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On the solution existence for collocation discretizations of time-fractional subdiffusion equations

Sebastian Franz, Natalia Kopteva

TL;DR

This work establishes existence and uniqueness for continuous collocation discretizations in time of time-fractional subdiffusion equations with Caputo derivatives, using both a Lax-Milgram variational framework and an eigenfunction expansion approach. By deriving a matrix representation and reducing the problem to small $m×m$ systems, the authors prove well-posedness for all orders $m≥1$ and all collocation point sets, and further extend the analysis to semilinear problems under Lipschitz conditions. A key analytical result shows that the matrix $M = D_1 W D_2 W^{-1}$ has no real negative eigenvalues, enabling robust solvability; this is supported by positivity of the characteristic polynomial coefficients and generalized Vandermonde determinants, with additional semi-computational validation up to $m≤20$ for common collocation schemes. The paper also provides a semi-computational spectrum analysis and demonstrates the methodology's compatibility with standard spatial discretizations, offering practical guidance for high-order time-stepping in fractional subdiffusion. Overall, the results give rigorous guarantees for the existence and uniqueness of high-order collocation solutions and a pathway to handle semilinear time-fractional problems.

Abstract

Time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain $m\times m$ matrices (where $m$ is the order of the collocation scheme), are verified both analytically, for all $m\ge 1$ and all sets of collocation points, and computationally, for all $ m\le 20$. The semilinear case is also addressed.

On the solution existence for collocation discretizations of time-fractional subdiffusion equations

TL;DR

This work establishes existence and uniqueness for continuous collocation discretizations in time of time-fractional subdiffusion equations with Caputo derivatives, using both a Lax-Milgram variational framework and an eigenfunction expansion approach. By deriving a matrix representation and reducing the problem to small systems, the authors prove well-posedness for all orders and all collocation point sets, and further extend the analysis to semilinear problems under Lipschitz conditions. A key analytical result shows that the matrix has no real negative eigenvalues, enabling robust solvability; this is supported by positivity of the characteristic polynomial coefficients and generalized Vandermonde determinants, with additional semi-computational validation up to for common collocation schemes. The paper also provides a semi-computational spectrum analysis and demonstrates the methodology's compatibility with standard spatial discretizations, offering practical guidance for high-order time-stepping in fractional subdiffusion. Overall, the results give rigorous guarantees for the existence and uniqueness of high-order collocation solutions and a pathway to handle semilinear time-fractional problems.

Abstract

Time-fractional parabolic equations with a Caputo time derivative of order are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain matrices (where is the order of the collocation scheme), are verified both analytically, for all and all sets of collocation points, and computationally, for all . The semilinear case is also addressed.
Paper Structure (8 sections, 7 theorems, 47 equations, 3 figures)

This paper contains 8 sections, 7 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Matrix representation of the collocation scheme \ref{['Col_method']}
  3. Well-posedness for the case ${\mathcal{L}}=c(t)$ without spatial derivatives
  4. Well-posedness by means of the Lax-Milgram Theorem
  5. Well-posedness by means of an eigenvalue test
  6. An analytical approach
  7. A semi-computational approach
  8. Semilinear case

Key Result

Lemma 2.2

With the notations matr1, matr2, matr3, the collocation scheme Col_method1 is equivalent to

Figures (3)

  • Figure 1: Eigenvalues for collocation methods using Chebyshev points $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)
  • Figure 2: Eigenvalues for collocation methods using equidistant points and $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)
  • Figure 3: Eigenvalues for collocation methods using Lobatto points and $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)

Theorems & Definitions (20)

  • Remark 2.1: Case $\theta_m<1$
  • Lemma 2.2
  • Remark 2.3: Case ${\mathcal{L}}={\mathcal{L}}(t)$
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2: Case ${\mathcal{L}}={\mathcal{L}}(t)$
  • Remark 3.3: Coercivity of $a(v,w):=\langle {\mathcal{L}} v, w\rangle$
  • Corollary 3.4: Quadratic collocation scheme
  • proof
  • ...and 10 more