Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology Abdon Atangana Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of Free State, Bloemfontein Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2018 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. 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Bilbow Typeset by VTeX I dedicate this book to the memories of my Lord Jesus Christ who died on the cross for our sins. To the memory of my late mother Ngono Antoinette who died in 2010; my grandmother who passed away in 2016. Preface The Lord God so loved the world and he gave it only begotten subject applied mathematics, such that whosoever learn it should not be ignorant, but use it, to understand the nature and also predict some natural phenomena accurately. And the Lord God said “My people are de- stroyed for lack of knowledge.” Well, applied mathematics is the gate and key of sciences, therefore negligence of applied mathematics injures the whole corpus of knowledge, since he who is ignorant of it cannot fully grasp other sciences or things of this world. As an example, how do we explain the movement of groundwater through the geological formation? This can only be achieved via mathematical equations. xv Acknowledgment The fear of the Lord is the foundation of wisdom, and the knowledge of the holy one is un- derstanding. I will first and foremost like to express my deepest thanks to the Lord Jesus, the King of kings, the Lord of lords, who died on the cross of Golgotha to save mankind from sins. I would thank God for his protection, heath, and wisdom. In the process of putting this book together, I become aware of how true this gift of writing is for me. The Lord God gave me the power to believe in my passion and peruse my dreams. I could not have completed this second book without faith I have in God the almighty. I am indebted to all my students: • To Miss Jessica Spannenberg my Ph.D. student for her assistance in proofreading. • To my Postdoctoral fellow Dr. Kolade Owolabi M. for his help in latex and proofreading. • To my Turkish Friend Dr. Ilknur Koca Hocam for her help with latex. • An excellent wife is the crown of her husband, who can find? For her worth is far above jewels. Thus whosoever find a good wife finds a good thing, and obtains favor of the Lord, that is why the Lord God, after overseeing the time-fractional differentiation of Abdon Atangana said, “It is time-locally but not time-fractionally good for Abdon Atangana to be alone, I will make him a helper fit for him.” And the rib that the Lord God had taken from Abdon Atangana, he made into a beautiful, caring, loving, wise, intelligent, God- fearing woman (Ernestine Alabaraoye) and brought her to Abdon Atangana. Then Abdon Atangana said, “This at last is bone of my bones, a controller of my fractal and chaotic be- haviors, flesh of my flesh, blood of my blood, love of my heard, my fractional derivative in Atangana–Baleanu sense, she will be called Ernestine Atangana, because she was taken out of me. Because of this, Abdon Atangana shall leave his father and mother and hold fast to his God-fearing wife and they shall become one flesh.” From my soul I would like deeply to thank my wife for her love, good food, prayers, and constant support. • Thanks to my two-year-old son Atangana Ndzengue Biliga Melchizedek who always helps me to wake up at night in order to complete this book. • To my family: Eleme Catherine, Ngono Adelaide, Eyebe Izidore, Ngono Antoinette, and all my nephews and nieces for their prayers. • To my fathers: Dr. Pierre Ndzengue, the ambassador of Cameroon to Japan, and Tara Noah Jean for their prayers and support. xvii CHAPTER 1 Aquifers and Their Properties 1.1 Introduction An aquifer is a geological formation in which groundwater flows through with ease. Aquifers should therefore have both permeability and porosity. Examples of these geological forma- tions which form aquifers include sandstone, conglomerate, fractured limestone, and uncon- solidated sand and gravel formations. Another example of an aquifer system is a fractured volcanic rock formation such as columnar basalt. The rubble zone where volcanic flow exists is usually both porous and permeable, thus allowing for good aquifer systems [1,2].Further- more, geological formations such as granite and schist have low porosity, and so are usually classified as poor aquifers. However, once they become fractured, they can produce good aquifers [1,2]. Furthermore, for a well to be productive, the well itself should be drilled into the ground to penetrate the aquifer. If groundwater is abstracted from the well at a rate faster than the aquifer is recovered, there is a decline in the water table, sometimes to a point that the well dries. As pumping occurs, the water table normally declines, resulting in a cone of depression at the well. Moreover, groundwater flow generally follows the slope of the water table, thus in this case, groundwater flows in the direction of the well being pumped. Not all aquifers can be seen as groundwater reservoirs [1–3]. These reservoirs are, however, found underground but only in cavernous geological formations where the formations surrounding fractures or cracks have undergone dissolution, forming open channels which allow rapid wa- ter movement similar to that of a river. Furthermore, since groundwater migrates at a slow rate through pore spaces of aquifer material, the only living organisms, that could float as it would in an actual river, are bacteria or viruses which are minute enough to migrate through pore spaces. Movement of groundwater through an aquifer occurs as groundwater is forced through a pore space of geological formations. Hence, porosity essentially defines an aquifer. To add, porosity of certain aquifers also allows them to act as good filters generating natural purifica- tion [2,3]. Since effort is required for forcing water movement through small pores, there is a loss in groundwater energy as it flows. This eventually results in decreased hydraulic head in the direction of groundwater flow. On the other hand, when pores are large in size, there is increased permeability, less energy loss, and rapid groundwater movement. Subsequently, groundwater migration is rapid for aquifers with large pores, such as in the case of the lower Portneuf River aquifer or in cases where porosity is a result of fractures which are intercon- nected. It is also significantly rapid in fractured rock aquifers such as the basalts of the eastern Snake River Plain. Despite being good aquifers, they are vulnerable to spreading of contami- nation which is challenging and often impossible to prevent [3]. Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology 1 DOI: 10.1016/B978-0-12-809670-3.00001-1 Copyright © 2018 Elsevier Inc. All rights reserved. 2Chapter1 1.2 Classification of Aquifers Hydrogeology is known as the field investigating the flow of water through aquifers. Terms related to aquifers include aquitard, which is a formation associated with low permeabil- ity along an aquifer; and aquiclude or aquifuge, which are associated with an impermeable formation existing above or below an aquifer [4–7]. In the case where the impermeable formation exists above the aquifer, the pressure created could cause the aquifer system to become confined.