Stationary solutions, boundary value problem

Another well-established formulation of classical mechanics (which is, of course, equivalent) is based on a boundary value problem. We seek a stationary solution of a functional of the path, S [9] ... 

Polyanin, A. D., Method for solution of some non-linear boundary value problems of a non-stationary diffusion-controlled (thermal) boundary layer, Int. J. Heat Mass Transfer, Vol. 25, No. 4, pp. 471 -85, 1982. 

In the case when the sutetance concentration at both ends of the segment (0,1] differs from zero, boundary layers appear in the neighborhood of both ends of the interval. The qualitative behavior of the solution for the stationary boundary value problem with distributed sources... 

In the study of stationary states in quantum chemistry, one would normally introduce boundary conditions, as for instance, (a) = (b) = 0 and solve the resulting eigenvalue problem. Solutions occur only for certain values of E = n, so-called eigenvalues, and the corresponding solutions (x) are called eigenfunctions. 

Recently Wake has applied a variational treatmrait to the stationary problem, deriving critical conditions both for the class A geometries and for the cube, square rod, and equicylinder in systems where the heat transf(H is resisted by conduction in the interior and by convection at the surface. Here the condition at the boundary becomes dO/dp + N6 = 0, where N is the Biot number hLIk The limit as bf- oo corresponds to the Frank-Kamenetskii solutions. Wake uses trigonometric, rather than polynomial, expressions for this tempoature field and proceeds to derive the conditions under which solutions of the time-dependent variational equations are just possible, associating these with a critical value of 6. Results for N = oo are listed as variational (2) in the Table. For the more rorai conditions of finite Biot numbers Wake compares his results for class A geometries with the analytical forms due to Thomas. Errors are less than 0.1 % though the computational effort required is substantial. 

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