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.

中文翻译：

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使用非局部分数算子的细胞溶质钙浓度分布的数学建模

本文的目的是研究由对流扩散方程控制的钙分布。数学和计算模型为理解依赖于细胞溶质钙浓度的钙信号提供了见解。此处模型包括重要的生理参数，如扩散系数、流速等。数学模型使用 Hilfer 导数进行细分，并构建了适当的边界条件。由于分数阶微分算子的非局部特性，即系统的下一状态不仅取决于其当前状态，而且取决于其所有先前状态，因此分数阶导数的使用比整数阶更有利。使用积分变换技术找到了在研究红细胞中细胞溶质钙扩散时出现的分数平流扩散方程的解析解。由于 Hilfer 导数是 Riemann-Liouville 和 Caputo 导数的推广，所以这两个导数也作为特例推导出来。已经完成数值模拟以观察表示模型的微分方程中涉及的导数的分数阶对钙浓度的影响，钙浓度是时间和距离的函数。钙的浓度分布由分数阶显着改变。已经完成数值模拟以观察表示模型的微分方程中涉及的导数的分数阶对钙浓度的影响，钙浓度是时间和距离的函数。钙的浓度分布由分数阶显着改变。已经完成数值模拟以观察表示模型的微分方程中涉及的导数的分数阶对钙浓度的影响，钙浓度是时间和距离的函数。钙的浓度分布由分数阶显着改变。