No abstract

This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with a non-classical boundary and an integral over-determination conditions. The existence, uniqueness and continuous dependence upon the data of the classical solution of the inverse problem is shown by using the generalised Fourier method. Furthermore in the numerical process, the boundary element method (BEM) together with the second-order Tikhonov regularization is employed with the choice of regularization parameter based on the generalised cross-validation (GCV) criterion. Numerical results are presented and discussed.

No abstract

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.

No abstract