Modulating functions-based method for parameters and source estimation in one-dimensional partial differential equations

Abstract

In this paper, modulating functions-based method is proposed for estimating space–time-dependent unknowns in one-dimensional partial differential equations. The proposed method simplifies the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of the modulating functions-based solution is proved. The wave and the fifth-order KdV equations are used as examples to show the effectiveness of the proposed method in both noise-free and noisy cases.

Keywords:

• Inverse problem
• parameter estimation
• source estimation
• joint estimation
• 1D-PDE
• modulating functions

AMS Subject Classifications:

• 65Lxx
• 65Mxx

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that have led to an improved version of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST).