The Ill-posedness and Fourier regularization for the backward heat conduction equation with uncertainty

The backward heat conduction problem (BHCP) is an important branch of inverse problems in mathematical physics. It is widely used and is one of the hot research fields nowadays. However, due to the ill-posedness of the inverse problem, the solution of the problem is difficult to obtain. Therefore, it is necessary to study regularization methods to solve such problems. This paper introduces the backward heat conduction equation (BHCE) with uncertainty and investigates its numerical solution. The uncertainty of parameters in the system is modeled by a fuzzy number. In the solution process, it is first proved that the equation is seriously ill-posed under the concept of granular differentiability. Second, a Fourier regularization method based on the fuzzy Fourier transform is proposed to solve the BHCE with uncertainty, and the granular representation of the approximate solution is given. Finally, some error estimates between the approximate solution and exact solution are provided under the condition of the prior bound assumptions and suitable choices of regularization parameters. Numerical examples also demonstrate the effectiveness and practicability of the proposed method.