Transformations of Singular Systems of Linear Partial Differential Equations

Abstract

Let S_D,L,A denote the regulated solution set of the linear partial differential equation where A is a matrix function of t,D is a constant diagonal matrix and L is a linear scalar differential operator which commutes with any A. If D is nonsin-gular, then for any pair of regulated matrices A and B there is a Fredholm map independent of L, from S _D,L,A to S _D,L,B which preserves initial values for all A and B. If S ^o _D,L,A denotes the subset of S _D,L,A with zero initial conditions, there is another Fredholm map independent of L, from S ^o _D,L,A to S ^o _D,L,B which preserves constant values of the solutions. These results are extended to the case where D is constant and diagonal, but may be singular

Keywords:

• Linear partial differential equations
• Fredholm transformations
• singular Danckwertz