Estimates of Stochastic Optimal Neumann Boundary Control Problems." SIAM Journal on Numerical Analysis 49, no. 4 (January 2011): 1532–52. http://dx.doi.org/10.1137/100801731.

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6

Eppler, Karsten, and Helmut Harbrecht. "Tracking Neumann Data for Stationary Free Boundary Problems." SIAM Journal on Control and Optimization 48, no. 5 (January 2010): 2901–16. http://dx.doi.org/10.1137/080733760.

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7

Werner, K. D. "Boundary value control problems involving the bessel differential operator." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 27, no. 4 (April 1986): 453–72. http://dx.doi.org/10.1017/s0334270000005075.

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AbstractIn this paper, we consider the hyperbolic partial differential equation wrr = wrr + 1/r wr − ν2 /r2w, where v ≥ 1/2 or ν = 0 is aprameter, with the Dirichlet, Neumann and mixed boundary conditions. The boundary controllability for such problems is investigated. The main resutl is that all “finite energy” intial states can be steered to the zero state in time T, using a control f ∈ L2 (0, T), provided T > 2. Furthermore, necessary conditions for controllability are also presented.

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8

Kowalewski, Adam, and Marek Miśkowicz. "Extremal Problems for Infinite Order Parabolic Systems with Boundary Conditions Involving Integral Time Lags." Pomiary Automatyka Robotyka 26, no. 4 (December 20, 2022): 37–42. http://dx.doi.org/10.14313/par_246/37.

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Extremal problems for integral time lag infinite order parabolic systems are studied in the paper. An optimal boundary control problem for distributed infinite order parabolic systems in which integral time lags appear in the Neumann boundary conditions is solved. Such equations constitute in a linear approximation a universal mathematical model for many diffusion processes (e.g., modeling and control of heat transfer processes). The time horizon is fixed. Using the Dubovicki-Milutin framework, the necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance indexes and constrained control are derived.

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9

Wong, Kar Hung. "On the computational algorithms for time-lag optimal control problems." Bulletin of the Australian Mathematical Society 32, no. 2 (October 1985): 309–11. http://dx.doi.org/10.1017/s0004972700009989.

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In this thesis we study the following two types of hereditary optimal control problems: (i) problems governed by systems of ordinary differential equations with discrete time-delayed arguments appearing in both the state and the control variables; (ii) problems governed by parabolic partial differential equations with Neumann boundary conditions involving time-delays.

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10

Krumbiegel, K., and J. Pfefferer. "Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations." Computational Optimization and Applications 61, no. 2 (December 2, 2014): 373–408. http://dx.doi.org/10.1007/s10589-014-9718-0.

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11

Mateos, Mariano, and Arnd Rösch. "On saturation effects in the Neumann boundary control of elliptic optimal control problems." Computational Optimization and Applications 49, no. 2 (November 12, 2009): 359–78. http://dx.doi.org/10.1007/s10589-009-9299-5.

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12

Mateos, Mariano, and Arnd RÃ¶sch. "On saturation effects in the Neumann boundary control of elliptic optimal control problems." PAMM 7, no. 1 (December 2007): 1060505–6. http://dx.doi.org/10.1002/pamm.200700674.

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13

Kowalewski, Adam. "Pointwise observation of the state given by parabolic system with boundary condition involving multiple time delays." Archives of Control Sciences 26, no. 2 (June 1, 2016): 189–97. http://dx.doi.org/10.1515/acsc-2016-0011.

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Abstract Various optimization problems for linear parabolic systems with multiple constant time delays are considered. In this paper, we consider an optimal distributed control problem for a linear parabolic system in which multiple constant time delays appear in the Neumann boundary condition. Sufficient conditions for the existence of a unique solution of the parabolic equation with the Neumann boundary condition involving multiple time delays are proved. The time horizon T is fixed. Making use of the Lions scheme [13], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic cost function with pointwise observation of the state and constrained control are derived.

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14

Kowalewski, Adam. "Pointwise observation of the state given by complex time lag parabolic system." Archives of Control Sciences 27, no. 1 (March 1, 2017): 77–89. http://dx.doi.org/10.1515/acsc-2017-0005.

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AbstractVarious optimization problems for linear parabolic systems with multiple constant time lags are considered. In this paper, we consider an optimal distributed control problem for a linear complex parabolic system in which different multiple constant time lags appear both in the state equation and in the Neumann boundary condition. Sufficient conditions for the existence of a unique solution of the parabolic time lag equation with the Neumann boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [13], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional with pointwise observation of the state and constrained control are derived. The example of application is also provided.

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15

Gonçalves, Etereldes, and Marcus Sarkis. "Analysis of Robust Parameter-Free Multilevel Methods for Neumann Boundary Control Problems." Computational Methods in Applied Mathematics 13, no. 2 (April 1, 2013): 207–35. http://dx.doi.org/10.1515/cmam-2013-0001.

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Abstract. We consider a linear-quadratic elliptic control problem (LQECP) where the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose multilevel preconditioners for the reduced system (the discrete Hessian system) resulting from the application of the Schur complement method to the discrete LQECP. In order to derive robust preconditioners with respect to stabilization parameters, we first show that the continuous reduced Hessian operator and the corresponding discrete Hessian matrix are associated to a linear combination of fractional negative Sobolev norms. Then we propose a preconditioner based on multilevel methods, including cases where the stabilization parameters are set equal to zero. We also present numerical experiments which agree with the theoretical results.

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16

Wang, Quanxiang. "Finite Volume Element Method for Solving the Elliptic Neumann Boundary Control Problems." Applied Mathematics 11, no. 12 (2020): 1243–52. http://dx.doi.org/10.4236/am.2020.1112085.

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17

Apel, Thomas, Johannes Pfefferer, and Arnd Rösch. "Finite element error estimates for Neumann boundary control problems on graded meshes." Computational Optimization and Applications 52, no. 1 (August 31, 2011): 3–28. http://dx.doi.org/10.1007/s10589-011-9427-x.

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18

Kowalewski, Adam. "Extremal Problems for Second Order Hyperbolic Systems with Boundary Conditions Involving Multiple Time-Varying Delays." Pomiary Automatyka Robotyka 27, no. 2 (June 16, 2023): 69–76. http://dx.doi.org/10.14313/par_248/69.

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Extremal problems for second order hyperbolic systems with multiple time-varying lags are presented. An optimal boundary control problem for distributed hyperbolic systems with boundary conditions involving multiple time-varying lags is solved. The time horizon is fixed. Making use of Dubovitski-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

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19

Huaizhong, Wang, and Li Yong. "Neumann Boundary Value Problems for Second-Order Ordinary Differential Equations Across Resonance." SIAM Journal on Control and Optimization 33, no. 5 (September 1995): 1312–25. http://dx.doi.org/10.1137/s036301299324532x.

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20

Apel, Thomas, Johannes Pfefferer, and Max Winkler. "Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra." Mathematical Methods in the Applied Sciences 39, no. 5 (August 9, 2015): 1206–32. http://dx.doi.org/10.1002/mma.3566.

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21

Chierici, Andrea, Valentina Giovacchini, and Sandro Manservisi. "Analysis and Computations of Optimal Control Problems for Boussinesq Equations." Fluids 7, no. 6 (June 14, 2022): 203. http://dx.doi.org/10.3390/fluids7060203.

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The main purpose of engineering applications for fluid with natural and mixed convection is to control or enhance the flow motion and the heat transfer. In this paper, we use mathematical tools based on optimal control theory to show the possibility of systematically controlling natural and mixed convection flows. We consider different control mechanisms such as distributed, Dirichlet, and Neumann boundary controls. We introduce mathematical tools such as functional spaces and their norms together with bilinear and trilinear forms that are used to express the weak formulation of the partial differential equations. For each of the three different control mechanisms, we aim to study the optimal control problem from a mathematical and numerical point of view. To do so, we present the weak form of the boundary value problem in order to assure the existence of solutions. We state the optimization problem using the method of Lagrange multipliers. In this paper, we show and compare the numerical results obtained by considering these different control mechanisms with different objectives.

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22

Pop, Nicolae, Miorita Ungureanu, and Adrian I. Pop. "An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction." Mathematics 9, no. 8 (April 19, 2021): 904. http://dx.doi.org/10.3390/math9080904.

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In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough.

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23

Tiba, Dan. "Finite Element Approximation for Shape Optimization Problems with Neumann and Mixed Boundary Conditions." SIAM Journal on Control and Optimization 49, no. 3 (January 2011): 1064–77. http://dx.doi.org/10.1137/100783236.

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24

Семисалов, Б. В. "A fast nonlocal algorithm for solving Neumann-Dirichlet boundary value problems with error control." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 4 (December 20, 2016): 500–522. http://dx.doi.org/10.26089/nummet.v17r446.

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Предложен метод численного решения краевых задач Неймана-Дирихле для уравнений эллиптического типа, обеспечивающий достижение требуемой точности с низким расходом памяти и машинного времени. Метод адаптирует свойства наилучших полиномиальных приближений для построения быстросходящихся алгоритмов без насыщения на основе нелокальных чебышевских приближений. Предложен новый подход к аппроксимации дифференциальных операторов и решению полученных задач линейной алгебры. Даны оценки погрешности численного решения. Обоснован и установлен экспериментально высокий порядок сходимости предложенного метода в задачах с $C^r$-гладкими и $C^{\infty}$-гладкими решениями. Получены выражения элементов массивов, аппроксимирующих операторы производных в задачах с различными граничными условиями. Эти выражения позволят читателю быстро реализовать метод с нуля. A method for searching numerical solutions to Neumann-Dirichlet boundary value problems for differential equations of elliptic type is proposed. This method allows reaching a desired accuracy with low consumption of memory and computer time. The method adapts the properties of best polynomial approximations for construction of algorithms without saturation on the basis of nonlocal Chebyshev approximations. A new approach to the approximation of differential operators and to solving the resulting problems of linear algebra is also proposed. Estimates of numerical errors are given. A high convergence rate of the proposed method is substantiated theoretically and is shown numerically in the case of problems with $C^r$-smooth and $C^{\infty}$-smooth solutions. Expressions for arrays approximating the differential operators in problems with various types of boundary conditions are obtained. These expressions allow the reader to quickly implement the method from scratch.

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25

Krumbiegel, K., C. Meyer, and A. Rösch. "A Priori Error Analysis for Linear Quadratic Elliptic Neumann Boundary Control Problems with Control and State Constraints." SIAM Journal on Control and Optimization 48, no. 8 (January 2010): 5108–42. http://dx.doi.org/10.1137/090746148.

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26

Tasso, Emanuele. "On the continuity of the trace operator in GSBV (Ω) and GSBD (Ω)." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 30. http://dx.doi.org/10.1051/cocv/2019014.

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In this paper, we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed (n − 1)-dimensional set Γ, with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.

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27

Chen, Goong, and Ying-Liang Tsai. "The boundary element numerical method for two-dimensional linear quadratic elliptic problems. I. Neumann control." Mathematics of Computation 49, no. 180 (1987): 479. http://dx.doi.org/10.1090/s0025-5718-1987-0906183-0.

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28

Lee, Hyung-Chun, and O. Yu Imanuvilov. "Analysis of Neumann Boundary Optimal Control Problems for the Stationary Boussinesq Equations Including Solid Media." SIAM Journal on Control and Optimization 39, no. 2 (January 2000): 457–77. http://dx.doi.org/10.1137/s0363012998347110.

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29

Apel, Thomas, Max Winkler, and Johannes Pfefferer. "Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains." IMA Journal of Numerical Analysis 38, no. 4 (October 9, 2017): 1984–2025. http://dx.doi.org/10.1093/imanum/drx059.

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30

Chen, Goong, and Ying-Liang Tsai. "The Boundary Element Numerical Method for Two-Dimensional Linear Quadratic Elliptic Problems: (I) Neumann Control." Mathematics of Computation 49, no. 180 (October 1987): 479. http://dx.doi.org/10.2307/2008323.

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31

Provatidis, Christopher G. "Comparison Between Bézier Extraction and Associated Bézier Elements in Eigenvalue Problems." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 17 (December 31, 2022): 605–15. http://dx.doi.org/10.37394/23203.2022.17.67.

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This paper shows that the control points which are implicitly encountered in the Bézier extraction during isogeometric analysis can be explicitly used to form Bézier elements of C^0-continuity in several ways, thus eventually leading to a superior accuracy and performance than the C^p-continuity. The study is reduced to the eigenvalue extraction in problems governed by the Helmholtz equation. Analysis is performed in conjunction with piecewise cubic interpolation for three benchmark tests in one and two dimensions. In the latter case a rectangular and a circular acoustical cavity under Neumann boundary conditions are analyzed. Several computational details are also discussed.

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32

V. Fardigola, Larissa. "Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control." Mathematical Control & Related Fields 3, no. 2 (2013): 161–83. http://dx.doi.org/10.3934/mcrf.2013.3.161.

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33

Cherniha, Roman. "Conditional symmetries for boundary value problems: new definition and its application for nonlinear problems with Neumann conditions." Miskolc Mathematical Notes 14, no. 2 (2013): 637. http://dx.doi.org/10.18514/mmn.2013.926.

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34

Zhou, Hua-Cheng, Ze-Hao Wu, Bao-Zhu Guo, and Yangquan Chen. "Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 7. http://dx.doi.org/10.1051/cocv/2022003.

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In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.

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35

Droniou, Jérome, Neela Nataraj, and Devika Shylaja. "Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 609–37. http://dx.doi.org/10.1515/cmam-2017-0054.

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AbstractThe article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

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36

Shi, Shengzhu, and Dazhi Zhang. "Local Null-Controllability for Some Quasi-Linear Phase-Field Systems with Neumann Boundary Conditions by one Control Force." Discrete Dynamics in Nature and Society 2022 (July 7, 2022): 1–16. http://dx.doi.org/10.1155/2022/7645304.

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In this article, we study the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller in the frame of classical solutions. In order to minimize the number of control forces, we prove the Carleman inequality for the associated linear system. By constructing a sequence of optimal control problems and an iteration method based on the parabolic regularity, we find a qualified control in Hölder space for the linear system. Based on the theory of Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough.

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37

Antil, Harbir, Deepanshu Verma, and Mahamadi Warma. "External optimal control of fractional parabolic PDEs." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 20. http://dx.doi.org/10.1051/cocv/2020005.

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In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

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38

Fardigola, Larissa, and Kateryna Khalina. "Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition." Mathematical Control & Related Fields 11, no. 1 (2021): 211–36. http://dx.doi.org/10.3934/mcrf.2020034.

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39

Krasnoshchok, Mykola. "Shape optimization in elliptic problem with nonlinear boundary conditions." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 35 (October 25, 2021): 57–66. http://dx.doi.org/10.37069/1683-4720-2021-35-5.

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Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. In practice we often meet problems, leading to the problem of finding an “optimal” shape of systems, the behaviour of which is described by elliptic equations. Problems of this type have been studied by many authors from mathematical as well as from computation point of view. When surfaces of a specimen which have been damaged by a corrosion aggressive attack are not accessible to direct inspection, one is forced to rely on over-determined measurements performed on the accessible part of the boundary. In this study, we consider such a non-destructive inspection technique modelled as a shape optimization problem, which consists of determining an unknown part of the boundary of a simply-connected bounded domain. The solution of Laplace equation $u$ satisfies nonlinear Robin condition on an unknown part of boundaty. We have Dirichlet and Neumann boundary conditions on the rest of boundary. We look for a curve, that minimizes the cost functional represented as a sum of regularizing term and $L_2$-norm of the difference of the state variable $u$ and some additional measurement on the part of admissible boundary. The existence of at least one curve is proved for an appropriate choice of the class of admissible curves. This will lead us to study properties of continuity of J and compactness of the set of controls. We shall characterize the shape derivative of the cost functional with respect to perturbations of the domain defined by a sufficiently smooth function. In order obtain necessary optimality condition we use the mapping method. This method transforms the unknown domain to a fixed one. The problem under consideration is essentially nonlinear. We prove Frechet differentiability of the solution operator.

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40

Nawaz, Yasir, Muhammad Shoaib Arif, Wasfi Shatanawi, and Muhammad Usman Ashraf. "A Fourth Order Numerical Scheme for Unsteady Mixed Convection Boundary Layer Flow: A Comparative Computational Study." Energies 15, no. 3 (January 27, 2022): 910. http://dx.doi.org/10.3390/en15030910.

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In this paper, a three-stage fourth-order numerical scheme is proposed. The first and second stages of the proposed scheme are explicit, whereas the third stage is implicit. A fourth-order compact scheme is considered to discretize space-involved terms. The stability of the fourth-order scheme in space and time is checked using the von Neumann stability criterion for the scalar case. The stability region obtained by the scheme is more than the one given by explicit Runge–Kutta methods. The convergence conditions are found for the system of partial differential equations, which are non-dimensional equations of heat transfer of Stokes first and second problems. The comparison of the proposed scheme is made with the existing Crank–Nicolson scheme. From this comparison, it can be concluded that the proposed scheme converges faster than the Crank–Nicolson scheme. It also produces less relative error than the Crank–Nicolson method for time-dependent problems.

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41

Mazón, José M., Marcos Solera, and Julián Toledo. "Gradient flows in metric random walk spaces." SeMA Journal 79, no. 1 (October 10, 2021): 3–35. http://dx.doi.org/10.1007/s40324-021-00272-z.

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AbstractRecently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each $$x\in X$$ x ∈ X , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

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Farshbaf-Shaker, M. Hassan, and Christian Heinemann. "Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 2 (January 26, 2018): 579–603. http://dx.doi.org/10.1051/cocv/2017041.

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Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variables. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M.H. Farshbaf−Shaker and C. Heinemann, Math. Models Methods Appl. Sci. 25 (2015) 2749–2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.

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PAPADAKIS, JOHN S., and EVANGELIA T. FLOURI. "A NEUMANN TO DIRICHLET MAP FOR THE BOTTOM BOUNDARY OF A STRATIFIED SUB-BOTTOM REGION IN PARABOLIC APPROXIMATION." Journal of Computational Acoustics 16, no. 03 (September 2008): 409–25. http://dx.doi.org/10.1142/s0218396x08003658.

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The acoustic propagation problem is modeled via the parabolic approximation. The physical domain consists of the water column with a horizontal water–bottom interface and the bottom region consists of N-strata with horizontal interfaces. The computational domain is restricted to the water column, while the stratified bottom region is modeled by a nonlocal boundary condition applied along the water–bottom interface, and having the form of a Neumann to Dirichlet map (NtD). The discrete analog of the NtD has been implemented in a finite difference scheme for the general wide angle PE model, and successfully tested for several benchmark problems. The stratification of the media can be either physical, e.g. sediment formulation in the bottom, or artificial/computational, e.g. forced by sparse distribution of environmental data measurements in the water column. It should be emphasized that the sound speed may vary from layer to layer, but is constant within each layer. The proposed NtD map can be used in geoacoustic inversion via the optimal control adjoint method.

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Provotorov, Vyacheslav V., Sergey M. Sergeev, and Van Nguyen Hoang. "Countable stability of a weak solution of a parabolic differential-difference system with distributed parameters on the graph." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 16, no. 4 (2020): 402–14. http://dx.doi.org/10.21638/11701/spbu10.2020.405.

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The article proposes an analog of E. Rothe’s method (semi-discretization with respect to the time variable) for construction convergent different schemes when analyzing the countable stability of a weak solution of an initial boundary value problem of the parabolic type with distributed parameters on a graph in the class of summable functions. The proposed method leads to the study of the input initial boundary value problem to analyze the boundary value problem in a weak setting for elliptical type equations with distributed parameters on the graph. By virtue of the specifics of this method, the stability of a weak solution is understood in terms of the spectral criterion of stability (Neumann’s countable stability), which establishes the stability of the solution with respect to each harmonic of the generalized Fourier series of a weak solution or a segment of this series. Thus, there is another possibility indicated, in addition to the Faedo—Galerkin method, for constructing approaches to the desired solution of the initial boundary value problem, to analyze its stability and the way to prove the theorem of the existence of a weak solution to the input problem. The approach is applied to finding sufficient conditions for the countable stability of weak solutions to other initial boundary value problems with more general boundary conditions — in which elliptical equations are considered with the boundary conditions of the second or third type. Further analysis is possible to find the conditions under which Lyapunov stability is established. The approach can be used to analyze the optimal control problems, as well as the problems of stabilization and stability of differential systems with delay. Presented method of finite difference opens new ways for approximating the states of a parabolic system, analyzing their stability in the numerical implementation and algorithmization of optimal control problems.

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Halidias, N. "Neumann boundary value problems with discontinuities." Applied Mathematics Letters 16, no. 5 (July 2003): 729–32. http://dx.doi.org/10.1016/s0893-9659(03)00074-0.

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Dipierro, Serena, Xavier Ros-Oton, and Enrico Valdinoci. "Nonlocal problems with Neumann boundary conditions." Revista Matemática Iberoamericana 33, no. 2 (2017): 377–416. http://dx.doi.org/10.4171/rmi/942.

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Da Lio, Francesca, and Francesco Palmurella. "Remarks on Neumann boundary problems involving Jacobians." Communications in Partial Differential Equations 42, no. 10 (September 8, 2017): 1497–509. http://dx.doi.org/10.1080/03605302.2017.1377231.

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Szymanska-Debowska, Katarzyna. "Solutions to nonlocal Neumann boundary value problems." Electronic Journal of Qualitative Theory of Differential Equations, no. 28 (2018): 1–14. http://dx.doi.org/10.14232/ejqtde.2018.1.28.

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Anderson, D. R., I. Rachůnková, and C. C. Tisdell. "Solvability of discrete Neumann boundary value problems." Journal of Mathematical Analysis and Applications 331, no. 1 (July 2007): 736–41. http://dx.doi.org/10.1016/j.jmaa.2006.09.002.

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Moradifam, Amir. "Least gradient problems with Neumann boundary condition." Journal of Differential Equations 263, no. 11 (December 2017): 7900–7918. http://dx.doi.org/10.1016/j.jde.2017.08.031.

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