Basic Equations and Special Functions of Mathematical Physics

Some boundary-value problems of the elastic and thermoelastic equilibrium of wedge-shaped bodies

“…As far as the boundary conditions are concerned, when (9 = Pj, ~ = ~j) on just one of the lateral surfaces of the CCP, for example, when p = P0 and conditions (2.7)b are specified when P = P0, everything can be proved in the same way as in the case of problems (2.6), (2.7)a, (2.8)a, (3.15) if the system of functions which is generated by problem (3.3), (3.4)b when p = P0, (3.4)a when P = Pl, (3.5)a is complete. Without touching on the question of investigating the completeness of this system of functions, we merely note that the system of functions generated by the boundary-value problem (3.3), (3.4)a, (3.5)a is complete [5].…”

“…Alm,, and Blm n are constants, and qJtmn is a non-trivial solution of the following Sturm-Liouville problem [5] ( If expression (3.6), taking formulae (3.7) into account, is substituted into the left-hand sides of equalities (2.5) (we recall that T = 0) and, taking formulae (3.2) account, equalities (3.1) are substituted into the right-hand sides of equalities (2.5), we obtain the following system of equations for the functions v&~ P) and VI& P> ( Proof of the equivalence of the boundary conditions (3.5) and (2.9) reduces to proof of the fact that system (3.12), with the corresponding boundary conditions, only has a trivial solution (with the exception of cases 1 and 2, which have been pointed out above) and the latter problem, in turn, reduces to an investigation of boundary-value problem (3.14). Finally, it can be stated that, if H = Hl(p)H2(13), the solution (3.10) has to be added to problem (2.6), (2.7)a, (2.8)b, (3.15), and the solution (3.11) to the problem (2.6), (2.7)b, (2.8)a, (3.15).…”