Analysis and numerical solution of nonlinear Volterra partial integrodifferential equations modeling swelling porous materials.

摘要

A nonlinear Volterra partial integrodifferential equation (VPIDE), derived using hybrid mixture theory and used to model swelling porous materials, is analyzed and solved numerically. The model application is an immersed porous material imbibing fluid through a cylinder's exterior boundary. A poignant example comes from the pharmaceutical industry where controlled release, drug-delivery systems are comprised of materials that permit nearly constant drug concentration profiles. In the considered application the release is controlled by the viscoelastic properties of a porous polymer network that swells when immersed in stomach fluid, consequently increasing the pore sizes and allowing the drug to escape. The VPIDE can be viewed as a combination of a non-linear diffusion equation and a constitutive equation modeling the viscoelastic effects. The viscoelastic model is expressed as an integral equation, thus adding an integral term to the non-linear partial differential equation. While this integral term poses both theoretical and numerical challenges, it provides fertile ground for interpretation and analysis.;Analysis of the VPIDE includes an existence and uniqueness proof which we establish under a given set of assumptions for the initial-boundary value problem. Additionally, a special case of the VPIDE is reduced to an ordinary differential equation via a derived similarity variable and solved. In order to solve the full VPIDE we derive a novel approach to constructing pseudospectral differentiation matrices in a polar geometry for computing the spatial derivatives. By construction, the norms of these matrices grow at the optimal rate of O (N2), for N-by- N matrices, versus O (N4) for conventional pseudospectral methods. This smaller norm offers an advantage over standard pseudospectral methods when solving time-dependent problems that require higher-resolution grids and, potentially, larger differentiation matrices. A method-of-lines approach is employed for the time-stepping using an implicit, fifth-order Runge-Kutta solver. After we show how to set up the equation and numerically solve it using this method, we show and interpret results for a variety of diffusion coefficients, permeability models, and parameters in order to study the model's behavior.