Numerical method of solving an inverse source problem for the convective transfer equation

Two inverse problems are considered for reconstruction of a source for the linear convective transfer equation. The first problem is to find a source that depends only on a spatial variable with final redefinition condition. The second problem consists in finding a source that depends only on time, according to an additional condition on the boundary of the region under consideration. To solve the first problem, we first discretize the derivative with respect to the spatial variable and the problem reduces to a differential-difference problem with respect to functions that depend on the time variable. For solving, we propose a special representation. As a result, the solution of the initial problem on each discrete value of the space variable reduces to solving two Cauchy problems and a linear equation with respect to the approximate value of the required source function for a discrete value of the spatial variable. For the numerical solution of the Cauchy problem, the implicit Euler method is used. To solve the second problem, the time discretization of the derivative is done and the problem reduces to a differential-difference problem with respect to functions that depend on the spatial variable. The resulting differential-difference problem is solved using a special representation. As a result, the solution of the second problem at each discrete value of the time variable reduces to solving two Cauchy problems and a linear equation with respect to the approximate value of the unknown source function with a discrete value of the time variable. To solve the Cauchy problem numerically, the implicit Euler method is again applied. In the proposed method, in contrast to the global regularization method, the regularization properties of the computational algorithm are employed and the solution is determined successively without the use of iterative methods. On the basis of the proposed method, numerical experiments were performed for model problems.