WAVELET ANALYSIS AND NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS

THE PRINCIPAL PURPOSE OF THIS PROPOSAL IS TO DEVELOP SOLUTION METHODS FOR INITIAL AND/OR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS BASED ON THE RECENTLY DISCOVERED WAVELET BASIS FUNCTIONS. THE BASIS FUNCTIONS HAVE PROPERTIES OF ORTHOGONALITY, LOCAL SUPPORT, SCALING BEHAVIOR, AND INHERENTLY PARALLELIZABLE ALGORITHMS FOR COMPUTER IMPLEMENTATION, WHICH MAKES THEM VERY SUITABLE FOR MULTILEVEL ANALYSIS OF DIFFERENTIAL EQUATIONS. AN ADDITIONAL PURPOSE OF THE PROPOSAL IS TO CONTINUE THE INVESTIGATION OF THE THEORETICAL AND NUMERICAL PROPERTIES OF WAVELETS. FROM THE INTRINSIC ASYMMETRY OF WAVELETS, THEY SEEM TO BE WELL-SUITED FOR THE SIMULATION OF HIGHLY ADVECTED PHENOMENA. THIS PROGRAM OF USING WAVELETS FOR NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS WILL BE CARRIED OUT IN SEVERAL PHASES. THE FIRST PHASE WILL CONCERN THE PROBLEM OF ONE AND TWO-DIMENSIONAL SCALAR EQUATIONS, BOTH LINEAR AND NON-LINEAR IN NATURE. THE LATER PHASES WILL INCLUDE VECTOR-VALUED PROBLEMS ARISING IN FLUID DYNAMICS, FIELD THEORY AND FINITE CLASTICITY. MORE SPECIFICALLY IN PHASE I WE SHALL ADDRESS THE SOLUTION OF LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS IN DIMENSION 1 AND 2, INCLUDING SITUATIONS WHERE THE SOLUTION EXHIBITS BOUNDARY LAYERS. WE SHALL ALSO INVESTIGATE THE WAVELET SOLUTION OF LINEAR AND NONLINEAR TIME-DEPENDENT PROBLEMS SUCH AS THE HEAT EQUATION, THE LINEAR ADVECTION EQUATION, THE BUCKLEY-LEVERETT EQUATIONS, AND THE KURAMOTO-SHIVASHINSKY EQUATION. COMPARISON WILL BE MADE WITH MORE CLASSICAL SOLUTION METHODS BASED, FOR INSTANCE, ON FINITE-DIFFERENCE AND FINITE ELEMENT METHODS. MOREOVER, THE DEPENDENCE OF THE WAVELET SYSTEMS ON THE WAVELET COEFFICIENT, IN PARTICULAR, AND THE DIFFERENTIABILITY PROPERTIES OF WAVELETS WILL BE STUDIED.