Optimal relaxation of bump-like solutions of the one-dimensional Cahn–Hilliard equation

Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation

We introduce a non-overlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension. We derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains, independently of the length of the time interval. These optimal transmission conditions are ...

Investigation of analytical and numerical solutions for one-dimensional independent-oftime Schrödinger Equation

In this paper, the numerical solution methods of one- particale, one – dimensional time- independentSchrodinger equation are presented that allows one to obtain accurate bound state eigen values andeigen functions for an arbitrary potential energy function V(x). These methods included the FEM(Finite Element Method), Cooly, Numerov and others. Here we considered the Numerov method inmore details...

Similarity solutions of the one-dimensional Fokker-Planck equation.

Group theoretic methods are used to construct new exact solutions for the one-dimensional Fokker-Planck equation corresponding to a class of non-linear forcing functions f(.‘cl. An important sub-class, corresponding tof(x) = a/x + /I-Y, a < 1, /? > 0 is shown to lead to stable solutions. A discussion is given on how generalized similarity methods could be applied to higher dimensional systems. ...

Travelling Wave Solutions of the One Dimensional Diffusion-reaction Equation

Relaxation to Equilibrium in the One-Dimensional Cahn-Hilliard Equation

We study the stability of a so-called kink profile for the one-dimensional Cahn– Hilliard problem on the real line. We derive optimal bounds on the decay to equilibrium under the assumption that the initial energy is less than three times the energy of a kink and that the initial Ḣ−1 distance to a kink is bounded. Working with the Ḣ−1 distance is natural, since the equation is a gradient flow w...