Scaling invariant transformation on optimal control problem

Abstract

Analytical solution by scale invariant approach to non-linear diffusion as a control problem is carefully studied. Non-linear diffusion is characterized by the partial differential equation. ∂u/∂t= ∂/∂х(D(u) ∂u/∂х) Where D(u) is the diffusion coefficient and when D(u) = uⁿ, this equation is known as the porous medium equation. Scaling grouping are introduced to reduce the original equations to ordinary differential equation. Self-similar solutions can be found for all n>0 in the case of the porous medium equation. As an illustration for the porous medium equation for n=5, numerical values are obtained for some parameters by the sealing invariant method.