Title page for ETD etd-06092008-113253

Document Type Doctoral Thesis
Author Minani, Froduald
Email minanif2006@yahoo.fr
URN etd-06092008-113253
Document Title Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis
Degree PhD (Mathematical Science)
Department Mathematics and Applied Mathematics
Advisor Name Title
Prof R Anguelov Supervisor
  • viscosity solutions
  • Hamilton-Jacobi
  • Hausdorff
Date 2008-04-11
Availability unrestricted

The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions.

The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions.

Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws.

In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickensí rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone.

In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size.

© University of Pretoria 2007

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