Adjoint equation method for solving the inverse diffusion source problem

The objects of the research are differential equations of diffusion (or thermal conductivity) kind. The subject of research is the algorithm for determining the function of the source or the initial conditions of the problem as per the experimentally measured values. The approach is based on a dual representation of functionals corresponding to experimentally observed quantities in the processes of mass and heat transfer. The inverse problem is formulated in the form of integral equations of the first kind, the core of which is the adjoint function (importance function) obtained as a solution of the adjoint (in the Lagrange sense) diffusion (thermal conductivity) equation with the detector sensitivity function in the right-hand side. Meanwhile, solving the adjoint equations by changing the variables is reduced to solving direct equations. To regularize the solution of the Volterra equation of the first kind corresponding to the problem of recovering the dependence of the boundary condition on time, the residual minimization for an overdetermined system of linear equations has been proposed to be used. The problem of reconstructing the dependence of the initial condition on the coordinate is formulated as a Fredholm equation of the first kind, the solution to which has been obtained using the Tikhonov regularization method. The results of model calculations are presented for the restoration of the time dependence of the sources set by a smooth function, a step function and a function with a harmonic component in the problem of one-dimensional diffusion in a homogeneous medium. These results prove that with the selected calculation parameters, the solutions obtained by the proposed method behave regularly and have quite an acceptable accuracy, even though the values of the sought-for function within the set search interval change by six orders of magnitude. This is seen by the authors as the main difference of their method from other approaches to solving this problem.