# Fractional Calculus and the ESR test

## TL;DR

This paper introduces a time-fractional PDE model for blood nutrient concentration affecting ESR, providing an analytic solution using Mittag-Leffler and Wright functions, extending prior models with fractional calculus.

## Contribution

It develops a novel fractional PDE model for ESR analysis and derives an explicit analytic solution, generalizing existing models by incorporating fractional derivatives.

## Key findings

- Analytic solution expressed via Mittag-Leffler and Wright functions.
- Model generalizes previous ESR models by including fractional derivatives.
- Provides a mathematical framework for more accurate blood nutrient modeling.

## Abstract

We consider a partial differential equation associated with a mathematical model describing the concentration of nutrients in blood which interferes directly on the erythrocyte sedimentation rate in the case of an average fluid velocity equal to zero. Introducing the fractional derivative in the Caputo sense, we propose a time-fractional mathematical model which contains, as a particular case, the model proposed by Sharma et al. Our main purpose is to obtain an analytic solution of this time-fractional partial differential equation in terms of the Mittag-Leffler function and Wright function.