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Variational Formulations for Fractional Integral and Differential Equations

  • Delfim F. M. Torres
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Abstract

Given a fractional-order linear equation Lαu = f, we define an appropriate symmetric bilinear form so that the fractional operator Lα is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.

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Cite

BibTeXRIS
BibTeX
@misc{torres2026variational,
  title = {{Variational Formulations for Fractional Integral and Differential Equations}},
  author = {Delfim F. M. Torres},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.22659},
  eprint = {2607.22659},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.22659},
}
RIS
TY  - UNPB
TI  - Variational Formulations for Fractional Integral and Differential Equations
AU  - Delfim F. M. Torres
PY  - 2026
DA  - 2026/07/01
PB  - arXiv
DO  - 10.48550/arxiv.2607.22659
UR  - https://arxiv.org/abs/2607.22659
AB  - Given a fractional-order linear equation L^αu = f, we define an appropriate symmetric bilinear form so that the fractional operator L^α is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.
ER  -