Bounds for Solutions of Some Non-Linear Parabolic Problems

Author: Dickson, Robert John

Year: 1954

Degree: Dissertation (Ph.D.)

Advisor: Bohnenblust, Henri Frederic

Committee Member: Unknown, Unknown

Option: Mathematics; Aeronautics

DOI: 10.7907/VMCG-3G45

Abstract

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Functions v(x,t) satisfying certain partial differential equations of the form v[subscript t]=F(x,t,v,v[subscript x],v[subscript xx] in the region R: 0 < x < 1, 0 < t [<=] T are studied. The principal results of Part I determine circumstances in which it can be asserted that v and v[subscript x] admit, in R, bounds which depend only on the bounds for the functions v(x,0), v(0,t), and v(1,t), and for the derivatives of these functions. The proofs employ certain elementary comparison theorems for solutions of partial differential inequalities. Some other applications of these theorems are also included in Part I.

In Part II analogous results are obtained for the system of first order ordinary differential equations which arises when the x-derivatives in the partial differential equation are replaced by divided differences. The bounds obtained in this case hold uniformly under refinement of the discretization.

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