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Journal articles on the topic 'Dirichlet boundary condition'

Author: Grafiati

Published: 6 September 2023

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1

Cao, Shunhua, and Stewart Greenhalgh. "Attenuating boundary conditions for numerical modeling of acoustic wave propagation." GEOPHYSICS 63, no. 1 (January 1998): 231–43. http://dx.doi.org/10.1190/1.1444317.

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Four types of boundary conditions: Dirichlet, Neumann, transmitting, and modified transmitting, are derived by combining the damped wave equation with corresponding boundary conditions. The Dirichlet attenuating boundary condition is the easiest to implement. For an appropriate choice of attenuation parameter, it can achieve a boundary reflection coefficient of a few percent in a one‐wavelength wide zone. The Neumann‐attenuating boundary condition has characteristics similar to the Dirichlet attenuating boundary condition, but it is numerically more difficult to implement. Both the transmitting boundary condition and the modified transmitting boundary condition need an absorbing boundary condition at the termination of the attenuating region. The modified transmitting boundary condition is the most effective in the suppression of boundary reflections. For multidimensional modeling, there is no perfect absorbing boundary condition, and an approximate absorbing boundary condition is used.

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2

Park, I. Y. "Quantum “violation” of Dirichlet boundary condition." Physics Letters B 765 (February 2017): 260–64. http://dx.doi.org/10.1016/j.physletb.2016.12.026.

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3

Diyab, Farah, and B. Surender Reddy. "Comparison of Laplace Beltrami Operator Eigenvalues on Riemannian Manifolds." European Journal of Mathematics and Statistics 3, no. 5 (October 23, 2022): 55–60. http://dx.doi.org/10.24018/ejmath.2022.3.5.143.

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Let $\Delta_{g}$ be the Laplace Beltrami operator on a manifold $M$ with Dirichlet (resp.,Neumann) boundary conditions. We compare the spectrum of on a Riemannian manifold for Neumann boundary condition and Dirichlet boundary condition . Then we construct aneffective method of obtaining small eigenvalues for Neumann's problem.

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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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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

Corrêa, Francisco Julio S. A., and Joelma Morbach. "A Dirichlet problem under integral boundary condition." Journal of Mathematical Analysis and Applications 478, no. 1 (October 2019): 1–13. http://dx.doi.org/10.1016/j.jmaa.2019.04.030.

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6

Amrouche, Cherif, and Šárka Nečasová. "Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition." Mathematica Bohemica 126, no. 2 (2001): 265–74. http://dx.doi.org/10.21136/mb.2001.134013.

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7

Yosaf, Asma, Shafiq Ur Rehman, Fayyaz Ahmad, Malik Zaka Ullah, and Ali Saleh Alshomrani. "Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation." Advances in Numerical Analysis 2016 (May 16, 2016): 1–12. http://dx.doi.org/10.1155/2016/8376061.

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The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.

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8

Hayasida, Kazuya, and Masao Nakatani. "On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition." Nagoya Mathematical Journal 157 (2000): 177–209. http://dx.doi.org/10.1017/s0027763000007248.

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The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex. In this article we eliminate the H-convexity condition from a portion Γ of the boundary and prove the existence theorem, where the boundary condition is satisfied on Γ in the weak sense.

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9

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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10

YAKUBOV, YAKOV. "COMPLETENESS OF ROOT FUNCTIONS AND ELEMENTARY SOLUTIONS OF THE THERMOELASTICITY SYSTEM." Mathematical Models and Methods in Applied Sciences 05, no. 05 (August 1995): 587–98. http://dx.doi.org/10.1142/s0218202595000346.

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In this paper we prove the completeness of the root functions (eigenfunctions and associated functions) of an elliptic system (in the sense of Douglis-Nirenberg) corresponding to the thermoelasticity system with the Dirichlet boundary value condition. The problem is considered in a domain with a non-smooth boundary. Then an initial boundary value problem corresponding to the thermoelasticity system with the Dirichlet boundary value condition is considered. We find sufficient conditions that guarantee an approximation of a solution to the initial boundary value problem by linear combinations of some “elementary solutions” to the thermoelasticity system.

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11

Hlaváček, Ivan, and Michal Křížek. "On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition." Applications of Mathematics 32, no. 2 (1987): 131–54. http://dx.doi.org/10.21136/am.1987.104242.

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12

Dukowicz, John K., Stephen F. Price, and William H. Lipscomb. "Incorporating arbitrary basal topography in the variational formulation of ice-sheet models." Journal of Glaciology 57, no. 203 (2011): 461–67. http://dx.doi.org/10.3189/002214311796905550.

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AbstractThere are many advantages to formulating an ice-sheet model in terms of a variational principle. In particular, this applies to the specification of boundary conditions, which might otherwise be problematic to implement. Here we focus primarily on the frictional basal sliding boundary condition in a non-Newtonian Stokes model. This type of boundary condition is particularly difficult because it is heterogeneous, requiring both a Dirichlet (no-penetration) condition normal to the bed, and a Neumann (frictional sliding) condition tangential to the bed. In general, Neumann conditions correspond to natural boundary conditions in a variational principle; that is, they arise naturally in the variational formulation and thus need not be explicitly specified. While the same is not necessarily true of Dirichlet conditions, it is possible to enforce a no-penetration condition using Lagrange multipliers within the variational principle so that the Dirichlet condition becomes a natural boundary condition. Thus, in the case of ice sheets, all relevant boundary conditions may be incorporated in the variational functional, making them particularly easy to discretize. For the Stokes model, the resulting basal boundary condition is valid for arbitrary topographic slopes. Here we apply the same methodology to the Blatter– Pattyn higher-order approximate model, which is ordinarily limited to small basal slopes by the smallaspect-ratio approximation. We introduce a modification that improves on the accuracy of the standard Blatter–Pattyn model for all values of the basal slope, as we demonstrate in the slow sliding regime for which analytical results are available. The remaining error is due to the effects of the small-aspect-ratio approximation in the Blatter–Pattyn model.

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13

Beals, Richard, and Nancy K. Stanton. "The Heat Equation for the -Neumann Problem, II." Canadian Journal of Mathematics 40, no. 2 (April 1, 1988): 502–12. http://dx.doi.org/10.4153/cjm-1988-021-8.

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Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

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14

Chien, Henry, Yupeng Fan, and Ziyun Zeng. "Application of the Chimera Method to Poisson’s Equation with the Homogeneous Dirichlet Boundary Condition." Journal of Physics: Conference Series 2287, no. 1 (June 1, 2022): 012004. http://dx.doi.org/10.1088/1742-6596/2287/1/012004.

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Abstract Establishing variational formulation is an effective way to study the existence and uniqueness of the solution of certain elliptic partial differential equation with boundary condition. For the solution of certain elliptic partial differential equation with boundary condition, we know that the numerical solution obtained by the finite element method approximates the solution of this equation. Moreover, to avoid gridding overly complex domains, we can use the Chimera method to decompose the domain into several overlapping sub-domains. In this paper, we study Poisson’s equation with the homogeneous Dirichlet boundary condition. By analyzing the existence and uniqueness of the solution of the corresponding variational formulation, we know the existence and uniqueness of the solution of Poisson’s equation with the homogeneous Dirichlet boundary condition. We use the Chimera method and the finite element method to deal with Poisson’s equation with the homogeneous Dirichlet boundary condition by constructing two iterative sequences and analyzing their properties.

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15

Barba-Ortega, J., J. Barón-Jaimez, and M. R. Joya. "Dirichlet boundary condition for the Ginzburg-Landau equations." Journal of Physics: Conference Series 466 (November 7, 2013): 012027. http://dx.doi.org/10.1088/1742-6596/466/1/012027.

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16

Schnurer, Oliver C. "A generalized Minkowski problem with Dirichlet boundary condition." Transactions of the American Mathematical Society 355, no. 2 (September 6, 2002): 655–63. http://dx.doi.org/10.1090/s0002-9947-02-03135-5.

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17

Benedicks, M., and W. F. Pfeffer. "The Dirichlet Problem With Denjoy-Perron Integrable Boundary Condition." Canadian Mathematical Bulletin 28, no. 1 (March 1, 1985): 113–19. http://dx.doi.org/10.4153/cmb-1985-013-1.

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AbstractThe Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

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18

TANIGUCHI, YUSUKE. "SCHRÖDINGER FUNCTIONAL FORMALISM WITH GINSPARG-WILSON FERMION." Modern Physics Letters A 22, no. 07n10 (March 28, 2007): 499–513. http://dx.doi.org/10.1142/s0217732307023080.

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In this proceeding we propose a new procedure to impose the Schrödinger functional Dirichlet boundary condition on the overlap Dirac operator and the domain-wall fermion using an orbifolding projection. With this procedure the zero mode problem with Dirichlet boundary condition can easily be avoided.

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19

Kone, Blaise, and Stanislas Ouaro. "On the Solvability of Discrete Nonlinear Two-Point Boundary Value Problems." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/927607.

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We prove the existence and uniqueness of solutions for a family of discrete boundary value problems by using discrete's Wirtinger inequality. The boundary condition is a combination of Dirichlet and Neumann boundary conditions.

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20

Ding, Ning, Jingjing Cai, and Li Xu. "A free boundary problem with nonlinear advection and Dirichlet boundary condition." Nonlinear Analysis: Real World Applications 69 (February 2023): 103719. http://dx.doi.org/10.1016/j.nonrwa.2022.103719.

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21

Xie, Hui, Jiming Song, Ming Yang, and Norio Nakagawa. "Boundary Integral Equation for Eddy-Current Problems with Dirichlet Boundary Condition." Journal of Nondestructive Evaluation 29, no. 4 (August 17, 2010): 214–21. http://dx.doi.org/10.1007/s10921-010-0079-z.

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22

Krutitskii, P. A. "Method of interior boundaries in a mixed problem of acoustic scattering." Mathematical Problems in Engineering 5, no. 2 (1999): 173–92. http://dx.doi.org/10.1155/s1024123x99001052.

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The mixed problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The Dirichlet boundary condition is given on some obstacles and the impedance boundary condition is specified on the rest. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called ‘Method of interior boundaries’, because additional boundaries are introduced inside scattering bodies, where impedance boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact our method holds for any positive wave numbers. The Neumann, Dirichlet, impedance problems and mixed Dirichlet–Neumann problem are particular cases of our problem.

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23

Nazarova, Kulzina Zh, Batirkhan Kh Turmetov, and Kairat Id Usmanov. "On a nonlocal boundary value problem with an oblique derivative." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 22, no. 1 (March 31, 2020): 81–93. http://dx.doi.org/10.15507/2079-6900.22.202001.81-93.

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The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the Rn space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique.

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24

LÓPEZ, RAFAEL. "WETTING PHENOMENA AND CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY." Reviews in Mathematical Physics 17, no. 07 (August 2005): 769–92. http://dx.doi.org/10.1142/s0129055x05002443.

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In a microscopic scale or microgravity environment, interfaces in wetting phenomena are usually modeled by surfaces with constant mean curvature (CMC surfaces). Usually, the condition regarding the constancy of the contact angle along the line of separation between different phases is assumed. Although the classical capillary boundary condition is the angle made at the contact line, configurations also occur in which a Dirichlet condition is appropriate. In this article, we discuss those with vanishing boundary conditions, such as those that occur on a thin flat portion of a plate of general shape covered with water. In this paper, we review recent works on the existence of CMC surfaces with non-empty boundary, with a special focus on the Dirichlet problem for the constant mean curvature equation.

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25

Li, Dong, Guixiang Xu, and Xiaoyi Zhang. "On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS." Canadian Journal of Mathematics 66, no. 5 (October 1, 2014): 1110–42. http://dx.doi.org/10.4153/cjm-2014-002-0.

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AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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26

Sun, Linlin, Wen Chen, and Alexander H. D. Cheng. "Evaluating the Origin Intensity Factor in the Singular Boundary Method for Three-Dimensional Dirichlet Problems." Advances in Applied Mathematics and Mechanics 9, no. 6 (November 28, 2017): 1289–311. http://dx.doi.org/10.4208/aamm.2015.m1153.

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AbstractIn this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

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27

Iwabuchi, Tsukasa. "On analyticity up to the boundary for critical quasi-geostrophic equation in the half space." Communications on Pure & Applied Analysis 21, no. 4 (2022): 1209. http://dx.doi.org/10.3934/cpaa.2022016.

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<p style='text-indent:20px;'>We study the Cauchy problem for the surface quasi-geostrophic equation with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show a natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.</p>

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28

Fašangová, Eva, and Eduard Feireisl. "The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 2 (1999): 319–29. http://dx.doi.org/10.1017/s0308210500021375.

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For a non-negative function ū(x), we study the long-time behaviour of solutions of the heat equationwith the Dirichlet or Neumann boundary conditions at x = 0. We find a critical parameter λD > 0 such that the solution subjected to the Dirichlet boundary condition tends to a spatially localized wave travelling to infinity in the space variable. On the other hand, there exists a λN > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dynamics is considerably influenced by the choice of boundary conditions.

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29

Liu, Peng, and JIANHUA YUE. "Comparison between Dirichlet boundary condition and mixed boundary condition in resistivity tomography through finiteelement simulation." European Journal of Electrical Engineering 20, no. 3 (June 27, 2018): 333–45. http://dx.doi.org/10.3166/ejee.20.333-345.

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30

Lozada-Cruz, German, Cosme Eustaquio Rubio-Mercedes, and Junior Rodrigues-Ribeiro. "Numerical Solution of Heat Equation with Singular Robin Boundary Condition." TEMA (São Carlos) 19, no. 2 (September 12, 2018): 209. http://dx.doi.org/10.5540/tema.2018.019.02.209.

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In this work we study the numerical solution of one-dimensional heatdiffusion equation with a small positive parameter subject to Robin boundary conditions. The simulations examples lead us to conclude that the numerical solutionsof the differential equation with Robin boundary condition are very close of theanalytic solution of the problem with homogeneous Dirichlet boundary conditionswhen tends to zero

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31

LI, JINGYU, and KAIJUN ZHANG. "REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER." Mathematical Models and Methods in Applied Sciences 21, no. 05 (May 2011): 1153–92. http://dx.doi.org/10.1142/s0218202511005283.

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We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

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32

Liu, Chein-Shan. "Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones." Journal of Mathematics Research 14, no. 4 (July 4, 2022): 1. http://dx.doi.org/10.5539/jmr.v14n4p1.

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In the paper, the eigenvalues of Sturm-Liouville problems (SLPs), generalized SLPs and periodic SLPs are solved. First, we propose a new method to transform the SLP with mixed boundary conditions to a {generalized} SLP for a transformed variable, for which the Dirichlet boundary conditions occur on two-side, but the coefficients are nonlinear functions of eigenvalue. To computing the eigenvalue and eigenfunction, we further recast the transformed system to an initial value problem for a new variable. In terms of the relative residual of two consecutive terminal values of the new variable a nonlinear equation is solved for seeking the eigenvalue by the fictitious time integration method (FTIM), which monotonically converges to the exact eigenvalue. We solve a numerically characteristic equation by the half-interval method (HIM) and a derivative-free iterative scheme LHL {(Liu, Hong $\&$ Li, 2021)} to achieve high precision eigenvalues. Next, the generalized SLP is transformed to a {new} one, so that the Dirichlet boundary condition happens on the right-end. By using the boundary shape function method and the uniqueness condition of the transformed variable, a definite initial value problem is derived for the new variable. To match the right-end Dirichlet boundary condition a numerically characteristic equation is {obtained and} solved by the HIM and LHL. Finally, new techniques for solving the periodic SLPs with three types periodic boundary conditions are proposed, which preserve the periodic boundary conditions with the aids of boundary shape functions. Three iterative algorithms are developed, which converge quickly.  All the proposed iterative algorithms are identified by testing some examples.

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33

Kot, Valery A. "INTEGRAL METHOD OF BOUNDARY CHARACTERISTICS: THE DIRICHLET CONDITION. PRINCIPLES." Heat Transfer Research 47, no. 11 (2016): 1035–55. http://dx.doi.org/10.1615/heattransres.2016014882.

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Kot, Valery A. "INTEGRAL METHOD OF BOUNDARY CHARACTERISTICS: THE DIRICHLET CONDITION. ANALYSIS." Heat Transfer Research 47, no. 10 (2016): 927–44. http://dx.doi.org/10.1615/heattransres.2016014883.

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35

Sharaf, Khadijah. "On a nonlinear problem with zero Dirichlet boundary condition." Applicable Analysis 96, no. 9 (August 16, 2016): 1466–82. http://dx.doi.org/10.1080/00036811.2016.1220548.

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36

Muñoz Rivera, Jaime E., and Yoshihiro Shibata. "A Linear Thermoelastic Plate Equation with Dirichlet Boundary Condition." Mathematical Methods in the Applied Sciences 20, no. 11 (July 25, 1997): 915–32. http://dx.doi.org/10.1002/(sici)1099-1476(19970725)20:11<915::aid-mma891>3.0.co;2-4.

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37

Jeong, Darae, Seungsuk Seo, Hyeongseok Hwang, Dongsun Lee, Yongho Choi, and Junseok Kim. "Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/359028.

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We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

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38

Samet, Bessem, and Calogero Vetro. "A Perturbed Cauchy Viscoelastic Problem in an Exterior Domain." Mathematics 11, no. 10 (May 14, 2023): 2283. http://dx.doi.org/10.3390/math11102283.

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A Cauchy viscoelastic problem perturbed by an inverse-square potential, and posed in an exterior domain of RN, is considered under a Dirichlet boundary condition. Using nonlinear capacity estimates specifically adapted to the non-local nature of the problem, the potential function and the boundary condition, we establish sufficient conditions for the nonexistence of weak solutions.

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Bacchelli, Valeria, Dario Pierotti, Stefano Micheletti, and Simona Perotto. "Parameter identification for the linear wave equation with Robin boundary condition." Journal of Inverse and Ill-posed Problems 27, no. 1 (February 1, 2019): 25–41. http://dx.doi.org/10.1515/jiip-2017-0093.

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Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

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Szajewska, Marzena, and Agnieszka Tereszkiewicz. "TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS." Acta Polytechnica 56, no. 3 (June 30, 2016): 245. http://dx.doi.org/10.14311/ap.2016.56.0245.

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Boundary value problems are considered on a simplex <em>F</em> in the real Euclidean space R<sup>2</sup>. The recent discovery of new families of special functions, orthogonal on <em>F</em>, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on <em>F</em>, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of <em>F</em> a Dirichlet condition is fulfilled and on the other Neumann’s works.

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Maulidi, Ikhsan, and Agah D. Garnadi. "Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition." Jurnal Matematika Integratif 13, no. 1 (July 12, 2017): 51. http://dx.doi.org/10.24198/jmi.v13.n1.11505.51-54.

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We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

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Maulidi, Ikhsan, and Agah D. Garnadi. "Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition." Jurnal Matematika Integratif 13, no. 1 (July 12, 2017): 51. http://dx.doi.org/10.24198/jmi.v13i1.11505.

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We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

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Gavrilyuk, Ivan P., Volodymyr L. Makarov, and Nataliya V. Mayko. "Weighted Estimates for Boundary Value Problems with Fractional Derivatives." Computational Methods in Applied Mathematics 20, no. 4 (October 1, 2020): 609–30. http://dx.doi.org/10.1515/cmam-2018-0305.

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AbstractWe consider the Dirichlet boundary value problem for linear fractional differential equations with the Riemann–Liouville fractional derivatives. By transforming the boundary value problem to the integral equation, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of the boundary value problems by a grid method is discussed and weighted estimates considering the boundary effect are obtained. It is shown that the accuracy (the convergence rate) near the boundary is better than inside the domain due to the influence of the Dirichlet boundary condition.

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McCartin, Brian J. "Eigenstructure of the equilateral triangle. Part III. The Robin problem." International Journal of Mathematics and Mathematical Sciences 2004, no. 16 (2004): 807–25. http://dx.doi.org/10.1155/s0161171204306125.

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Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored.

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Williams, Hollis. "Neumann and Dirichlet boundary conditions for the one-dimensional wave equation." Physics Education 57, no. 5 (July 29, 2022): 055028. http://dx.doi.org/10.1088/1361-6552/ac832d.

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Abstract The notion of a boundary condition is typically considered to be somewhat advanced and not suitable to be introduced in high school level physics. In this article, we give a simple visual demonstration of the difference between Dirichlet and Neumann boundary conditions for a string which oscillates according to the one-dimensional wave equation. The classical Dirichlet problem for a vibrating string in the plane is mathematically deep, but we will avoid the mathematical issues here and only focus on some key physical points which are relevant for understanding how waves propagate along strings.

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Alghanemi, Azeb, Slim Chaabane, Hichem Chtioui, and Abdellahi Soumaré. "Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems." Mathematics 11, no. 8 (April 20, 2023): 1955. http://dx.doi.org/10.3390/math11081955.

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We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of Rn,n≥3 and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri-Coron and demands new constructions and extra ideas.

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Cano-Casanova, Santiago. "Positive weak solutions for heterogeneous elliptic logistic BVPs with glued Dirichlet-Neumann mixed boundary conditions." AIMS Mathematics 8, no. 6 (2023): 12606–21. http://dx.doi.org/10.3934/math.2023633.

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<abstract><p>This article concerns the existence of positive weak solutions of a heterogeneous elliptic boundary value problem of logistic type in a very general annulus. The novelty of this work lies in considering non-classical mixed glued boundary conditions. Namely, Dirichlet boundary conditions on a component of the boundary, and glued Dirichlet-Neumann boundary conditions on the other component of the boundary. In this paper we perform a complete analysis of the existence of positive weak solutions of the problem, giving a necessary condition on the $ \lambda $ parameter for the existence of them, and a sufficient condition for the existence of them, depending on the $ \lambda $-parameter, the spatial dimension $ N \geq 2 $ and the exponent $ q > 1 $ of the reaction term. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results obtained in this paper are pioners in the field, because up the knowledge of the autor, this is the first time where this kind of logistic problems have been analyzed.</p></abstract>

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Izsák, Ferenc, and Gábor Maros. "Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions." Fractional Calculus and Applied Analysis 23, no. 2 (April 28, 2020): 378–89. http://dx.doi.org/10.1515/fca-2020-0018.

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AbstractFractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. The existence-uniqueness result is stated also for two-dimensional domains. Finally, a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.

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Ahmad, Daud, M. Khalid Mahmood, Qin Xin, Ferdous M. O. Tawfiq, Sadia Bashir, and Arsha Khalid. "A Computational Model for q -Bernstein Quasi-Minimal Bézier Surface." Journal of Mathematics 2022 (September 20, 2022): 1–21. http://dx.doi.org/10.1155/2022/8994112.

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A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q -Bernstein polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q -Bernstein–Bézier minimal surface.

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Zhan, Huashui. "Diffusion Convection Equation with Variable Nonlinearities." Journal of Function Spaces 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/2397474.

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The paper studies diffusion convection equation with variable nonlinearities and degeneracy on the boundary. Unlike the usual Dirichlet boundary value, only a partial boundary value condition is imposed. If there are some restrictions in the diffusion coefficient, the stability of the weak solution based on the partial boundary value condition is obtained. In general, we may obtain a local stability of the weak solutions without any boundary value condition.

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