Abstract:We study a new non-classical class of variational problems that is motivated by some recent research on the non-linear revenue problem in the field of economics. This class of problem can be set up as a maximising problem in the calculus of variations (CoV) or optimal control. However, the state value at the final fixed time, ( ), y T is a priori unknown and the integrand is a function of the unknown ( ).y T This is a non-standard CoV problem. In this paper we apply the new costate boundary conditions ( ) p T in the formulation of the CoV problem. We solve a sample example in this problem class using the numerical shooting method to solve the resulting TPBVP, and incorporate the free ( ) y T as an additional unknown. Essentially the same results are obtained using symbolic algebra software.
The scalar scattering of a plane wave by a strictly convex obstacle with impedance boundary conditions is considered. A uniform bound of the total cross section for all values of the frequency is presented. The high-frequency limit of the transport cross section is calculated and presented as a classical functional of the variational calculus.
We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover, for exterior Helmholtz equation we have constructed an explicit normal derivative of the Dirichlet Green function. In the same way the Dirichlet-to-Neumann operator is constructed. We proved that normalized coefficients are uniformly bounded from zero.
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