Recent zbMATH articles in MSC 31https://zbmath.org/atom/cc/312021-01-08T12:24:00+00:00WerkzeugOn modified Bitsadze-Samarskiy problem.https://zbmath.org/1449.310012021-01-08T12:24:00+00:00"Kovaleva, L. A."https://zbmath.org/authors/?q=ai:kovaleva.l-aSummary: We study the non-local boundary value problem which is an analogue of the Bitsadze-Samarskiy problem. For the two-dimensional case we reduce this problem to the local boundary value problem, more exactly to the Dirichlet problem for the analogue of the Laplace equation on the stratified set. Using the Poincare-Perron method we establish that the solution is the upper envelope of the set of subharmonic functions taking given values on the boundary.A discontinuous Galerkin method by patch reconstruction for biharmonic problem.https://zbmath.org/1449.653212021-01-08T12:24:00+00:00"Li, Ruo"https://zbmath.org/authors/?q=ai:li.ruo"Ming, Pingbing"https://zbmath.org/authors/?q=ai:ming.pingbing"Sun, Zhiyuan"https://zbmath.org/authors/?q=ai:sun.zhiyuan"Yang, Fanyi"https://zbmath.org/authors/?q=ai:yang.fanyi"Yang, Zhijian"https://zbmath.org/authors/?q=ai:yang.zhijianSummary: We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn-Hilliard conditions.https://zbmath.org/1449.653362021-01-08T12:24:00+00:00"Shivanian, Elyas"https://zbmath.org/authors/?q=ai:shivanian.elyas"Abbasbandy, Saeid"https://zbmath.org/authors/?q=ai:abbasbandy.saeidSummary: In this study, we develop an approximate formulation for a generalization form of biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as Cahn-Hilliard type boundary conditions with application to spinodal decomposition. Since the rigorous steps to analyze such a problem is of high-order derivatives, implementing multiple boundary conditions and especially when the geometry of domain of the problem is complex. In PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high-order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by an erudite application of PSMRPI on nodal points near the boundaries of the domain. The main results of generalized biharmonic problem are demonstrated by some examples to show validity and trustworthy of PSMRPI technique.Cyclicity in the harmonic Dirichlet space.https://zbmath.org/1449.460242021-01-08T12:24:00+00:00"Abakumov, Evgueni"https://zbmath.org/authors/?q=ai:abakumov.evgeny-v"El-Fallah, Omar"https://zbmath.org/authors/?q=ai:el-fallah.omar"Kellay, Karim"https://zbmath.org/authors/?q=ai:kellay.karim"Ransford, Thomas"https://zbmath.org/authors/?q=ai:ransford.thomas-jSummary: The harmonic Dirichlet space is the Hilbert space of functions \(f\in L^2(\mathbb{T})\) such that \[ \Vert f\Vert^2_{\mathcal{D}(\mathbb{T})}:= \sum_{n\in\mathbb{Z}}(1+\vert n\vert)\vert\widehat f(n)\vert^2 <\infty. \] We give sufficient conditions for \(f\) to be cyclic in \(\mathcal{D}(\mathbb{T})\), that is, for \(\{\zeta^nf(\zeta):n\ge 0\}\) to span a dense subspace of \(\mathcal{D}(\mathbb{T})\).
For the entire collection see [Zbl 1404.42002].The Dirichlet problem for the Poisson type equations in the plane.https://zbmath.org/1449.300852021-01-08T12:24:00+00:00"Gutlyanskiĭ, V.Ya."https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-v"Ryazanov, V. I."https://zbmath.org/authors/?q=ai:ryazanov.vladimir-iSummary: We present a new approach to the study of semilinear equations of the form \(\text{div} [A(z)\Delta u] = f (u)\), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions \(A(z)\), whereas its reaction term \(f (u)\) is a continuous non-linear function. We establish a theorem on the existence of weak \(C(\overline{D})\cap W^{1,2}_{\text{loc}} (D)\) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains \(D\) without degenerate boundary components and give applications to equations of mathematical physics in anisotropic media.On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces.https://zbmath.org/1449.320122021-01-08T12:24:00+00:00"Unal, C."https://zbmath.org/authors/?q=ai:unal.cihan|unal.cemal"Aydin, I."https://zbmath.org/authors/?q=ai:aydin.ilknur|aydin.ismailLet \(p:\mathbb{R}^n\longrightarrow[1,+\infty)\) be a measurable function and let \(\vartheta:\mathbb{R}^n\longrightarrow(0,+\infty)\) be locally integrable. Denote by \(L^p_\vartheta(\mathbb{R}^n)\) the space of all measurable functions \(f\) such that \(\int_{\mathbb{R}^n}|f(x)|^{p(x)}\vartheta(x)dx<+\infty\) and let \(W^{1,p}_\vartheta(\mathbb{R}^n):=\{f\in L^p_\vartheta(\mathbb{R}^n): \partial{f}/\partial{x_j}\in L^p_\vartheta(\mathbb{R}^n),\;j=1,\dots,n\}\). The authors study the space \(W^{1,p}_\vartheta(\mathbb{R}^n)\) and various capacities associated with this space.
Reviewer: Marek Jarnicki (Kraków)