Dehghan and Shakeri [15] were first to apply exponential transformation to the Lane-Emden equation in order to address the difficulty of a singular point at x = 0 and solve the resulting nonsingular problem using the variational iteration method.
In order to find the path of a singular control, Pontryagin's maximum principle [34] is utilized and the adjoint variables must satisfy the adjoint equations given by frac{dlambda_{1}}{dt}=-frac{partial H}{partial x}, quadquad frac{dlambda _{2}}{dt}=- frac{partial H}{partial y}.
Looking at Fig. 9, the corresponding shift in the s-value falls within the order of the singular perturbation parameter ε, as expected by a singular perturbation analysis.
Bauman (1991) argued that the key task of modernity is to make order of disorder, often by enforcement of a singular legislative logic.
The main consideration of the present work is the construction of a new approximation of a singular integral in order to use it for a numerical solution of singular integral equations.
As the order of singular optimality turns out to be two, an explicit formula for the singular optimal control is also presented.
In this section we recall briefly the order of singular controls and the higher order optimality conditions.
