Sultan Sial

Energy minimization using Sobolev gradients: application to phase separation and ordering

A common problem in physics and engineering is the calculation of the minima of energy functionals. The theory of Sobolev gradients provides an efficient method for seeking the critical points of such a functional. We apply the method to functionals describing coarse-grained Ginzburg-Landau models commonly used in pattern formation and ordering processes.

Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg-Landau functionals

The Sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain Ginzburg-Landau type functionals [S. Sial, J. Neuberger, T. Lookman, A. Saxena, Energy minimization using Sobolev gradients: application to phase separation and ordering, J. Comput. Phys. 189 (2003) 88-97]. In this article a Sobolev gradient method for the related time evolution is discussed.

A Sobolev gradient algorithm for minimum energy states of s-wave superconductors—finite element setting

The application of a Sobolev gradient method for finding vortices in s-wave superconductors via minimization of their Landau–Ginzburg energies is demonstrated in a finite element setting. It is seen that the method is highly efficient while at the same time retaining the simplicity of the steepest descent algorithm.

Approximating time evolution related to Ginzburg-Landau functionals via Sobolev gradient methods in a finite-element setting

Sobolev gradients have previously [1] been used to approximate time evolution related to a model A functional in a finite-difference setting in this journal. Here a related approach in a finite-element setting is discussed.

Energy minimization related to the nonlinear Schrödinger equation

In this the window of the Sobolev gradient technique to the problem of minimizing a Schrodinger functional associated with a nonlinear Schrodinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrodinger functionals.

Application of Sobolev gradient method to Poisson-Boltzmann system

The idea of a weighted Sobolev gradient, introduced and applied to singular differential equations in [1], is extended to a Poisson-Boltzmann system with discontinuous coefficients. The technique is demonstrated on fully nonlinear and linear forms of the Poisson- Boltzmann equation in one, two, and three dimensions in a finite difference setting. A comparison between the weighted gradient and FAS multigrid is given for large jump size in the coefficient function.

On solitons: the biomolecular nonlinear transmission line models with constant and time variable coefficients

Abstract The article studies the dynamics of solitons in electrical microtubule model, which describes the propagation of waves in nonlinear dynamical system. Microtubules are not only a passive support of a cell but also they have highly dynamic structures involved in cell motility, intracellular transport and signaling. The underlying model has been considered with constant and variable coefficients of time function. The solitary wave ansatz has been applied successfully to extract these solitons. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of these models.

Recursive form of Sobolev gradient method for ODEs on long intervals

The Sobolev gradient method has been shown to be effective at constructing finite-dimensional approximations to solutions of initial-value problems. Here we show that the efficiency of the algorithm as often used breaks down for long intervals. Efficiency is recovered by solving from left to right on subintervals of smaller length. The mathematical formulation for Sobolev gradients over non-uniform one-dimensional grids is given so that nodes can be added or removed as required for accuracy. A recursive variation of the Sobolev gradient algorithm is presented which constructs subintervals according to how much work is required to solve them. This allows efficient solution of initial-value problems on long intervals, including for stiff ODEs. The technique is illustrated by numerical solutions for the prototypical problem u′=u, equation for flame-size, and the van der Pols equation.

Approximate solutions to Poisson-Boltzmann systems with Sobolev gradients

A weighted Sobolev gradient approach [1] is presented to a nonlinear PBE [2] with discontinuous coefficient functions. A comparison is given between the weighted and unweighted Sobolev gradient in the finite element setting in two and three dimensions. Behavior of the various Sobolev gradients is discussed for large jump size in the coefficient. A comparison with Newtons method is given where the failure of Newtons method is demonstrated for a test problem.

Numerical solution of Burgers’ equation by the Sobolev gradient method

Abstract Burgers’ equation is solved numerically with Sobolev gradient methods. A comparison is shown with other numerical schemes presented in this journal, such as modified Adomian method (MAM) [1] and by a variational method (VM) which is based on the method of discretization in time [2] . It is shown that the Sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms.