Research

Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann

Research Interests

  • Computational and Applied Mathematics

  • Modelling, Numerical Analysis, Scientific Computing and PDEs

  • Nonlocal Models and Applications of Nonlocal Operators

  • Fractional Calculus, Sobolev Spaces, Calculus of Variations and PDEs

  • Numerical Approximation of Nonlocal and Fractional Operators and PDEs

Publications

  1. X. Feng and M. Sutton, Finite Element Methods for Approximating Weak Fractional Derivatives and Fractional Differential Equations, In preparation.

  2. X. Feng and M. Sutton, On A New Class of Fractional Calculus of Variations and Related Fractional Differential Equations, Submitted (2021) (also see arXiv:2107.04134).

  3. X. Feng and M. Sutton, On New Families of Fractional Sobolev Spaces, In review (2020) (also see arXiv:2007.10245*).

  4. X. Feng and M. Sutton, A New Theory of Fractional Differential Calculus, Anal. Appl. 19(4) (2021) 715-750 (also see arXiv:2007.10244*).

  5. X. Feng and M. Sutton, A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces: One-dimensional Case, arXiv:2004.10833 Research notes (2020).


* The two papers, "A New Theory of Fractional Differential Calculus" and "On New Families of Fractional Sobolev Spaces" are companion papers. The arXiv preprint, titled "A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces: One-dimensional Case" is a combination of these two submissions. The arXiv preprint contains a plethora of background information from classical fractional calculus and numerous details that may aid a newcomer to any of the related topics or the interested reader.

Presentations

  1. A New Finite Element Method for Fractional Calculus of Variations Problems. Joint Mathematics Meetings 2022, Seattle, WA, January 2022.

  2. A New Finite Element Method for Fractional Calculus of Variations Problems. Fall 2021 Finite Element Circus, University Park, PA, November 2021.

  3. Weak Solutions to Fractional Differential Equations and Their Finite Element Approximation. Math Presentations Seminar, Knoxville, TN, October 2021.

  4. New Families of Fractional PDEs Arising from Fractional Calculus of Variations. 44th SIAM SEAS Annual Meeting, Auburn, AL, Septemeber 2021.

  5. One-side Fractional Differential Equations and Their Finite Element Approximations. Math Presentations Seminar, Knoxville, TN, September 2021.

  6. A New Theory of Weak Fractional Calculus, Sobolev Spaces, and the Fractional Calculus of Variations. SIAM Annual Meeting, Spokane, WA, July 2021.

  7. A New Finite Element Method for Fractional Calculus of Variations Problems. Spring 2021 Finite Element Circus, April 2021.

  8. Weak Solutions to Fractional Differential Equations. UTK Graduate Research Presentation Competition, March 2021.

  9. A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces. One Nonlocal World, January 2021.

  10. A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces. The Second Joint SIAM/CAIMS Annual Meeting, Toronto, Canada, July 2020.

  11. New Families of Fractional PDEs Arising from Fractional Calculus of Variations and Fractional IVPs. Differential Equations Seminar, Knoxville, TN, March 2020.

  12. A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces. Computational and Applied Mathematics Research Seminar, Knoxville, TN, March 2020.

  13. A New Theory of Fractional Differential Calculus, Fractional Sobolev Spaces, and Fractional PDEs. SIAM Graduate Student Research Showcase, Knoxville, TN, November 2019.

  14. An Introduction to Fractional Calculus. Maryville College Invited Talk, Maryville, TN, October 2019.

  15. A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces. 43rd SIAM SEAS Annual Meeting, Knoxville, TN, September 2019.

  16. Fractional Differential Calculus, Fractional Calculus of Variations and Differential Equations and their Numerical Solutions. Oral Specialty Exam, Knoxville, TN, May 2019.

  17. Path Optimization in a Bounded Domain. RISE Above Research Conference, Johnson City, TN, April 2016.

  18. Path Optimization in a Bounded Domain. National Conference on Undergraduate Research, Asheville, NC, April 2016.

Conferences, Lectures, and Workshops

  • Joint Mathematics Meetings 2022 - Seattle, Washington - January 2022

  • Fall 2021 Finite Element Circus - Penn State University - University Park, PA - November 2021

  • 44th SIAM SEAS Annual Meeting, Auburn University - Auburn, AL - September 2021 (Originally scheduled March 2020)

  • SIAM Annual Meeting (AN21), Spokane, Washington - July 2021 (Virtual)

  • 50th John H. Barrett Memorial Lectures - A3N2M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models, University of Tennessee, Knoxville - Knoxville, TN - May 2021 (Virtual - Postponed from 2020)

  • Spring 2021 Finite Element Circus - Virtual - April 2021

  • SIAM Conference on Computational Science and Engineering (CSE21), Fort Worth, Texas - March 2021 (Virtual)

  • One Nonlocal World Opening Event, Virtual - January 2021

  • 4th Conference on Numerical Methods for Fractional-Derivative Problems, Beijing Computational Science Research Center - Beijing, China - October 2020 (Virtual)

  • 2020 SACNAS - The National Diversity in STEM Virtual Conference, October 2020 (Virtual)

  • 3rd Annual Meeting of the SIAM Texas-Liousiana Section, Texas A&M University - College Station, TX - October 2020 (Virtual)

  • The Second Joint SIAM/CAIMS Annual Meeting (AN20), Toronto, Canada - July 2020 (Virtual)

  • The Second International Conference on Computational Methods and Applications in Engineering, Mississippi State University - Starkville, MS - May 2020 (Postponed)

  • Householder Lecture, University of Tennessee, Knoxville - Knoxville, TN - November 2019

  • 43rd SIAM SEAS Annual Meeting, University of Tennessee, Knoxville - Knoxville, TN - September 2019

  • Householder Lecture, Oak Ridge National Laboratory - Oak Ridge, TN - February 2019

  • Summer School on Fractional and Other Nonlocal Models 2018, Basque Center for Applied Mathematics (BCAM) - Bilbao, Spain - May 2018

Wonderful Quotes

The following words have stuck with me for sometime and/or had a lasting impact on the way I relate to mathematics.

  • Math is the vocabulary of your own intuition. - Dan Meyer

  • Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann

  • Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were. - Andrew Wiles

The photo in the banner above was taken on a brutally cold and windy morning on Roan Mountain. January 2017.