We study a singular one-dimensional parabolic problem with initial data in the BV space, the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not satisfied in a pointwise manner. We study the facet creation process and the extinction of solutions caused by the evolution of facets. Our major tool is the comparison principle provided by the theory of viscosity solutions developed in [10].
We perform a time-space discretisation, known as the leapfrog method, for nonlinear stochastic functional wave equations driven by multiplicative time-space white noise. To prove its stability we apply Cairoli's maximal inequalities for two-parameter martingales and provide a lemma for estimating solutions to a class of stochastic wave equations and a Gronwall-type inequality over cones. The method converges in L 2 at a rate of O(√ h), where h is a time-space step size.
The method of lines (MOL) for diffusion equations with Neumann boundary conditions is considered. These equations are transformed by a discretization in space variables into systems of ordinary differential equations. The proposed ODEs satisfy the mass conservation law. The stability of solutions of these ODEs with respect to discreteL2norms and discreteW1,∞norms is investigated. Numerical examples confirm the parabolic behaviour of this model and very regular dynamics.
We present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.
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