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Two methods addressing variable-exponent fractional initial and boundary value problems and Abel integral equation

Xiangcheng Zheng

TL;DR

This work develops two complementary analytic frameworks—convolution and perturbation methods—to analyze variable-exponent fractional models, including initial/boundary value problems and Abel integral equations. The convolution method reformulates leading-variable-exponent problems into a transformed model via a generalized identity function, enabling proofs of well-posedness, regularity, and inverse problems, as well as numerical schemes with provable convergence. The perturbation method provides an alternative when convolution is inapplicable (e.g., two-sided space-fractional or distributed-variable models) by decomposing the kernel into a base part and a low-order perturbation, yielding coercivity and contraction-based well-posedness results. The Abel integral equation is treated under both approaches, with results covering $0< ext{alpha}_0<1$, $ ext{alpha}_0=0$, and $ ext{alpha}_0=1$, and additional data-constraint requirements identified for endpoint exponents. Together, the methods offer a robust toolkit for rigorous analysis and numerical approximation of variable-exponent nonlocal models and their integral equations, with implications for applied contexts where exponents vary in time or space.

Abstract

Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the sundiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relation between the convolution method and the perturbation method is discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.

Two methods addressing variable-exponent fractional initial and boundary value problems and Abel integral equation

TL;DR

This work develops two complementary analytic frameworks—convolution and perturbation methods—to analyze variable-exponent fractional models, including initial/boundary value problems and Abel integral equations. The convolution method reformulates leading-variable-exponent problems into a transformed model via a generalized identity function, enabling proofs of well-posedness, regularity, and inverse problems, as well as numerical schemes with provable convergence. The perturbation method provides an alternative when convolution is inapplicable (e.g., two-sided space-fractional or distributed-variable models) by decomposing the kernel into a base part and a low-order perturbation, yielding coercivity and contraction-based well-posedness results. The Abel integral equation is treated under both approaches, with results covering , , and , and additional data-constraint requirements identified for endpoint exponents. Together, the methods offer a robust toolkit for rigorous analysis and numerical approximation of variable-exponent nonlocal models and their integral equations, with implications for applied contexts where exponents vary in time or space.

Abstract

Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the sundiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relation between the convolution method and the perturbation method is discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.
Paper Structure (27 sections, 12 theorems, 222 equations, 1 figure)

This paper contains 27 sections, 12 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. A generalized identity function
  5. Convolution method
  6. A transformed model
  7. Well-posedness of transformed model
  8. Well-posedness of original model
  9. Solution regularity
  10. An inverse problem
  11. Numerical approximation
  12. A numerically-feasible formulation
  13. Semi-discrete scheme and error estimate
  14. Fully-discrete scheme and error estimate
  15. Numerical experiments
  16. ...and 12 more sections

Key Result

Theorem 1

Suppose $f\in W^{1,1}(L^2)$ and $u_0\in \check H^2$, then model (Model2)--(ibc) admits a unique solution in $W^{1,p}(L^2)\cap L^p(\check H^2)$ for $1< p<\frac{1}{1-\alpha_0}$ and

Figures (1)

  • Figure 1: (left) Plots of $U_n$ at $(\frac{1}{2},\frac{1}{2})$ under different $\alpha_0$; (right) Errors under different $\alpha_0$ and $\tau$.

Theorems & Definitions (24)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 14 more