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The KWW Relaxation Spectrum Determination Using the Post-Widder Inversion Formula
1Department of Technology Fundamentals, Faculty of Production Engineering, University of Life Sciences in Lublin, 20-612 Lublin, Poland.
Materials (Basel, Switzerland)
|August 13, 2026
Summary
This study introduces a new method to recover the relaxation spectrum for the Kohlrausch-Williams-Watts (KWW) model using the Post-Widder Laplace transform. The approach simplifies calculations by avoiding modulus differentiation, enabling accurate KWW spectrum approximation.
Area of Science:
- Materials Science
- Chemical Physics
- Applied Mathematics
Background:
- The Kohlrausch-Williams-Watts (KWW) model is crucial for describing relaxation phenomena in disordered materials.
- Recovering the relaxation spectrum from the relaxation modulus is a challenging inverse problem.
- Existing methods often involve complex differentiation, limiting practical application.
Purpose of the Study:
- To develop an efficient and accurate method for recovering the relaxation spectrum of the KWW model.
- To utilize the Post-Widder Laplace transform inversion rule for direct spectrum approximation.
- To avoid numerical or analytical differentiation of the relaxation modulus.
Main Methods:
- Derivation of an analytical formula for high-order Post-Widder approximations of the KWW relaxation spectrum.
- Development of a recurrence formula for generating a sequence of approximate spectrum models.
- Implementation of an adaptive rule for fast convergence of the model sequence.
- Numerical studies across a wide range of relaxation times and stretching exponents.
Main Results:
- An analytical formula was derived for approximating the KWW relaxation spectrum directly from the relaxation modulus.
- A recurrence relation was established, enabling the generation of approximate spectrum models.
- The algorithm demonstrates fast convergence, achieving high-order models within few iterations.
- Numerical simulations confirm excellent approximation accuracy for various KWW spectra.
Conclusions:
- The proposed Post-Widder inversion method offers a robust and efficient way to determine the KWW relaxation spectrum.
- This approach simplifies the recovery process by relying solely on algebraic calculations with relaxation modulus values.
- The method provides a valuable tool for analyzing relaxation dynamics in diverse materials.
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