In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω× [0, T ] which is decomposed into an overlapping collection of cylindrical subregions of the form Ωl ×[0, T ], fo...

Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf3)C. We consider non-hermitian re...

Pseudo-monotonicity seems to be a good notion to deal with convergence in non-linear terms of partial differential equations. J.-L. Lions [16] used two different definitions of pseudo-monotonicity for elliptic and parabolic problems, and derived associated existence results. Nonlinear elliptic-parabolic equations are intermediate equations for which an intermediate pseudo-monotonicity is define...

In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned lin...

The DeGiorgi-Nash-Moser estimate plays a crucial role in the study of quasilinear elliptic and parabolic equations. In the present paper we shall show that this fundamental estimate holds for solutions of a linear parabolic Volterra inte-grodiierential equation: @u @t = @ @x i a ij (x; t) @u @x j + Z t 0 @ @x i b ij (x; t;) @u @x j dd; where fa ij g and fb ij g are only assumed to be measurable...

The paper deals with the existence of solutions of some parabolic bilateral problems approximated by the renormalized solutions of some parabolic equations.

Abstract. We propose a new notion of weak solutions (dissipative solutions) for nonisotropic, degenerate, second order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic,...

Based on the concept of domain decomposition we construct a class of alternating group explicit method for fourth order parabolic equations. Furthermore, an exponential type alternating group explicit method for 2D convection-diffusion equations is derived, which is effective in convection dominant cases. Both of the two methods have the property of unconditional stability and intrinsic paralle...

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative pr...

1. Global Energy Problems and the Role of Renewables 1.1. Energy demand 1.2. Energy, environmental problems and sustainable development 1.3. Water demand and consumption 1.4. Water and energy 1.5. Desalination and energy 1.6. Renewable energy technologies 2. Solar Collectors 2.1. Parabolic trough collectors 2.2. Fresnel collectors 2.3. Parabolic dish reflector 2.4. Heliostat field collector 3. ...