Journal of Scientific Computing

, Volume 79, Issue 2, pp 700–717

# Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach

• Zhaopeng Hao
• Moongyu Park
• Guang Lin
• Zhiqiang Cai
Article

## Abstract

This paper develops a Galerkin approach for two-sided fractional differential equations with variable coefficients. By the product rule, we transform the problem into an equivalent formulation which additionally introduces the fractional low-order term. We prove the existence and uniqueness of the solutions of the Dirichlet problems of the equations with certain diffusion coefficients. We adopt the Galerkin formulation, and prove its error estimates. Finally, several numerical examples are provided to illustrate the fidelity and accuracy of the proposed theoretical results.

## Keywords

Fractional diffusion equation Two-sided fractional derivative Galerkin methods Error estimate

## Mathematics Subject Classification

26A33 65M06 65M12 65M55 65T50

## Notes

### Acknowledgements

Hao would like to acknowledge the support by National Natural Science Foundation of China (No. 11671083), China Scholarship Council (No. 201506090065). Lin gratefully acknowledges the support from National Science Foundation (DMS-1555072, DMS-1736364, and DMS-1821233). Cai would like to acknowledge the support by the NSF Grant DMS-1522707.

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## Authors and Affiliations

• Zhaopeng Hao
• 1
• Moongyu Park
• 2
• Guang Lin
• 3
• 4
• Zhiqiang Cai
• 3
1. 1.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA
3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
4. 4.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA