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Journal articles on the topic 'Dirichlet solution'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Nugraheni, Sekar, and Christiana Rini Indrati. "SOLUSI LEMAH MASALAH DIRICHLET PERSAMAAN DIFERENSIAL PARSIAL LINEAR ELIPTIK ORDER DUA." Journal of Fundamental Mathematics and Applications (JFMA) 1, no. 1 (2018): 1. http://dx.doi.org/10.14710/jfma.v1i1.2.

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The weak solution is one of solutions of the partial differential equations, that is generated from derivative of the distribution. In particular, the definition of a weak solution of the Dirichlet problem for second order linear elliptic partial differential equations is constructed by the definition and the characteristics of Sobolev spaces on Lipschitz domain in R^n. By using the Lax Milgram Theorem, Alternative Fredholm Theorem and Maximum Principle Theorem, we derived the sufficient conditions to ensure the uniqueness of the weak solution of Dirichlet problem for second order linear ellip

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2

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 (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 s

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3

Afrouzi, G. A. "The existence of positive solutions for an elliptic boundary value problem." International Journal of Mathematics and Mathematical Sciences 2005, no. 13 (2005): 2005–10. http://dx.doi.org/10.1155/ijmms.2005.2005.

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By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

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4

Karachik, Valery. "Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball." Mathematics 9, no. 16 (2021): 1907. http://dx.doi.org/10.3390/math9161907.

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In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a

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5

Grzywny, Tomasz, Moritz Kassmann, and Łukasz Leżaj. "Remarks on the Nonlocal Dirichlet Problem." Potential Analysis 54, no. 1 (2020): 119–51. http://dx.doi.org/10.1007/s11118-019-09820-9.

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AbstractWe study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.

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6

Machů, Hana. "Filippov solutions of vector Dirichlet problems." Mathematica Slovaca 70, no. 2 (2020): 401–16. http://dx.doi.org/10.1515/ms-2017-0359.

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Abstract If in the right-hand sides of given differential equations occur discontinuities in the state variables, then the natural notion of a solution is the one in the sense of Filippov. In our paper, we will consider this type of solutions for vector Dirichlet problems. The obtained theorems deal with the existence and localization of Filippov solutions, under effective growth restrictions. Two illustrative examples are supplied.

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7

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

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8

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 l

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9

Algazin, O. D. "Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip." Mathematics and Mathematical Modeling, no. 3 (August 3, 2018): 1–12. http://dx.doi.org/10.24108/mathm.0318.0000120.

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In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tr

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10

LaChapelle, J. "Path Integral Solution of the Dirichlet Problem." Annals of Physics 254, no. 2 (1997): 397–418. http://dx.doi.org/10.1006/aphy.1996.5650.

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11

Tanaka, Kazuaki. "A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations." Japan Journal of Industrial and Applied Mathematics 38, no. 3 (2021): 731–56. http://dx.doi.org/10.1007/s13160-021-00456-0.

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AbstractThis paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution u and a numerically computed approximate solution $${\hat{u}}$$ u ^ , we evaluate the number of sign-changes of u (the number of nodal domains) and determine the location of zero level-sets of u (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen–C

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12

Christiansen, Søren. "First kind integral equations for the numerical solution of the plane Dirichlet problem." Applications of Mathematics 34, no. 4 (1989): 285–302. http://dx.doi.org/10.21136/am.1989.104357.

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13

Wigley, Neil M. "Corner Behavior of Solutions of Semilinear Dirichlet Problems." Canadian Journal of Mathematics 37, no. 6 (1985): 1025–46. http://dx.doi.org/10.4153/cjm-1985-055-x.

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In recent years there has been considerable attention paid to the behavior of solutions of elliptic boundary value problems in domains with piecewise smooth boundary. In two dimensions the study concerns the behavior of a solution near a corner, and in three (or more) dimensions two cases have been given considerable attention: a conical vertex on the boundary, or an edge.The solution of such a problem may be singular at the nonsmooth boundary points. The standard example in two dimensions is a solution in polar coordinates of the Dirichlet problem near a corner of interior angle πα;u = r1/α s

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14

El-shenawy, Atallah, and Elena A. Shirokova. "The Approximate Solution of 2D Dirichlet Problem in Doubly Connected Domains." Advances in Mathematical Physics 2018 (July 9, 2018): 1–6. http://dx.doi.org/10.1155/2018/6951513.

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We propose a new method for constructing an approximate solution of the two-dimensional Laplace equation in an arbitrary doubly connected domain with smooth boundaries for Dirichlet boundary conditions. Using the fact that the solution of the Dirichlet problem in a doubly connected domain is represented as the sum of a solution of the Schwarz problem and a logarithmic function, we reduce the solution of the Schwartz problem to the Fredholm integral equation with respect to the boundary value of the conjugate harmonic function. The solution of the integral equation in its turn is reduced to sol

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15

Zhao, Kaihong, and Yongkun Li. "Existence and Global Exponential Stability of Equilibrium Solution to Reaction-Diffusion Recurrent Neural Networks on Time Scales." Discrete Dynamics in Nature and Society 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/624619.

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The existence of equilibrium solutions to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales is proved by the topological degree theory and M-matrix method. Under some sufficient conditions, we obtain the uniqueness and global exponential stability of equilibrium solution to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales by constructing suitable Lyapunov functional and inequality skills. One example is given to illustrate the effectiveness of our results.

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16

Aldashev, S. A. "CORRECTNESS OF THE LOCAL BOUNDARY VALUE PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS." Vestnik of Samara University. Natural Science Series 22, no. 1-2 (2017): 7–17. http://dx.doi.org/10.18287/2541-7525-2016-22-1-2-7-17.

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Correctness of boundary value problems in a plane for elliptical equations has been studied properly using the method of the theory of analytic functions. At investigation of analogous problems, when the number of independent variables is more than two, there arise principle difficulties. Quite good and convenient method of singular integral equations has to be abandoned because there is no complete theory of multidimensional singular integral equations. Boundary value problems for second-order elliptical equations in domains with edges have been studied properly earlier. Explicit classical so

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17

Ashyralyev, Allaberen, and Fatma Songul Ozesenli Tetikoglu. "FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions." Abstract and Applied Analysis 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/454831.

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A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. The first and second-orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.

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18

Avci, Mustafa, and Alexander Pankov. "Multivalued elliptic operators with nonstandard growth." Advances in Nonlinear Analysis 7, no. 1 (2018): 35–48. http://dx.doi.org/10.1515/anona-2016-0043.

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AbstractThe paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known {p(x)} growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.

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19

Liu, Hairong, Tian Long, and Xiaoping Yang. "The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group." Communications in Contemporary Mathematics 21, no. 01 (2019): 1750069. http://dx.doi.org/10.1142/s0219199717500699.

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We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be cons

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20

Rukavishnikov, Viktor A., and Elena I. Rukavishnikova. "On the Dirichlet Problem with Corner Singularity." Mathematics 8, no. 11 (2020): 1870. http://dx.doi.org/10.3390/math8111870.

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We consider the Dirichlet problem for an elliptic equation with a singularity. The singularity of the solution to the problem is caused by the presence of a re-entrant corner at the boundary of the domain. We define an Rν-generalized solution for this problem. This allows for the construction of numerical methods for finding an approximate solution without loss of accuracy. In this paper, the existence and uniqueness of the Rν-generalized solution in set W∘2,α1(Ω,δ) is proven. The Rν-generalized solution is the same for different parameters ν.

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21

Huang, Jinjin, and Lei Qiao. "The Dirichlet Problem on the Upper Half-Space." Abstract and Applied Analysis 2012 (2012): 1–5. http://dx.doi.org/10.1155/2012/203096.

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22

Wubneh, Kahsay Godifey. "Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions." Bulletin of Mathematical Sciences and Applications 22 (October 2020): 1–9. http://dx.doi.org/10.18052/www.scipress.com/bmsa.22.1.

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In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield

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23

Motreanu, Dumitru. "Quasilinear Dirichlet problems with competing operators and convection." Open Mathematics 18, no. 1 (2020): 1510–17. http://dx.doi.org/10.1515/math-2020-0112.

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Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

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24

Motreanu, Dumitru. "Quasilinear Dirichlet problems with competing operators and convection." Open Mathematics 18, no. 1 (2020): 1510–17. http://dx.doi.org/10.1515/math-2020-0112.

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Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

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25

Sadiku, M. N. O., and D. T. Hunt. "Solution of Dirichlet problems by the Exodus method." IEEE Transactions on Microwave Theory and Techniques 40, no. 1 (1992): 89–95. http://dx.doi.org/10.1109/22.108327.

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26

Estrada, Ricardo. "Dirichlet Convolution Inverses and Solution of Integral Equations." Journal of Integral Equations and Applications 7, no. 2 (1995): 159–66. http://dx.doi.org/10.1216/jiea/1181075867.

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27

Sanchez, Luis. "Positive solutions for a class of semilinear two-point boundary value problems." Bulletin of the Australian Mathematical Society 45, no. 3 (1992): 439–51. http://dx.doi.org/10.1017/s0004972700030331.

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We study the existence of positive solutions of the periodic, Neumann or Dirichlet problem for the semilinear equation u″ + f(t, u) = 0, 0 ≤ t ≤ T, where f is a Carathéodory function. Our assumptions in each case are such that the problem possesses a lower solution or an upper solution.

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28

Savaşanerİl, Nurcan Baykuş, and Havva Delİbaş. "Analytic Solution for the Dirichlet Problem in 2-D." Journal of Computational and Theoretical Nanoscience 15, no. 2 (2018): 611–15. http://dx.doi.org/10.1166/jctn.2018.7133.

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A broad class of steady-state physical problems can be reduced to finding the harmonic functions that satisfy certain boundary conditions. A fundamental equation of applied mathematics is Laplace equation. This equation models important phenomena in engineering and physics, Laplace equation with satisfied boundary values is known as the Dirichlet problem. In this study, an alternative method is presented for the solution of the Dirichlet problem in a cut-ring region and the solution function of the problem is based on the Green function, and therefore on elliptic functions.

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29

Tang, Moxun. "Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 3 (2003): 705–17. http://dx.doi.org/10.1017/s0308210500002614.

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Let B be the unit ball in Rn, n ≥ 3. Let 0 < p < 1 < q ≤ (n + 2)/(n − 2). In 1994, Ambrosetti et al. found that the semilinear elliptic Dirichlet problem admits at least two solutions for small λ > 0 and no solution for large λ. In this paper, we prove that there is a critical number Λ > 0 such that this problem has exactly two solutions for λ ∈ (0, Λ), exactly one solution for λ = Λ and no solution for λ > Λ.

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30

Ergashev, Tuhtasin G. "The Dirichlet problem for elliptic equation with several singular coefficients." e-Journal of Analysis and Applied Mathematics 1, no. 1 (2018): 81–99. http://dx.doi.org/10.2478/ejaam-2018-0006.

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AbstractRecently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables.

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31

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; \var

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32

Morris, Quinn, Ratnasingham Shivaji, and Inbo Sim. "Existence of positive radial solutions for a superlinear semipositone p-Laplacian problem on the exterior of a ball." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 2 (2018): 409–28. http://dx.doi.org/10.1017/s0308210517000452.

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We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

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33

Kokilashvili, Vakhtang, and Vakhtang Paatashvili. "The Dirichlet Problem for Harmonic Functions in the Smirnov Class with Variable Exponent." gmj 14, no. 2 (2007): 289–99. http://dx.doi.org/10.1515/gmj.2007.289.

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Abstract A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.

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34

Papageorgiou, Nikolaos S., Vicenţiu D. Rădulescu, and Dušan D. Repovš. "Nonlinear Dirichlet problems with unilateral growth on the reaction." Forum Mathematicum 31, no. 2 (2019): 319–40. http://dx.doi.org/10.1515/forum-2018-0114.

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AbstractWe consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, {p=2}), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the m

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35

Boateng, Francis Ohene, Joseph Ackora-Prah, Benedict Barnes, and John Amoah-Mensah. "A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 706–22. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3893.

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In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear ellipticÂ partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solutio

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36

Krutitskii, P. A. "The Dirichlet Problem for the EquationΔu−k2u=0in the Exterior of Nonclosed Lipschitz Surfaces". International Journal of Mathematics and Mathematical Sciences 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/302628.

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We study the Dirichlet problem for the equationΔu−k2u=0in the exterior of nonclosed Lipschitz surfaces inR3. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

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37

Montenegro, Marcelo, and Antonio Suárez. "Existence of a positive solution for a singular system." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 140, no. 2 (2010): 435–47. http://dx.doi.org/10.1017/s0308210509000705.

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Maso, Gianni Dal, and Annalisa Malusa. "Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 1 (1995): 99–114. http://dx.doi.org/10.1017/s0308210500030778.

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Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

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39

YAKUBOV, YAKOV. "COMPLETENESS OF ROOT FUNCTIONS AND ELEMENTARY SOLUTIONS OF THE THERMOELASTICITY SYSTEM." Mathematical Models and Methods in Applied Sciences 05, no. 05 (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

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40

Liu, Lijuan. "Existence of positive solutions to the fractional Laplacian with positive Dirichlet data." Filomat 34, no. 6 (2020): 1795–807. http://dx.doi.org/10.2298/fil2006795l.

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We consider the fractional Laplacian with positive Dirichlet data { (-?)?/2 u = ?up in ?, u > 0 in ?, u = ? in Rn\?, where p > 1,0 < ? < min{2,n}, ? ? Rn is a smooth bounded domain, ? is a nonnegative function, positive somewhere and satisfying some other conditions. We prove that there exists ?* > 0 such that for any 0 < ? < ?*, the problem admits at least one positive classical solution; for ? > ?*, the problem admits no classical solution. Moreover, for 1 < p ? n+?/n-?, there exists 0 < ?? ? ?* such that for any 0 < ? < ??, the problem admits a second pos

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41

Bobkov, Vladimir, and Sergey Kolonitskii. "On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 5 (2019): 1163–73. http://dx.doi.org/10.1017/prm.2018.88.

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AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based o

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42

Gorban, Yu, and А. Soloviova. "A GENERALIZED SOLUTION OF THE DIRICHLET PROBLEM FOR A MODEL ANISOTROPIC WEIGHTED EQUATION." Bulletin Taras Shevchenko National University of Kyiv. Mathematics Mechanics, no. 1 (41) (2020): 11–15. http://dx.doi.org/10.17721/1684-1565.2020.01-41.03.11-15.

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The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.

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Coclite, Mario Michele. "Two perturbation results for nondegenerate solutions of some semilinear Dirichlet problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 1-2 (1989): 1–11. http://dx.doi.org/10.1017/s0308210500024963.

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SynopsisThe stability of nondegenerate solutions of some semilinear Dirichlet problems is studied. Two specific situations are considered: firstly, a singular perturbation of the differential operator; secondly, a perturbation of the nonlinear term using a term which also depends on the gradient of the solution.

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Loghin, Daniel. "Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 2 (2018): 175–92. http://dx.doi.org/10.2478/auom-2018-0024.

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Abstract The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which empl

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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 (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

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Krutitskii, P. A. "The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/269607.

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We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integra

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Li, Xinfu, Guang Zhang, and Ying Wang. "Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation." Discrete Dynamics in Nature and Society 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/503496.

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A discrete logistic steady-state equation with both positive and negative birth rate of population will be considered. By using sub- and upper-solution method, the existence of bounded positive solutions and the existence and uniqueness of positive solitons will be established. To this end, the Dirichlet eigenvalue problem with positive and negative coefficients is considered, and a general sub- and upper-solution theorem is also obtained.

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Su, Baiyun. "Dirichlet Problem for the Schrödinger Operator in a Half Space." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/578197.

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For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, a solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

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Utkina, E. A. "A uniqueness theorem for solution of one Dirichlet problem." Russian Mathematics 55, no. 5 (2011): 51–55. http://dx.doi.org/10.3103/s1066369x11050082.

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N. Emenogu, George. "Solution of Dirichlet Boundary Value Problem by Mellin Transform." IOSR Journal of Mathematics 9, no. 6 (2014): 01–06. http://dx.doi.org/10.9790/5728-0960106.

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