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Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Rasheed Sarbast Kamal, Modanli Mahmut, Abdulazeez Sadeq Taha

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 3

33--42

This paper examines a third-order fractional partial differential equation (FPDE) in the Caputo sense. The Theta difference method (TDM) is utilized to investigate the problem, and a first-order difference scheme is developed. Stability estimates are obtained by applying the Von Neumann analysis method. A test problem is presented as an application, and numerical results are obtained using Matlab software. Error estimates, as well as exact and approximate solutions are presented in a data analysis table. The simulation results are shown through error analysis tables and figures.

Rational operation of variable declining rate filters

Dąbrowski W.

Environment Protection Engineering

2011

Vol. 37, nr 4

35-53

An approximate solution to the system of equations governing the flow distribution among variable declining rate (VDR) filters results in flow rates through filters being elements of a geometrical progression. Based on this approximation, it was deduced how to operate a plant in order to keep the same flow rates through VDR filters for various total head losses of flow. These principles of operation were carefully verified using the accurate di Bernardo mathematical model of VDR filter plants. It was deduced that the longest filter runs result from such an operation of a plant for which the ratio of the highest to the average flow rates through a filter and simultaneously the affordable total head loss of flow through the plant are the highest.