A robust collocation method for singularly perturbed discontinuous coefficients parabolic differential difference equations

ABSTRACT

The singularly perturbed parabolic equations with non-smooth convection coefficient, discontinuous source term, and a large negative spatial shift are studied in this paper. The implicit Euler method for the temporal direction and the fitted extended cubic B-spline collocation method for the spatial direction are applied to formulate a parameter-uniform numerical method. The proposed scheme is proven to be accurate with a linear order of convergence. Two test examples are examined to verify the presented numerical method. The computational findings obtained by the developed method confirm the theoretical estimates. The obtained findings provide more accurate results than methods found in the literature.

KEYWORDS:

• Singularly perturbed problem
• parabolic differential equations
• Extended cubic B-spline method
• discontinuous coefficients
• large negative shift

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their helpful suggestions that helped to improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

This research received no specific funding from public, private, or non-profit funding agencies.

Additional information

Notes on contributors

Wondimagegnehu Simon Hailu

Wondimagegnehu Simon Hailu is a PhD candidate. He is currently employed as a lecturer of mathematics at Dilla University in Ethiopia. His research interest spans the areas of the development of the numerical methods for singularly perturbed ordinary and partial differential equations. He has more than 4 research articles published in various national and international journals.

Gemechis File Duressa

Gemechis File Duressa holds an M.Sc. from Addis Ababa University in Ethiopia and a Ph.D. from the National Institute of Technology in Warangal, India. He is currently employed as a full professor of mathematics and Director of the School of Graduate Studies at Jimma University in Ethiopia. He has over 184 research papers published in various national and international journals. He serves as a referee for several prestigious international journals. He is the research supervisor for Wondimagegnehu Simon Hailu.