Dickson, Robert John (1954) Bounds for solutions of some non-linear parabolic problems. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-12102003-104645
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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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||1 January 1954|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||12 Dec 2003|
|Last Modified:||26 Dec 2012 03:12|
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