Hölder-Type Approximation for the Spatial Source Term of a Backward Heat Equation

Abstract

We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation

with a homogeneous Cauchy boundary condition, where ϕ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term f and provide Hölder-type error estimates in both L ² and H ¹ norms. Numerical experiments are provided.

Keywords:

• Backward
• Fourier series
• Heat source
• Interpolation
• Regularization

AMS Subject Classification:

• 35K05
• 42A16
• 65D05
• 65N21

ACKNOWLEDGMENTS

The work was done when M. N. Minh and P. T. Nam were students in Vietnam National University at HoChiMinh City.