[Paper Review] Hermite functions with discontinuous coefficients for the solution of fractal diffusion retrospective problems

Oleg Yaremko|arXiv (Cornell University)|Sep 18, 2013

Mathematical and Theoretical Analysis3 references3 citations

TL;DR

This paper proposes a novel method for solving fractal diffusion retrospective inverse problems in piecewise-homogeneous media using Hermite functions with discontinuous coefficients. By generalizing classical Hermite polynomials to account for spatially discontinuous diffusion coefficients, the authors derive a series solution based on fractional-order time derivatives and operator transforms, proving existence and uniqueness of the solution for the inverse problem.

ABSTRACT

In this article we study the retrospective inverse problem. The retrospective inverse problem consists of in the reconstruction of a priori unknown initial condition of the dynamic system from its known final condition. Existence and uniqueness of the solution is proved.

Motivation & Objective

• To address the ill-posed nature of retrospective inverse problems in fractal diffusion, where initial conditions must be reconstructed from final-state observations.
• To extend classical Hermite function methods to systems with discontinuous diffusion coefficients, modeling piecewise-homogeneous media.
• To establish a formal solution for the inverse problem using operator transforms and generalized Hermite polynomials.
• To prove existence and uniqueness of the solution under conditions of unlimited solvability for the underlying boundary value problem.
• To generalize the solution framework to include fractional-order time derivatives (α) and hyperbolic limits (α=2).

Proposed method

• Utilizes the operator transform method to map the inverse problem into a spectral domain using eigenfunctions of Sturm-Liouville problems on piecewise-homogeneous domains.
• Introduces Hermite functions with discontinuous coefficients derived from the Taylor expansion of the Mittag-Leffler function in the spectral domain.
• Applies the inverse operator transform to reconstruct the initial condition as a series involving derivatives of the final-state data at t=0.
• Derives a generalized form of Hermite polynomials: $ H_j(x) = Σ_{k=0}^{[j/2]} \frac{(-1)^k j!}{\Gamma(k\alpha+1)(j-2k)!} x^{j-2k} $, where α governs the fractal behavior.
• Establishes a solution representation: $ f(x) = \sum_{j=0}^{\infty} \frac{u_j(\tau) \tau^{\beta j}}{j!} H_j\left(\frac{x}{\tau^\beta}\right) $, with $ \beta = \alpha/2 $, valid for fractal and hyperbolic cases.
• Validates the method through analogies with the hyperbolic case (α=2), recovering known solutions such as $ f(x) = \frac{u(\tau, \tau+x) + u(\tau, \tau-x)}{2} $.

Experimental results

Research questions

• RQ1 How can Hermite functions be generalized to handle discontinuous diffusion coefficients in fractal diffusion problems?
• RQ2 What conditions ensure the existence and uniqueness of a solution to the retrospective inverse problem in piecewise-homogeneous media?
• RQ3 How does the introduction of fractional time derivatives (via α) affect the structure of the solution and its convergence?
• RQ4 Can the operator transform method be adapted to derive a formal solution for inverse problems with discontinuous coefficients?
• RQ5 What is the relationship between the generalized Hermite functions and classical Hermite polynomials in the limit α→2?

Key findings

• The solution to the retrospective fractal diffusion problem is expressed as an infinite series involving generalized Hermite functions with discontinuous coefficients.
• Existence and uniqueness of the solution are rigorously proven under the condition of unlimited solvability of the underlying boundary value problem.
• For the hyperbolic case (α=2), the solution reduces to $ f(x) = \frac{u(\tau, \tau+x) + u(\tau, \tau-x)}{2} $, recovering a known result.
• The generalized Hermite polynomials are defined as $ H_j(x) = \sum_{k=0}^{[j/2]} \frac{(-1)^k j!}{\Gamma(k\alpha+1)(j-2k)!} x^{j-2k} $, which reduce to classical Hermite polynomials when α=2.
• The method successfully generalizes classical inverse problem solutions by replacing standard Hermite functions with those adapted to discontinuous coefficients.
• The inverse Dirichlet problem for the half-plane is solved using a similar framework, yielding $ f(y) = \sum_{j=0}^{\infty} \frac{u^{(j)}(l,0)}{j!} \cdot \frac{(y+li)^j + (y-li)^j}{2} $, with $ f(y) = \text{Re}\, u(l, y+li) $ in the analytic continuation case.