Identifying a control function in parabolic partial differential equations from overspecified boundary data

Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter p(t)p(t) in the quasilinear parabolic equation ut=uxx+p(t)u+ϕut=uxx+p(t)u+ϕ, in [0,1]×(0,T][0,1]×(0,T] with known initial and boundary conditions and subject to an additional condition in the form of ∫01k(x)u(x,t)dx=E(t),0≤t≤T which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method.