# Numerical study of unsteady diffusion in circle

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## Abstract

In this study, we present a discretization scheme based on generalized integral representation method for a numerical evaluation of an unsteady diffusion problem in a circular domain. The scheme employs the fundamental solution of the associated steady-state diffusion operator along with piecewise constant approximation for the unknown function. By its construction, our scheme does not require continuity of the approximate solution across the computational elements and thus is flexible for various partitions of the problem domain. Therefore, for numerical validation, we provide examples in the unit circle partitioned via three different manners. Examples on triangulated partitions of the unit square are also included. The derivation of the numerical scheme is straightforward and it is easy-to-program.

## Keywords

Numerical solution of diffusion problem Unsteady diffusion in a circle Steady-state fundamental solution Generalized integral representation method (GIRM) Numerical scheme based on GIRM## Mathematics Subject Classification

65N99 65L99## Notes

### Acknowledgements

The authors thank the anonymous reviewer for his comments and suggestions, which helped to improving the quality of the manuscript. The authors thank E. Avirmed for the help of code debugging and conducting tests. This work has been done within the framework of the Project P2017-2412 supported by the National University of Mongolia.

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