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Space-Dependent Fractional Evolution Equations: A New Approach

Tiago Augusto dos Santos Boza, Paulo Mendes de Carvalho Neto

TL;DR

This paper develops an abstract framework for space-dependent fractional evolution equations with Caputo time derivatives $cD_t^{\alpha(x)}$ and bounded operators $A$ on a Banach space, establishing existence and uniqueness for both linear and semilinear problems. It constructs the variable-order calculus toolkit, proves core inequalities, and shows well-posedness via fixed-point arguments on small time intervals before patching to a global solution, with a Mittag-Leffler-type representation under commutativity conditions. The theory is then applied to epidemiological modeling, notably by formulating fractional and space-dependent SIR models that incorporate memory effects and nonlocal spatial transmission, offering a richer description of heterogeneous populations. The results provide a rigorous foundation for analyzing space-varying memory in diffusion-like processes and lay out promising directions for numerical methods and data-driven validation in epidemic dynamics and other applications with spatial heterogeneity. Overall, the work advances the mathematical treatment of variable-order fractional dynamics with bounded operators and paves the way for extending to unbounded operators and more complex systems.

Abstract

Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish existence and uniqueness results for solutions in both linear and semilinear settings. Our findings provide deeper insights into how spatially varying fractional derivatives influence the behavior of differential equations, shedding light on their mathematical properties and potential applications.

Space-Dependent Fractional Evolution Equations: A New Approach

TL;DR

This paper develops an abstract framework for space-dependent fractional evolution equations with Caputo time derivatives and bounded operators on a Banach space, establishing existence and uniqueness for both linear and semilinear problems. It constructs the variable-order calculus toolkit, proves core inequalities, and shows well-posedness via fixed-point arguments on small time intervals before patching to a global solution, with a Mittag-Leffler-type representation under commutativity conditions. The theory is then applied to epidemiological modeling, notably by formulating fractional and space-dependent SIR models that incorporate memory effects and nonlocal spatial transmission, offering a richer description of heterogeneous populations. The results provide a rigorous foundation for analyzing space-varying memory in diffusion-like processes and lay out promising directions for numerical methods and data-driven validation in epidemic dynamics and other applications with spatial heterogeneity. Overall, the work advances the mathematical treatment of variable-order fractional dynamics with bounded operators and paves the way for extending to unbounded operators and more complex systems.

Abstract

Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish existence and uniqueness results for solutions in both linear and semilinear settings. Our findings provide deeper insights into how spatially varying fractional derivatives influence the behavior of differential equations, shedding light on their mathematical properties and potential applications.

Paper Structure

This paper contains 12 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Motivation and Challenges in Solving Space-Dependent Fractional Differential Equations
  3. Summary of the Manuscript
  4. Prerequisites on the Theory of Variable Order
  5. Well-Posedness of the Linear Problem
  6. Well-Posedness of the Semilinear Problem
  7. Application of the Theory
  8. The Classical SIR Model
  9. Addressing Limitations of the Classical SIR Model
  10. A Fractional SIR Model for Epidemic Dynamics
  11. A Generalized SIR Model with Fractional Differentiation Depending on Spatial Position
  12. Final Conclusions

Key Result

Lemma 11

For $0<\theta<\gamma<1$, we have that for every $y\in[e^{1/(1-\theta)},e^{1/(1-\gamma)}]$.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 40 more