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+? u t = u x x + p ( t ) u + ? , 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 ? 0 1 k ( x ) u ( x , t ) d x = 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 Adomians 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.
control function adomian decomposition method closed form solution quasilinear parabolic partial differential equations energy overspecification