Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator

Ritu Agarwal; null Kritika; Sunil Dutt Purohit; Devendra Kumar

doi:10.3934/dcdss.2021017

Article Contents

2021, Volume 14, Issue 10: 3387-3399. Doi: 10.3934/dcdss.2021017

This issue Previous Article Applying quantum calculus for the existence of solution of

$ q $

-integro-differential equations with three criteria Next Article Analysis and new applications of fractal fractional differential equations with power law kernel

Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator

1.
Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, INDIA
2.
Department of HEAS(Mathematics), RTU, Kota-324010, INDIA
3.
Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India

^* Corresponding author: Devendra Kumar
^* Corresponding author: Devendra Kumar

Received: September 30, 2019

Revised: November 30, 2019

Early access: March 2021

Published: October 2021

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Abstract

The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow velocity etc. The mathematical model is fractionalised using Hilfer derivative and appropriate boundary conditions have been framed. The use of fractional order derivative is more advantageous than the integer order because of the non-local property of the fractional order differentiation operator i.e. the next state of the system depends not only upon its current state but also upon all of its preceeding states. Analytic solution of the fractional advection diffusion equation arising in study of diffusion of cytosolic calcium in RBC is found using integral transform techniques. Since, the Hilfer derivative is generalisation of Riemann- Liouville and Caputo derivatives so, these two are also deduced as special cases. The numerical simulation has been done to observe the effects of the fractional order of the derivatives involved in the differential equation representing the model over the concentration of calcium which is function of time and distance. The concentration profile of calcium is significantly changed by the fractional order.

Keywords:

Fractional advection diffusion equation,
cytosolic calcium concentration,
Hilfer derivative,
Fourier transform,
Laplace transform.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Graph between $ C(x, t) $ and $ t $ for various values of $ \xi $ for $ \eta = 0 $ which corresponds to the Riemann- Liouville derivative

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Figure 2. Graph between $ C(x, t) $ and t for various values of $ \xi $ for $ \eta = 1 $ which corresponds to Caputo derivative

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Figure 3. Graph between $ C(x, t) $ and t for various values of $ \xi $ for $ \eta = 0.90 $

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