The Mittag-Leffler Function for Re-Evaluating the Chlorine Transport Model: Comparative Analysis

Abstract

This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

“…Accordingly, the successful management of ensuring the quality of drinking water requires adherence to the aforementioned chlorine concentration and ensuring that this percentage does not rise above the specified limit' [2]. For more on the industrial relevance of chlorine, one may read the recent work by Aljohani et al [3], where various uses of chlorine were stated, including its great applications in water sciences, insecticides and drugs manufacturing, and textile industries to mention just a few.…”

“…The just stated model in equations (1)-( 3) was initially studied in [4] through the application of separation of variable method, and in [1,2] using the combination of Laplace transform method and the complex analysis' residue approach via the use of root-finding numerical schemes, and then in [3] where yet the application of separation of variable method is sought through the use of Mittag-Leffler functions after the incorporation of variable-order partial derivative. However, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of bi-layered cylindrical pipe, having considered a composite conducting cylindrical pipe with perfect interfacial conditions imposed in-between the two layers.…”

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

“…Besides, several authors ([3]- [8]) implemented different analytical and numerical methods to treating the system (1)- (5). The LT method was widely applied to solve various physical models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, it requires massive calculations as seen in Refs.…”

A Novel Ansatz Method for Solving the Neutron Diffusion System in Cartesian Geometry

“…Although the LT method was shown as an effective and efficient approach to solve several scientific models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], it needs massive computational calculations when applied on the present system, especially when using the residues method as can be observed in Khaled [2], Bakodah and Ebaid [16], and Ebaid et al [20]. However, researchers in the field of Mathematics and Physics would prefer simple but effective methods to solve the governing system of the studied phenomena.…”