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Nonlinear Boundary Conditions in Sobolev Spaces

Description: The method of dual steepest descent is used to solve ordinary differential equations with nonlinear boundary conditions. A general boundary condition is B(u) = 0 where where B is a continuous functional on the nth order Sobolev space Hn[0.1J. If F:HnCO,l] —• L2[0,1] represents a 2 differential equation, define *(u) = 1/2 IIF < u) li and £(u) = 1/2 l!B(u)ll2. Steepest descent is applied to the functional 2 £ a * + £. Two special cases are considered. If f:lR —• R is C^(2), a Type I boundary condition is defined by B(u) = f(u(0),u(1)). Given K: [0,1}xR—•and g: [0,1] —• R of bounded variation, a Type II boundary condition is B(u) = ƒ1/0K(x,u(x))dg(x).
Date: December 1984
Creator: Richardson, Walter Brown
Partner: UNT Libraries