Systems of First Order Partial Differential Equations and Monotone Iterative Technique

Abstract

This chapter analyzes systems of first order partial differential equations and the monotone iterative technique. Because of the fact that first order partial differential equations arise naturally in modeling growth population of cells that constantly change in their properties, a study of existence, uniqueness, and stability properties was initiated. In one study, monotone iterative technique was employed to obtain improvable upper and lower bounds for solutions. The chapter extends such results for systems of first order partial differential equations. If the coefficients of the gradient terms are different, proving existence results for the system by the method of characteristics seems to be difficult. However, if the monotone iterative technique is employed, this difficulty can be eliminated, because in this case, the study of the given system can be reduced to the study of linear uncoupled systems. For this purpose, the chapter first investigates comparison results and then develops a monotone technique in the context of quasi-solutions and mixed monotone operators. One of the comparison results proved provides bounds for solutions in terms of solutions of ordinary differential equations, which in turn contains as a very special case, the well-known Haar's lemma.