A Reconstructed Potential Function in the Diffusion Equation Based on Terminal Observation Data
The reconstruction of spatial correlation potential function in diffusion equation is studied by using terminal time observation data.Taking the one-dimensional spatial domain diffusion model as an example,a monotonicity method for reconstructing the potential function is studied.This method is also applicable to problems involving multi-dimensional spatial diffusion models.In terms of theoretical analysis,the extremum principle and regularity es-timation of the solution for the forward problem of the diffusion equation are first derived.Then a bounded operator is constructed according to the diffusion equation,and its monotonicity is proved.Furthermore,the uniqueness of the reconstructed potential function is proved by using the monotonicity of the operator and the fixed point iteration.Fi-nally,based on the operator semigroup theory,the stability of the reconstructed potential function in Hilbert space is verified under the condition that the terminal time is large enough.In terms of numerical experiments,an appropriate iterative algorithm is designed based on theoretical analysis.Three typical numerical examples are selected for numer-ical experiments.The experimental results show that the algorithm is stable and effective,and the accuracy of theo-retical results such as monotonicity,uniqueness and stability is verified.Through theoretical analysis and numerical experiments,it is feasible to reconstruct the monotonicity method of the potential function of the diffusion equation.
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