## Abstract

We investigate an inverse and ill-posed problem for the two-dimensional inhomogeneous heat equation in the presence of a general source term. The goal here consists of recovering not only the temperature distribution but also the thermal flux from the measure data. With the appearance of the general source term, this model gets far worse than its homogeneous counterpart. Based on an analysis of the instability caused in the solution, we propose a kernel regularization scheme to stabilize the investigated problem. The Hölder-type convergence estimates are achieved under some appropriate a priori assumptions. We further numerically demonstrate the theoretical results by proposing a robust algorithm based on the 2-D fast Fourier transform. The numerical outputs exemplify the feasibility and efficiency of the proposed method.

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## Funding

The work of the first-name author, Tran Nhat Luan, is supported by the Institute for Computational Science and Technology (ICST), Ho Chi Minh City, and Department of Science and Technology, Ho Chi Minh City (under the grant no. 456/QÐ-KHCNTT).

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Communicated by: Lothar Reichel

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Luan, T.N., Khanh, T.Q. Determination of temperature distribution and thermal flux for two-dimensional inhomogeneous sideways heat equations.
*Adv Comput Math* **46, **54 (2020). https://doi.org/10.1007/s10444-020-09796-w

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### Keywords

- Two-dimensional sideways heat equation
- Kernel regularization method
- Stable estimate
- Hölder-type error estimates

### Mathematics subject classification (2010)

- 35K05
- 35Q79
- 35R25
- 35R30
- 80A23