Search results for "Boundary value problem"
showing 10 items of 551 documents
A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential …
2002
Abstract In this article, we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Peclet number. From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve …
Reduction of a Non—Linear Parabolic Initial—Boundary Value Problem to Cauchy Problem for a System of ODEs
2004
We consider the boundary value problem for a parabolic equation in the form $$\frac{{\partial {\text{u}}}}{{\partial t}} = \frac{1}{{p(x)}}\frac{\partial }{{\partial x}}\left( {p(x)f'(u)\frac{{\partial u}}{{\partial x}}} \right) + F(u),x \in (0,l),t0,$$ (1) $$u(0,x) = {u_0}(x),$$ (2) $$\frac{{\partial u}}{{\partial x}}{|_{x = 0}} = {f_1}\left( {{u_1}} \right),$$ (3) $$\frac{{\partial u}}{{\partial x}}{|_{x = 1}} = {f_2}\left( {{u_2}} \right),$$ (4) where u = u(t,x) is the unknown function, f 1, f 2, F, f are nonlinear functions and f′ (u) > 0, $${u_1} = {u_1}\left( t \right) \equiv u\left( {t,0} \right),{u_2} = {u_2}(t) \equiv u\left( {t,l} \right),f'\left( u \right) \equiv df(u)/du,p(x) \g…
On Some Properties of the Dirichlet Problem at Resonance
2008
Abstract The boundary value problem at resonance 𝑥″ + 𝑥 = 𝑞 sin 𝑡 + 𝑓(𝑡,𝑥,𝑥′), 𝑥(0) = 0, 𝑥(π) = 0, is considered, where 𝑓 : [0,π] × 𝑹2 → 𝑹 is a bounded Carathéodory function, 𝑞 is a parameter. We state the multiplicity results without assuming that 𝑓 has limits.
EXISTENCE OF THREE SOLUTIONS FOR A MIXED BOUNDARY VALUE PROBLEM WITH THE STURM-LIOUVILLE EQUATION
2012
Abstract. The aim of this paper is to establish the existence of threesolutions for a Sturm-Liouville mixed boundary value problem. The ap-proach is based on multiple critical points theorems. 1. IntroductionThe aim of this paper is to establish, under a suitable set of assumptions, theexistence of at least three solutions for the following Sturm-Liouville problemwith mixed boundary conditions(RS λ )ˆ−(pu ′ ) ′ +qu = λf(t,u) in I =]a,b[u(a) = u ′ (b) = 0,where λ is a positive parameter and p, q, f are regular functions. To be precise,if f : [a,b] × R→ Ris a L 2 -Carath´eodory function and p,q ∈ L ∞ ([a,b]) suchthatp 0 := essinf t∈[a,b] p(t) > 0, q 0 := essinf t∈[a,b] q(t) ≥ 0,then we prove …
Identification of small inhomogeneities: Asymptotic factorization
2007
We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral …
Multiple solutions for a discrete boundary value problem involving the p-Laplacian.
2008
Multiple solutions for a discrete boundary value problem involving the p-Laplacian are established. Our approach is based on critical point theory.
Finite element analysis of varitional crimes for a quasilinear elliptic problem in 3D
2000
We examine a finite element approximation of a quasilinear boundary value elliptic problem in a three-dimensional bounded convex domain with a smooth boundary. The domain is approximated by a polyhedron and a numerical integration is taken into account. We apply linear tetrahedral finite elements and prove the convergence of approximate solutions on polyhedral domains in the $W^1_2$ -norm to the true solution without any additional regularity assumptions.
Uniqueness of solutions for some elliptic equations with a quadratic gradient term
2008
We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by −Δu + λ |∇u| 2 u r = f (x) ,λ , r >0. The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they could be applied to obtain uniqueness results for nonlinear equations having the p-Laplacian operator as the principal part. Our results improve those already known, even…
A Meshfree Solver for the MEG Forward Problem
2015
Noninvasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the Method of Fundamental Solutions (MFS) as a meshfree alternative to the Boundary Element Method (BEM). The solution of the MEG forward problem is obtained, via the Method of Particular Solutions (MPS), by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell’s equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The p…
DORA algorithm for network flow models with improved stability and convergence properties
2001
A new methodology for the solution of shallow water equations is applied for the computation of the unsteady-state flow in an urban drainage network. The inertial terms are neglected in the momentum equations and the solution is decoupled into one kinematic and one diffusive component. After a short presentation of the DORA (Double ORder Approximation) methodology in the case of a single open channel, the new methodology is applied to the case of a sewer network. The transition from partial to full section and vice versa is treated without the help of the Preissmann approximation. The algorithm also allows the computation of the diffusive component in the case of vertical topographic discon…