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Journal articles on the topic 'Homogeneous Neumann boundary conditions'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 March 2023

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1

Jiang, Jun, Jinfeng Wang, and Yingwei Song. "The Influence of Dirichlet Boundary Conditions on the Dynamics for a Diffusive Predator–Prey System." International Journal of Bifurcation and Chaos 29, no. 09 (2019): 1950113. http://dx.doi.org/10.1142/s021812741950113x.

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A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

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2

Baddour, R. E., and W. Parsons. "A Comparison of Dirichlet and Neumann Wavemakers for the Numerical Generation and Propagation of Nonlinear Long-Crested Surface Waves." Journal of Offshore Mechanics and Arctic Engineering 126, no. 4 (2004): 287–96. http://dx.doi.org/10.1115/1.1835987.

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We are studying numerically the problem of generation and propagation of long-crested gravity waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite depth. A nonorthogonal curvilinear coordinate system, which follows the free surface, is constructed and the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm. “Wavemakers” are modeled using both the Dirichlet and Neumann lateral boundary conditions and a full comparison is given. Overall, the Dirichlet model was more stable than the Neumann model, with an upper limit steepness S=2A/λ of 0.08 using good resolution compared with the Neumann’s maximum of 0.05.

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3

Fiebig-Wittmaack, Mclitta. "Multiplicity of solutions of a nonlinear boundary problem with homogeneous neumann boundary conditions." Applicable Analysis 29, no. 3-4 (1988): 253–68. http://dx.doi.org/10.1080/00036818808839784.

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4

ENACHE, CRISTIAN. "LOWER BOUNDS FOR BLOW-UP TIME IN SOME NON-LINEAR PARABOLIC PROBLEMS UNDER NEUMANN BOUNDARY CONDITIONS." Glasgow Mathematical Journal 53, no. 3 (2011): 569–75. http://dx.doi.org/10.1017/s0017089511000139.

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AbstractThis paper deals with some non-linear initial-boundary value problems under homogeneous Neumann boundary conditions, in which the solutions may blow up in finite time. Using a first-order differential inequality technique, lower bounds for blow-up time are determined.

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5

Otero, Jose, Ernesto Hernandez, Ruben Santiago, Raul Martinez, Francisco Castillo, and Joaquin Oseguera. "Non-parabolic interface motion for the one-dimensional Stefan problem: Neumann boundary conditions." Thermal Science 21, no. 6 Part B (2017): 2699–708. http://dx.doi.org/10.2298/tsci151218311o.

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In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.

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6

YIN, Zhaoyang. "Global existence for quasilinear parabolic systems with homogeneous Neumann boundary conditions." Nonlinear Differential Equations and Applications NoDEA 13, no. 2 (2006): 235–48. http://dx.doi.org/10.1007/s00030-005-0038-z.

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7

Solà-Morales, J., and M. València. "Trend to spatial homogeneity for solutions to semilinear damped wave equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 105, no. 1 (1987): 117–26. http://dx.doi.org/10.1017/s0308210500021958.

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SynopsisThe semilinear damped wave equationssubject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

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8

Amo-Navarro, Jesús, Ricardo Vinuesa, J. Alberto Conejero, and Sergio Hoyas. "Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives." Mathematics 9, no. 19 (2021): 2508. http://dx.doi.org/10.3390/math9192508.

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In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows.

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9

Kirk, Colleen M., and W. Edward Olmstead. "Thermal blow-up in a finite strip with superdiffusive properties." Fractional Calculus and Applied Analysis 21, no. 4 (2018): 949–59. http://dx.doi.org/10.1515/fca-2018-0052.

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Abstract We investigate the problem of a high-energy source localized within a one-dimensional superdiffusive medium of finite length. The problem is modeled by a fractional diffusion equation with a nonlinear source term. For the boundary conditions, we consider both the case of homogeneous Dirichlet conditions and the case of homogeneous Neumann conditions. We investigate this model to determine whether or not blow-up occurs. It is demonstrated that a blow-up may or may not occur for the Dirichlet case. On the other hand, a blow-up is unavoidable for the Neumann case.

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10

Subani, Norazlina, Muhammad Aniq Qayyum Mohamad Sukry, Muhammad Arif Hannan, Faizzuddin Jamaluddin, and Ahmad Danial Hidayatullah Badrolhisam. "Analysis of Different Boundary Conditions on Homogeneous One-Dimensional Heat Equation." Malaysian Journal of Science Health & Technology 7, no. 1 (2021): 15–21. http://dx.doi.org/10.33102/mjosht.v7i1.153.

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Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

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11

Рудаков, Игорь, Igor Rudakov, Алексей Лукавый, and Aleksey Lukavyy. "Periodic solutions of a quasilinear wave equations with variable coefficients." Bulletin of Bryansk state technical university 2014, no. 3 (2014): 147–55. http://dx.doi.org/10.12737/23145.

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12

Gromyk, A., I. Konet, and T. Pylypyuk. "PARABOLIC BOUNDARY VALUE PROBLEMS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED SOLID CYLINDER." Bukovinian Mathematical Journal 8, no. 2 (2020): 40–55. http://dx.doi.org/10.31861/bmj2020.02.04.

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The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous wedge-shaped solid cylinder were constructed at first time by the method of integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices). The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable $\varphi \in (0; \varphi_0)$, a Fourier integral transform on the Cartesian segment $(-l_1;l_2)$ by an applicative variable $z$ and a hybrid integral transform of the Hankel type of the first kind on a segment $(0;R)$ of the polar axis with $n$ points of conjugation by an radial variable $r$ were used to construct solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image. The structure of the solution of the problem in the case of setting the Neumann boundary conditions on the wedge edges is analyzed. Exact analytical formulas for the components of the main solutions are written and the theorem on the existence of a single bounded classical solution of the problem is formulated. The obtained solutions are algorithmic in nature and can be used (using numerical methods) in solving applied problems.

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13

Htoo, Khin Phyu Phyu, and Eiji Yanagida. "Singular solutions of a superlinear parabolic equation with homogeneous Neumann boundary conditions." Nonlinear Analysis: Theory, Methods & Applications 151 (March 2017): 96–108. http://dx.doi.org/10.1016/j.na.2016.11.015.

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14

Subani, Norazlina, Faizzuddin Jamaluddin, Muhammad Arif Hannan Mohamed, and Ahmad Danial Hidayatullah Badrolhisam. "Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions." Journal of Physics: Conference Series 1551 (May 2020): 012002. http://dx.doi.org/10.1088/1742-6596/1551/1/012002.

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15

Kita, Kosuke, and Mitsuharu Ôtani. "On a comparison theorem for parabolic equations with nonlinear boundary conditions." Advances in Nonlinear Analysis 11, no. 1 (2022): 1165–81. http://dx.doi.org/10.1515/anona-2022-0239.

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Abstract In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

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16

Lasiecka, I., and R. Triggiani. "Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data." Journal of Differential Equations 94, no. 1 (1991): 112–64. http://dx.doi.org/10.1016/0022-0396(91)90106-j.

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17

Pereira, Antonio L. "Generic hyperbolicity for scalar parabolic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, no. 6 (1993): 1031–40. http://dx.doi.org/10.1017/s0308210500029711.

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SynopsisFor the reaction diffusion equationwith homogeneous Neumann boundary conditions, we give results on the generic hyperbolicity of equilibria with respect to a for fixed f and with respect to f for fixed a.

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18

Ma, Heping. "Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli." Discrete Dynamics in Nature and Society 2021 (August 5, 2021): 1–13. http://dx.doi.org/10.1155/2021/9323338.

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In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

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19

Guo, Gai Hui, and Bing Fang Li. "Turing Instability and Hopf Bifurcation for the General Brusselator System." Advanced Materials Research 255-260 (May 2011): 2126–30. http://dx.doi.org/10.4028/www.scientific.net/amr.255-260.2126.

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The Brusselator system subject to homogeneous Neumann boundary conditions is investigated. It is firstly shown that the homogeneous equilibrium solution becomes Turing unstable or diffusively unstable when parameters are chosen properly. Then the existence of Hopf bifurcation to the ODE and PDE models is obtained. Examples of numerical simulations are also shown to support and supplement the analytical results.

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20

Goudjo, Côme, Babacar Lèye, and Mamadou Sy. "Weak Solution to a Parabolic Nonlinear System Arising in Biological Dynamic in the Soil." International Journal of Differential Equations 2011 (2011): 1–24. http://dx.doi.org/10.1155/2011/831436.

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We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction-diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.

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21

d'Avenia, Pietro, Lorenzo Pisani, and Gaetano Siciliano. "Nonautonomous Klein–Gordon–Maxwell systems in a bounded domain." Advances in Nonlinear Analysis 3, S1 (2014): s37—s45. http://dx.doi.org/10.1515/anona-2014-0009.

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AbstractThis paper deals with the Klein–Gordon–Maxwell system in a bounded spatial domain with a nonuniform coupling. We discuss the existence of standing waves in equilibrium with a purely electrostatic field, assuming homogeneous Dirichlet boundary conditions on the matter field and nonhomogeneous Neumann boundary conditions on the electric potential. Under suitable conditions we prove existence and nonexistence results. Since the system is variational, we use Ljusternik–Schnirelmann theory.

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22

Lankeit, Johannes. "Long-term behaviour in a chemotaxis-fluid system with logistic source." Mathematical Models and Methods in Applied Sciences 26, no. 11 (2016): 2071–109. http://dx.doi.org/10.1142/s021820251640008x.

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We consider the coupled chemotaxis Navier–Stokes model with logistic source terms: [Formula: see text] [Formula: see text] [Formula: see text] in a bounded, smooth domain [Formula: see text] under homogeneous Neumann boundary conditions for [Formula: see text] and [Formula: see text] and homogeneous Dirichlet boundary conditions for [Formula: see text] and with given functions [Formula: see text] satisfying certain decay conditions and [Formula: see text] for some [Formula: see text]. We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state [Formula: see text].

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23

Payne, L. E., and G. A. Philippin. "Blow-up phenomena in parabolic problems with time-dependent coefficients under Neumann boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 3 (2012): 625–31. http://dx.doi.org/10.1017/s0308210511000485.

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This paper deals with the blow-up of solutions to a class of parabolic problems with time-dependent coefficients under homogeneous Neumann boundary conditions. For one set of problems in this class we show that no global solution can exist. For another we derive lower bounds for the time of blow-up when blow-up occurs.

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24

Metcalfe, Jason, and Jacob Perry. "Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions." Communications on Pure & Applied Analysis 11, no. 2 (2012): 547–56. http://dx.doi.org/10.3934/cpaa.2012.11.547.

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25

Hussein, S. O., and Taysir E. Dyhoum. "Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions." Applied Mathematics and Computation 430 (October 2022): 127285. http://dx.doi.org/10.1016/j.amc.2022.127285.

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26

Godoy, Tomas. "Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions." Opuscula Mathematica 43, no. 1 (2023): 19–46. http://dx.doi.org/10.7494/opmath.2023.43.1.19.

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Let \(\Omega\) be a \(C^{2}\) bounded domain in \(\mathbb{R}^{n}\) such that \(\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\), where \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint closed subsets of \(\partial\Omega\), and consider the problem\(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\tau\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=\eta\) on \(\Gamma_{2}\), where \(0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})\), \(\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}\), and \(g:\Omega \times(0,\infty)\rightarrow\mathbb{R}\) is a nonnegative Carath�odory function. Under suitable assumptions on \(g\) and \(\eta\) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow \(g\) to be singular at \(s=0\) and also at \(x\in S\) for some suitable subsets \(S\subset\overline{\Omega}\). The Dirichlet problem \(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\sigma\) on \(\partial\Omega\) is also studied in the case when \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\).

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27

Gharahi, Alireza, and Peter Schiavone. "The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects." Mathematics and Mechanics of Solids 26, no. 1 (2020): 30–44. http://dx.doi.org/10.1177/1081286520935508.

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We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

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28

Algazin, O. D., and A. V. Kopaev. "A Mixed Boundary Value Problem for the Laplace Equation in a semi-infinite Layer." Mathematics and Mathematical Modeling, no. 5 (February 6, 2021): 1–12. http://dx.doi.org/10.24108/mathm.0520.0000229.

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The paper offers a solution of the mixed Dirichlet-Neumann and Dirichlet-Neumann-Robin boundary value problems for the Laplace equation in the semi-infinite layer, using the previously obtained solution of the mixed Dirichlet-Neumann boundary value problem for a layer.The functions on the right-hand sides of the boundary conditions are considered to be functions of slow growth, in particular, polynomials. The solution to boundary value problems is also sought in the class of functions of slow growth. Continuing the functions on the right-hand sides of the boundary conditions on the upper and lower sides of the semi-infinite layer from the semi-hyperplane to the entire hyperplane, we obtain the Dirichlet-Neumann problem for the layer, the solution of which is known and written in the form of a convolution. If the right-hand sides of the boundary conditions are polynomials, then the solution is also a polynomial. To the solution obtained it is necessary to add the solution of the problem for a semi-infinite layer with homogeneous boundary conditions on the upper and lower sides and with an inhomogeneous boundary condition of Dirichlet, Neumann or Robin on the lateral side. This solution is written as a series. If we take a finite segment of the series, then we obtain a solution that exactly satisfies the Laplace equation and the boundary conditions on the upper and lower sides of the semi-infinite layer and approximately satisfies the boundary condition on the lateral side.An example of solving the Dirichlet-Neumann and Dirichlet-Neumann-Robin problems is considered, describing the temperature field of a semi-infinite plate the upper side of which is heat-isolated, on the lower side the temperature is set in the form of a polynomial, and the lateral side is either heat-isolated, or holds a zero temperature, or has heat exchange with a zero-temperature environment. For the first two Dirichlet-Neumann problems, the solution is obtained in the form of polynomials. For the third Dirichlet-Neumann-Robin problem, the solution is obtained as a sum of a polynomial and a series. If in this solution the series is replaced by a finite segment, then an approximate solution of the problem will be obtained, which approximately satisfies the Robin condition on the lateral side of the semi-infinite layer.

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29

Klimek, Malgorzata, Mariusz Ciesielski, and Tomasz Blaszczyk. "Exact and Numerical Solution of the Fractional Sturm–Liouville Problem with Neumann Boundary Conditions." Entropy 24, no. 2 (2022): 143. http://dx.doi.org/10.3390/e24020143.

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In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. Finally, we present the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule.

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30

Mihăilescu, Mihai, та Gheorghe Moroşanu. "Eigenvalues of −Δp − Δq Under Neumann Boundary Condition". Canadian Mathematical Bulletin 59, № 3 (2016): 606–16. http://dx.doi.org/10.4153/cmb-2016-025-2.

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AbstractThe eigenvalue problem −Δpu − Δqu = λ|u|q−2u with p ∊ (1,∞), q ∊ (2,∞), p ≠ q subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ℝN with N ≥ 2. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval (λ1, ∞) plus an isolated point λ = 0. This comprehensive result is strongly related to our framework, which is complementary to the well-known case p = q ≠ 2 for which a full description of the set of eigenvalues is still unavailable.

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31

FAGUNDES, F. N., T. L. ANTONACCI OAKES, B. B. DILEM, and J. A. NOGUEIRA. "ON THE EFFECTS OF THE NEUMANN BOUNDARY CONDITIONS IN THE COLEMAN–WEINBERG MECHANISM." International Journal of Modern Physics A 25, no. 07 (2010): 1389–403. http://dx.doi.org/10.1142/s0217751x10048093.

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We investigate the effects of the homogeneous Neumann boundary conditions in the scalar electrodynamics with self-interaction. We show that if the length of the finite region is small enough ([Formula: see text], where Mϕ is the mass of the scalar field generated by the Coleman–Weinberg mechanism) the spontaneous symmetry breaking will not be induced and the vector field will not develop mass, however the scalar field will.

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32

Kawohl, Bernhard. "On the convexity and symmetry of solutions to an elliptic free boundary problem." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 42, no. 1 (1987): 57–68. http://dx.doi.org/10.1017/s1446788700033954.

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AbstractWe study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

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33

Yan, Shuling, Xinze Lian, Weiming Wang, and Youbin Wang. "Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model." Discrete Dynamics in Nature and Society 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/170501.

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We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

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34

Korzyuk, V. I., and J. V. Rudzko. "Method of reflections for the Klein–Gordon equation." Doklady of the National Academy of Sciences of Belarus 66, no. 3 (2022): 263–68. http://dx.doi.org/10.29235/1561-8323-2022-66-3-263-268.

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Using the method of reflections, the solutions of the first and second mixed problem for the homogenous Klein–Gordon equation in a quarter plane and of the first mixed problem for the homogenous Klein–Gordon equation in a halfstrip are written out in an explicit analytical form. The Cauchy conditions of these problems are inhomogeneous, but the Dirichlet boundary condition (or the Neumann boundary condition) is homogeneous. Conditions are formulated, under which the solutions to these problems are classical.

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35

Marinoschi, Gabriela. "Well-posedness of singular diffusion equations in porous media with homogeneous Neumann boundary conditions." Nonlinear Analysis: Theory, Methods & Applications 72, no. 7-8 (2010): 3491–514. http://dx.doi.org/10.1016/j.na.2009.12.033.

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36

Revelli, R., and L. Ridolfi. "Generalized collocation method for two-dimensional reaction-diffusion problems with homogeneous Neumann boundary conditions." Computers & Mathematics with Applications 56, no. 9 (2008): 2360–70. http://dx.doi.org/10.1016/j.camwa.2008.05.041.

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37

Tellini, Andrea. "High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions." Journal of Mathematical Analysis and Applications 467, no. 1 (2018): 673–98. http://dx.doi.org/10.1016/j.jmaa.2018.07.034.

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38

Ouedraogo, Arouna, and Tiyamba Valea. "Existence and uniqueness of renormalized solution to multivalued homogeneous Neumann problem with L1-data." Gulf Journal of Mathematics 13, no. 2 (2022): 42–66. http://dx.doi.org/10.56947/gjom.v13i2.590.

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In this paper, we discuss the existence and uniqueness of renormalized solution to nonlinear multivalued elliptic problem β(u)-div a(x, Du) ∈ f with homogeneous Neumann boundary conditions and L1 -data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. Some a-priori estimates are used to obtain our results.

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39

Hromyk, Andrii, Ivan Konet, and Tetiana Pylypiuk. "Parabolic Boundary Value Problems in a Piecewise Homogeneous Wedge-Shaped Cylindrical-Circular Layer with a Cavity." Mathematical and computer modelling. Series: Physical and mathematical sciences 22 (December 6, 2022): 14–29. http://dx.doi.org/10.32626/2308-5878.2022-23.14-29.

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The unique exact analytical solutions of parabolic boundary value prob-lems of mathematical physics in piecewise homogeneous by the radial variable zwedge-shaped by the angular variable cylindrical-circular layer with a cavity were constructed at first time by the method of classical integral and hybrid in-tegral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article.The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible com-binations (Dirichlet—Neumann, Neumann—Dirichlet) are considered.Finite integral Fourier transform by an angular variable, a finite inte-gral Fourier transform on the Cartesian segment by an applicative variable and a hybrid integral transform of the Weber type on the polar axis with npoints of conjugation by a radial variable were used to construct solutions of investigated boundary value problems.The consistent application of integral transforms by geometric variables al-lows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st or-der differential equation whose unique solution iswritten in a closed form.The consistent application of inverse integral transforms to the ob-tained solution in the space of images restores the solutions of the consid-ered parabolic boundary value problems through their integral image in an explicit formin the space of the originals.At the same time, the main solutions to the problems were obtained in an explicit form.

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WU, HAO, MAURIZIO GRASSELLI, and SONGMU ZHENG. "CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS." Mathematical Models and Methods in Applied Sciences 17, no. 01 (2007): 125–53. http://dx.doi.org/10.1142/s0218202507001851.

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This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

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Zhang, Cun-Hua, and Xiang-Ping Yan. "Normal Forms of Hopf Bifurcation for a Reaction-Diffusion System Subject to Neumann Boundary Condition." Journal of Applied Mathematics 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/657307.

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A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensional spatial domain(0,lπ)withl>0is considered. According to the normal form method and the center manifold theorem for reaction-diffusion equations, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of system near the constant steady state(0,0)are obtained.

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Arumugam, Gurusamy, and André H. Erhardt. "Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion." Journal of Elliptic and Parabolic Equations 6, no. 2 (2020): 685–709. http://dx.doi.org/10.1007/s41808-020-00078-6.

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Abstract This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

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GRISO, GEORGES. "INTERIOR ERROR ESTIMATE FOR PERIODIC HOMOGENIZATION." Analysis and Applications 04, no. 01 (2006): 61–79. http://dx.doi.org/10.1142/s021953050600070x.

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In a previous paper about homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary, we proved that the global error is of order ε1/2. Now, for an open set Ω with sufficiently smooth boundary [Formula: see text] and homogeneous Dirichlet or Neumann limit conditions, we show that in any open set strongly included in Ω the error is of order ε. If the open set Ω ⊂ ℝn is of polygonal (n = 2) or polyhedral (n = 3) boundary, we also give the global and interior error estimates.

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CHEN, SHANSHAN, JUNPING SHI, and JUNJIE WEI. "GLOBAL STABILITY AND HOPF BIFURCATION IN A DELAYED DIFFUSIVE LESLIE–GOWER PREDATOR–PREY SYSTEM." International Journal of Bifurcation and Chaos 22, no. 03 (2012): 1250061. http://dx.doi.org/10.1142/s0218127412500617.

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In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

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Feng, Xiao-zhou, and Zhi-guo Wang. "Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-(n+1) Functional Response." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/851028.

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This paper discusses a predator-prey system with Holling-(n+1) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution(u~,v~)is asymptotically stable when the parameterksatisfies some conditions.

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Idrissa Ibrango and Stanislas Ouaro. "ENTROPY SOLUTIONS FOR NONLINEAR ELLIPTIC ANISOTROPIC PROBLEMS WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITION." Journal of Applied Analysis & Computation 6, no. 2 (2016): 271–92. http://dx.doi.org/10.11948/2016022.

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Börsch-Supan, Wolfgang, and Melitta Fiebig-Wittmaack. "Stability of stationary solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions." Journal of Differential Equations 94, no. 1 (1991): 55–66. http://dx.doi.org/10.1016/0022-0396(91)90102-f.

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YIN, ZHAOYANG. "ON THE GLOBAL EXISTENCE OF SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS." Glasgow Mathematical Journal 47, no. 2 (2005): 237–48. http://dx.doi.org/10.1017/s0017089505002442.

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López-Soriano, Rafael, and Alejandro Ortega. "A strong maximum principle for the fractional laplace equation with mixed boundary condition." Fractional Calculus and Applied Analysis 24, no. 6 (2021): 1699–715. http://dx.doi.org/10.1515/fca-2021-0073.

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Abstract In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet–Neumann boundary data which extends the one proved by J. Dávila (cf. [11]) to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non–local counterpart to a Hopf’s Lemma for fractional elliptic problems with mixed boundary data.

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Chekurin, Vasyl, and Lesya Postolaki. "Application of the Least Squares Method in Axisymmetric Biharmonic Problems." Mathematical Problems in Engineering 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/3457649.

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An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.

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