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Готові списки джерел за темами / Nonlocal Neumann boundary conditions / Статті в журналах

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Статті в журналах з теми "Nonlocal Neumann boundary conditions"

Автор: Grafiati

Опубліковано: 10 березня 2023

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1

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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2

You, Huaiqian, Xin Yang Lu, Nathaniel Trask, and Yue Yu. "An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems." ESAIM: Mathematical Modelling and Numerical Analysis 55 (2021): S811—S851. http://dx.doi.org/10.1051/m2an/2020058.

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Анотація:

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞(Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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3

You, Huaiqian, XinYang Lu, Nathaniel Task, and Yue Yu. "An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 4 (June 18, 2020): 1373–413. http://dx.doi.org/10.1051/m2an/2019089.

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Анотація:

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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4

Turmetov, B. Kh, and V. V. Karachik. "NEUMANN BOUNDARY CONDITION FOR A NONLOCAL BIHARMONIC EQUATION." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 14, no. 2 (2022): 51–58. http://dx.doi.org/10.14529/mmph220205.

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Анотація:

The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained. First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.

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5

Bogoya, Mauricio, and Cesar A. Gómez S. "On a nonlocal diffusion model with Neumann boundary conditions." Nonlinear Analysis: Theory, Methods & Applications 75, no. 6 (April 2012): 3198–209. http://dx.doi.org/10.1016/j.na.2011.12.019.

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6

Aksoylu, Burak, and Fatih Celiker. "Nonlocal problems with local Dirichlet and Neumann boundary conditions." Journal of Mechanics of Materials and Structures 12, no. 4 (May 20, 2017): 425–37. http://dx.doi.org/10.2140/jomms.2017.12.425.

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7

Gomez, C. A., and J. A. Caicedo. "ON A RESCALED NONLOCAL DIFFUSION PROBLEM WITH NEUMANN BOUNDARY CONDITIONS." Advances in Mathematics: Scientific Journal 10, no. 8 (August 7, 2021): 3013–22. http://dx.doi.org/10.37418/amsj.10.8.2.

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Анотація:

In this work, we consider the rescaled nonlocal diffusion problem with Neumann Boundary Conditions \[ \begin{cases} u_t^{\epsilon}(x,t)=\displaystyle\frac{1}{\epsilon^2} \int_{\Omega}J_{\epsilon}(x-y)(u^\epsilon(y,t)-u^\epsilon(x,t))dy\\ \qquad \qquad+\displaystyle\frac{1}{\epsilon}\int_{\partial \Omega}G_\epsilon(x-y)g(y,t)dS_y,\\ u^\epsilon(x,0)=u_0(x), \end{cases} \] where $\Omega\subset\mathbb{R}^{N}$ is a bounded, connected and smooth domain, $g$ a positive continuous function, $J_\epsilon(z)=C_1\frac{1}{\epsilon^N}J(\frac{z}{\epsilon}), G_\epsilon(x)=C_1\frac{1}{\epsilon^N}G(\frac{x}{\epsilon}),$ $J$ and $G$ well defined kernels, $C_1$ a normalization constant. The solutions of this model have been used without prove to approximate the solutions of a family of nonlocal diffusion problems to solutions of the respective analogous local problem. We prove existence and uniqueness of the solutions for this problem by using the Banach Fixed Point Theorem. Finally, some conclusions are given.

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8

Andreu, F., J. M. Mazón, J. D. Rossi, and J. Toledo. "A nonlocal p-Laplacian evolution equation with Neumann boundary conditions." Journal de Mathématiques Pures et Appliquées 90, no. 2 (August 2008): 201–27. http://dx.doi.org/10.1016/j.matpur.2008.04.003.

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9

Agarwal, Praveen, Jochen Merker, and Gregor Schuldt. "Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs." Axioms 10, no. 2 (April 24, 2021): 74. http://dx.doi.org/10.3390/axioms10020074.

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Анотація:

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

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10

Karachik, Valery, Batirkhan Turmetov, and Hongfen Yuan. "Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball." Mathematics 10, no. 7 (April 3, 2022): 1158. http://dx.doi.org/10.3390/math10071158.

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Анотація:

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green’s functions.

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11

Zhang, Wei, Jiang Yang, Jiwei Zhang, and Qiang Du. "Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain." Communications in Computational Physics 21, no. 1 (December 5, 2016): 16–39. http://dx.doi.org/10.4208/cicp.oa-2016-0033.

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AbstractThis paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain. Two classes of artificial boundary conditions (ABCs) are designed, namely, nonlocal analog Dirichlet-to-Neumann-type ABCs (global in time) and high-order Padé approximate ABCs (local in time). These ABCs reformulate the original problem into an initial-boundary-value (IBV) problem on a bounded domain. For the global ABCs, we adopt a fast evolution to enhance computational efficiency and reduce memory storage. High order fully discrete schemes, both second-order in time and space, are given to discretize two reduced problems. Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

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12

Boussaïd, Samira. "UNIVERSAL ATTRACTOR FOR A NONLOCAL REACTION-DIFFUSION PROBLEM WITH DYNAMICAL BOUNDARY CONDITIONS." Advances in Mathematics: Scientific Journal 11, no. 9 (September 29, 2022): 789–801. http://dx.doi.org/10.37418/amsj.11.9.4.

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Анотація:

A nonlocal reaction-diffusion equation is presented in this article, based on a model proposed by J. Rubinstein and P. Sternberg [6] with a nonlinear strictly monotone operator. A dynamical boundary condition is considered, rather then the usual ones such as Neumann or Dirichlet boundary conditions. The well-posedness and the existence of a universal attractor of this problem, which describes the long time behavior of the solution, are established.

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13

Yildirim, Ozgur. "On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9, no. 1 (January 31, 2019): 60–72. http://dx.doi.org/10.11121/ijocta.01.2019.00592.

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In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed.

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14

Montagu, E. L., and John Norbury. "Solution Structure for Nonautonomous Nonlocal Elliptic Equations with Neumann Boundary Conditions." Integral Transforms and Special Functions 13, no. 5 (January 1, 2002): 461–70. http://dx.doi.org/10.1080/10652460213527.

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15

Barles, Guy, Christine Georgelin, and Espen R. Jakobsen. "On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations." Journal of Differential Equations 256, no. 4 (February 2014): 1368–94. http://dx.doi.org/10.1016/j.jde.2013.11.001.

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16

Chabrowski, J. "On bi-nonlocal problem for elliptic equations with Neumann boundary conditions." Journal d'Analyse Mathématique 134, no. 1 (February 2018): 303–34. http://dx.doi.org/10.1007/s11854-018-0011-5.

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17

Patlashenko, Igor, and Dan Givoli. "Non-Reflecting Finite Element Schemes for Three-Dimensional Acoustic Waves." Journal of Computational Acoustics 05, no. 01 (March 1997): 95–115. http://dx.doi.org/10.1142/s0218396x97000071.

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The finite element solution of problems involving three-dimensional acoustic waves in an infinite wave guide, and in the infinite medium around a structure is considered. Such problems are typical in structural acoustics, and this paper concentrates on the efficient numerical treatment of the infinite acoustic medium away from the structure. The unbounded domain is truncated by means of an artificial boundary ℬ. On ℬ, non-reflecting boundary conditions are used; these are either nonlocal Dirichlet-to-Neumann conditions, or their localized counterparts. For the high-order localized conditions, special three-dimensional finite elements are constructed for use in the layer adjacent to ℬ. The performance of the nonlocal and localized boundary conditions is compared via numerical experiments involving a three-dimensional wave guide.

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18

Niculescu, Constantin P., and Ionel Rovenţa. "Large Solutions for Semilinear Parabolic Equations Involving Some Special Classes of Nonlinearities." Discrete Dynamics in Nature and Society 2010 (2010): 1–11. http://dx.doi.org/10.1155/2010/491023.

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Анотація:

We consider a new class of nonlinearities for which a nonlocal parabolic equation with Neumann boundary conditions has finite time blow-up solutions. Our approach is inspired by previous work done by Jazar and Kiwan (2008) and El Soufi et al. (2007).

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19

Andreu, F., J. M. Mazón, J. D. Rossi, and J. Toledo. "Local and nonlocal weighted $p$-Laplacian evolution equations with Neumann boundary conditions." Publicacions Matemàtiques 55 (January 1, 2011): 27–66. http://dx.doi.org/10.5565/publmat_55111_03.

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20

Yang, Fei-Ying, Wan-Tong Li, and Shigui Ruan. "Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions." Journal of Differential Equations 267, no. 3 (July 2019): 2011–51. http://dx.doi.org/10.1016/j.jde.2019.03.001.

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21

Turmetov, Batirkhan, Valery Karachik, and Moldir Muratbekova. "On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions." Mathematics 9, no. 17 (August 24, 2021): 2020. http://dx.doi.org/10.3390/math9172020.

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Анотація:

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

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22

Koshanova, M., М. Muratbekova, and B. Turmetov. "SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION FOR THE NONLOCAL POISSON EQUATION." BULLETIN Series of Physics & Mathematical Sciences 71, no. 3 (September 30, 2020): 74–83. http://dx.doi.org/10.51889/2020-3.1728-7901.10.

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Анотація:

In this paper, we study new classes of boundary value problems for a nonlocal analogue of the Poisson equation. The boundary conditions, as well as the nonlocal Poisson operator, are specified using transformation operators with orthogonal matrices. The paper investigates the questions of solvability of analogues of boundary value problems of the Dirichlet and Neumann type. It is proved that, as in the classical case, the analogue of the Dirichlet problem is unconditionally solvable. For it, theorems on the existence and uniqueness of the solution to the problem are proved. An explicit form of the Green's function, a generalized Poisson kernel, and an integral representation of the solution are found. For an analogue of the Neumann problem, an exact solvability condition is found in the form of a connection between integrals of given functions. The Green's function and an integral representation of the solution of the problem under study are also constructed.

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23

Afrouzi, Ghasem A., Z. Naghizadeh, and Nguyen Thanh Chung. "Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions." Boletim da Sociedade Paranaense de Matemática 40 (January 18, 2022): 1–11. http://dx.doi.org/10.5269/bspm.44144.

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Анотація:

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

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24

Heidari, Samira, and Abdolrahman Razani. "Infinitely many solutions for nonlocal elliptic systems in Orlicz–Sobolev spaces." Georgian Mathematical Journal 29, no. 1 (October 28, 2021): 45–54. http://dx.doi.org/10.1515/gmj-2021-2110.

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Abstract Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory.

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25

Karachik, Valery, and Batirkhan Turmetov. "On solvability of some nonlocal boundary value problems for biharmonic equation." Mathematica Slovaca 70, no. 2 (April 28, 2020): 329–42. http://dx.doi.org/10.1515/ms-2017-0355.

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Анотація:

Abstract In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

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26

Bouziani, Abdelfatah. "Initial-boundary value problem with a nonlocal condition for a viscosity equation." International Journal of Mathematics and Mathematical Sciences 30, no. 6 (2002): 327–38. http://dx.doi.org/10.1155/s0161171202004167.

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Анотація:

This paper deals with the proof of the existence, uniqueness, and continuous dependence of a strong solution upon the data, for an initial-boundary value problem which combine Neumann and integral conditions for a viscosity equation. The proof is based on an energy inequality and on the density of the range of the linear operator corresponding to the abstract formulation of the studied problem.

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27

Bogoya, Mauricio, Raul Ferreira, and Julio D. Rossi. "Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models." Proceedings of the American Mathematical Society 135, no. 12 (December 1, 2007): 3837–47. http://dx.doi.org/10.1090/s0002-9939-07-09205-2.

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28

Slepčev, Dejan. "Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions." Nonlinear Analysis: Theory, Methods & Applications 52, no. 1 (January 2003): 79–115. http://dx.doi.org/10.1016/s0362-546x(02)00098-6.

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29

Gómez, Cesar A., and Julio D. Rossi. "A nonlocal diffusion problem that approximates the heat equation with Neumann boundary conditions." Journal of King Saud University - Science 32, no. 1 (January 2020): 17–20. http://dx.doi.org/10.1016/j.jksus.2017.08.008.

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30

Liao, Menglan, and Wenjie Gao. "Blow-up phenomena for a nonlocal p-Laplace equation with Neumann boundary conditions." Archiv der Mathematik 108, no. 3 (November 15, 2016): 313–24. http://dx.doi.org/10.1007/s00013-016-0986-z.

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31

PATLASHENKO, IGOR, and DAN GIVOLI. "OPTIMAL LOCAL NONREFLECTING BOUNDARY CONDITIONS FOR TIME-DEPENDENT WAVES." Journal of Computational Acoustics 08, no. 01 (March 2000): 157–70. http://dx.doi.org/10.1142/s0218396x00000108.

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Анотація:

Nonreflecting Boundary Conditions (NRBCs) are often used on artificial boundaries as a method for the numerical solution of wave problems in unbounded domains. Recently, a two-parameter hierarchy of optimal local NRBCs of increasing order has been developed for elliptic problems, including the problem of time-harmonic acoustic waves. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition in the L2 norm for functions which can be Fourier-decomposed. The optimal NRBCs are combined with finite element discretization in the computational domain. Here this approach is extended to time-dependent acoustic waves. In doing this, the Semi-Discrete DtN approach is used as the starting point. Numerical examples involving propagating disturbances in two dimensions are given.

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32

Ashyralyyev, C., and A. Cay. "Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 5–17. http://dx.doi.org/10.31489/2020m3/5-17.

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Анотація:

In modeling various real processes, an important role is played by methods of solution source identification problem for partial differential equation. The current paper is devoted to approximate of elliptic over determined problem with integral condition for derivatives. In the beginning, inverse problem is reduced to some auxiliary nonlocal boundary value problem with integral boundary condition for derivatives. The parameter of equation is defined after solving that auxiliary nonlocal problem. The second order of accuracy difference scheme for approximately solving abstract elliptic overdetermined problem is proposed. By using operator approach existence of solution difference problem is proved. For solution of constructed difference scheme stability and coercive stability estimates are established. Later, obtained abstract results are applied to get stability estimates for solution Neumann-type overdetermined elliptic multidimensional difference problems with integral conditions. Finally, by using MATLAB program, we present numerical results for two dimensional and three dimensional test examples with short explanation on realization on computer.

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33

Esposito, Giampiero, and Giuseppe Pollifrone. "Noncovariant Gauges in Simple Supergravity." International Journal of Modern Physics D 06, no. 04 (August 1997): 479–90. http://dx.doi.org/10.1142/s0218271897000285.

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Анотація:

A gauge-averaging functional of the axial type is studied for simple supergravity at one loop about flat Euclidean four-space bounded by a three-sphere, or two concentric three-spheres. This is a generalization of recent work on the axial gauge in quantum supergravity on manifolds with boundary. Ghost modes obey nonlocal boundary conditions of the spectral type, in that half of them obey Dirichlet or Neumann conditions at the boundary. In both cases, they give a vanishing contribution to the one-loop divergence. The admissibility of noncovariant gauges at the classical level is also proved.

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34

Yuan, Yueding, and Zhiming Guo. "Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/378172.

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We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

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35

Xiang, Zhaoyin, Qiong Chen, and Chunlai Mu. "Blowup properties for several diffusion systems with localised sources." ANZIAM Journal 48, no. 1 (July 2006): 37–56. http://dx.doi.org/10.1017/s1446181100003400.

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Анотація:

AbstractThis paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

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36

Hameed, Raad, Boying Wu, and Jiebao Sun. "Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions." Boundary Value Problems 2013, no. 1 (2013): 34. http://dx.doi.org/10.1186/1687-2770-2013-34.

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37

Cortazar, Carmen, Manuel Elgueta, Julio D. Rossi, and Noemi Wolanski. "How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems." Archive for Rational Mechanics and Analysis 187, no. 1 (November 3, 2007): 137–56. http://dx.doi.org/10.1007/s00205-007-0062-8.

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38

Qu, Chengyuan, Xueli Bai, and Sining Zheng. "Blow-up versus extinction in a nonlocal p -Laplace equation with Neumann boundary conditions." Journal of Mathematical Analysis and Applications 412, no. 1 (April 2014): 326–33. http://dx.doi.org/10.1016/j.jmaa.2013.10.040.

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39

Ahmad, Bashir, Sotiris K. Ntouyas, and Jessada Tariboon. "Fractional Differential Equations with Nonlocal Integral and Integer–Fractional-Order Neumann Type Boundary Conditions." Mediterranean Journal of Mathematics 13, no. 5 (September 21, 2015): 2365–81. http://dx.doi.org/10.1007/s00009-015-0629-9.

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40

Wang, Yulan, Zhaoyin Xiang, and Jinsong Hu. "Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption." Journal of Applied Mathematics 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/648067.

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Анотація:

We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.

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41

Tian, Huimin, and Lingling Zhang. "Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions." Open Mathematics 18, no. 1 (January 1, 2020): 1552–64. http://dx.doi.org/10.1515/math-2020-0088.

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Анотація:

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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42

Turkyilmazoglu, Mustafa. "Hyperbolic Partial Differential Equations with Nonlocal Mixed Boundary Values and their Analytic Approximate Solutions." International Journal of Computational Methods 15, no. 02 (September 28, 2017): 1850003. http://dx.doi.org/10.1142/s0219876218500032.

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Анотація:

Partial differential equations of hyperbolic type when considered with mixed Dirichlet/Neumann constraints as well as nonlocal conservation conditions model many physical phenomena. The prime motivation of the current work is to apply the recently developed meshfree method to such differential equations. The scheme is built on series expansion of the solution via proper base functions akin to the Galerkin approach. In many cases, the simple polynomials are adequate to convert the hyperbolic partial differential equation and boundary conditions of nonlocal kind into easily treatable algebraic equations concerning the coefficients of the series. If the sought solutions are polynomials of any degree, then the method has the ability of resolving the equations in an exact manner. The validity, applicability, accuracy and performance of the method are illustrated on some well-analyzed hyperbolic equations available in the open literature.

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43

Bouziani, Abdelfatah. "On the Solvability of a Nonlocal Problem Arising in Dynamics of Moisture Transfer." gmj 10, no. 4 (December 2003): 607–22. http://dx.doi.org/10.1515/gmj.2003.607.

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Abstract In the recent years, evolution problems with an integral term in the boundary conditions have received a great deal of attention. Such problems, in general, are nonself-adjoint, and this poses the basic source of difficulty, which can considerably complicate the application of standard functional and numerical techniques. To avoid these complications, we have introduced a nonclassical function space to establish a priori estimates without any additional complication as compared to the classical evolution problems. As an example of the applicability of this way of solving problems of this type, we investigate an initial-boundary value problem for a pseudoparabolic equation which combines Neumann and integral conditions.

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44

Boni, Théodore K., and Diabaté Nabongo. "Quenching for Discretizations of a Nonlocal Parabolic Problem with Neumann Boundary Condition." Cubo (Temuco) 12, no. 1 (2010): 23–40. http://dx.doi.org/10.4067/s0719-06462010000100004.

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45

Poláčik, Peter, and Vladimir Šošovička. "Stable periodic solutions of a spatially homogeneous nonlocal reaction–diffusion equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 867–84. http://dx.doi.org/10.1017/s0308210500023118.

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Анотація:

Nonlocal reaction–diffusion equations of the form ut = uxx + F(u, α(u)), where are considered together with Neumann or Dirichlet boundary conditions. One of the main results deals with linearisation at equilibria. It states that, for any given set of complex numbers, one can arrange, choosing the equation properly, that this set is contained in the spectrum of the linearisation. The second main result shows that equations of the above form can undergo a supercritical Hopf bifurcation to an asymptotically stable periodic solution.

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46

Berikelashvili, Givi, and Nodar Khomeriki. "On a numerical solution of one nonlocal boundary-value problem with mixed Dirichlet–Neumann conditions." Lithuanian Mathematical Journal 53, no. 4 (October 2013): 367–80. http://dx.doi.org/10.1007/s10986-013-9214-8.

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47

Turmetov, Batirkhan, Maira Koshanova, and Moldir Muratbekova. "On some analogues of periodic problems for Laplace equation with an oblique derivative under boundary conditions." e-Journal of Analysis and Applied Mathematics 2020, no. 1 (January 1, 2020): 13–27. http://dx.doi.org/10.2478/ejaam-2020-0002.

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AbstractIn this paper, we study solvability of new classes of nonlocal boundary value problems for the Laplace equation in a ball. The considered problems are multidimensional analogues (in the case of a ball) of classical periodic boundary value problems in rectangular regions. To study the main problem, first, for the Laplace equation, we consider an auxiliary boundary value problem with an oblique derivative. This problem generalizes the well-known Neumann problem and is conditionally solvable. The main problems are solved by reducing them to sequential solution of the Dirichlet problem and the problem with an oblique derivative. It is proved that in the case of periodic conditions, the problem is conditionally solvable; and in this case the exact condition for solvability of the considered problem is found. When boundary conditions are specified in the anti-periodic conditions form, the problem is certainly solvable. The obtained general results are illustrated with specific examples.

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48

Zaheer-ud-Din, Muhammad Ahsan, Masood Ahmad, Wajid Khan, Emad E. Mahmoud, and Abdel-Haleem Abdel-Aty. "Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media." Mathematics 8, no. 11 (November 17, 2020): 2045. http://dx.doi.org/10.3390/math8112045.

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Анотація:

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

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49

Ashyralyev, Allaberen, and Elif Ozturk. "The Numerical Solution of the Bitsadze-Samarskii Nonlocal Boundary Value Problems with the Dirichlet-Neumann Condition." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/730804.

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Анотація:

We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem with the Dirichlet-Neumann condition for the multidimensional elliptic equation. The first and second orders of accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

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50

Turmetov, B. Kh, and V. V. Karachik. "On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 651–67. http://dx.doi.org/10.35634/vm210409.

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Анотація:

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

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