In the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in Alvino et al.
2. Steady-State Temperature Problems The above problems for the Laplace equation are illustrated by the steady-state solutions of the 2-D and 3-D heat equation. By a steady-state function u, we mean a function that is independent on time t. Thus, u t ≡ 0. In particular if u satis...
coupled with fast matrix–vector multiply and compression algorithms, and iterative solvers such as gmres, provide spectacularly effective computational tools for solving a range of linear boundary value problems, for example in potential theory, elasticity, and acoustic and electromagnetic wave scattering ...
24.2Steadystatesolutionsinhigherdimensions Laplace’sEquationarisesasasteadystateproblemfortheHeatorWaveEquationsthatdonotvarywithtimeso that ∂u ∂t =0= ∂ 2 u ∂t 2 . 2D: ∆u= ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 =0.(24.2) 2 3D: ∆u= ∂ 2 u ∂x 2 + ∂ 2 u...
Two basic solvers (Euler and Talbot) are included, along with *symbolic* versions of those solvers. The symbolic solutions take substantially longer to calculate, but are capable of any desired accuracy. Also, the symbolic versions require the Symbolic Toolbox, whereas the basic versions do not....
It is worth noting that convergence rates of DG methods for eigenvalue problems on planar domains have been studied, for example in [30], and higher-order SFEMs for eigenvalue problems were proposed and analyzed in [31]. Particularly observed in [31] was a non-synchronization phenomenon ...
This restriction can sometimes be useful, as it allows to prove quasi-concavity of solutions to simple problems such as {{{−∆u + u = 1 {{{ u > 0 { u=0 in Ω, in Ω, on ∂Ω in Example 2.2. Let us also mention that the Hölder continuity of f can be weakened ...
We demonstrate solutions by two sets of twenty-four sample problems in order to analyze the sensitivity of the model to changes in the mean and variance of delivery time. In the first set of problems, we do not restrict the variance to be less than initial variance; in the second set of...
Under Assumption 2.1 with (iii) replaced by (2.7), there exists a sequence of solutions for (1.1) whose C1(Ω)-norms diverge to infinity. Next, we consider a sequence of solutions converging to zero. To this end, we introduce the assumption below. Assumption 2.5. Suppose that 1 < q ...
We also observe that similar problems arise not only in hydrodynamics but also in elasticity and electromagnetism theories. The described technique, namely, the possibility of passing from two to three independent variables by agiven transformation enables us to find purely physical solutions for awide...