Inverse time-dependent source problem for the heat equation with nonlocal boundary conditions

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.