2 research outputs found
Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations
In this paper, by employing the asymptotic expansion method, we prove the
existence and uniqueness of a smoothing solution for a time-dependent nonlinear
singularly perturbed partial differential equation (PDE) with a small-scale
parameter. As a by-product, we obtain an approximate smooth solution,
constructed from a sequence of reduced stationary PDEs with vanished high-order
derivative terms. We prove that the accuracy of the constructed approximate
solution can be in any order of this small-scale parameter in the whole domain,
except a negligible transition layer. Furthermore, based on a simpler link
equation between this approximate solution and the source function, we propose
an efficient algorithm, called the asymptotic expansion regularization (AER),
for solving nonlinear inverse source problems governed by the original PDE. The
convergence-rate results of AER are proven, and the a posteriori error
estimation of AER is also studied under some a priori assumptions of source
functions. Various numerical examples are provided to demonstrate the
efficiency of our new approach