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Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

Ravshan Ashurov, Damir Shamuratov

Abstract

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.

Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

Abstract

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time . The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order . The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.
Paper Structure (8 sections, 14 theorems, 162 equations)

This paper contains 8 sections, 14 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic behavior of the multinomial Mittag-Leffler function
  4. Lower bound for an infinitesimal denominator
  5. Existence
  6. Uniqueness
  7. Special case of the operator $A(x,D)$
  8. Conclusion

Key Result

Lemma 2.1

20 Let $\sigma > \frac{|\alpha|}{m} + \frac{N}{2m}.$ Then the operator $D^{\alpha}\widehat{A}^{-\sigma}$ acts (completely) continuously from $L_{2}(\Omega)$ into $C(\Omega)$, and the estimate holds.

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 13 more