Derivative with a New Parameter

Dr. Abdon Atangana is Academic Head of Department and Professor of Applied Mathematics at the University of the Free State, Bloemfontein, Republic of South Africa. He obtained his honours and master’s degrees from the Department of Applied Mathematics at the UFS with distinction. He obtained his PhD in applied mathematics from the Institute for Groundwater Studies. He was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, and he was awarded The World Academy of Sciences (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries. In 2018 Dr. Atangana was elected as a member of the African Academy of Sciences and in 2021 a member of The World Academy of Sciences. He also ranked number one in the world in mathematics, number 186 in the world in all fields, and number one in Africa in all fields, according to the Stanford University list of top 2% scientists in the world. He was one of the first recipients of the Obada Award in 2018. Dr. Atangana published a paper that was ranked by Clarivate in 2017 as the most cited mathematics paper in the world. Dr. Atangana serves as an editor for 20 international journals, lead guest editor for 10 journals, and is also a reviewer of more than 200 international accredited journals. His research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Dr. Atangana is a pioneer in research on fractional calculus with non-local and non-singular kernels popular in applied mathematics today. He is the author of numerous books, including Integral Transforms and Engineering: Theory, Methods, and Applications, Taylor and Francis/CRC Press; Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, Taylor and Francis/CRC Press; Numerical Methods for Fractional Differentiation, Springer; Fractional Stochastic Differential Equations, Springer; Fractional Order Analysis, Wiley; Applications of Fractional Calculus to Modeling in Dynamics and Chaos, Taylor and Francis/CRC Press; Fractional Operators with Constant and Variable Order with Application to Geo-hydrology, Elsevier/Academic Press; Derivative with a New Parameter: Theory, Methods, and Applications, Elsevier/Academic Press; and New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, Elsevier/Academic Press; among others.

Chapter 1. History of derivatives from Newton to Caputo
1.1: Introduction of calculus
1.2: Definition of local and fractional derivative
1.3: Definitions and Properties of their anti-derivatives (Integral)
1.4: Limitations and strength of local and fractional derivatives
1.5: Classification of fractional derivatives
Chapter 2:  Local derivative with new parameter
2.1: Definition and anti-derivative
2.2: Properties of local derivative with new parameter
2.3: Definition Partial derivative with new parameter
2.4: Properties of partial derivatives with new parameters
Chapter 3: Novel integral transform
3.1: Definition and properties of beta-Laplace transform
3.2: Definition and properties of beta-Sumudu transform
3.3: Definition and properties of beta-Fourier transform
Chapter 4: Method for partial differential with beta derivative
4.1:  Homotopy  decomposition method
4.2: Variational iteration method
4.3:  Sumudu decomposition method
4.4: Laplace decomposition method
4.5: Numerical method
Chapter 5:  Applications of local derivative with new parameter
5.1: Model of groundwater flow within the confined aquifer
5.2: Model of groundwater flow equation within a leaky aquifer
5.3: Model of Lassa fever or Lassa hemorrhagic fever
5.4: Model of Ebola hemorrhagic fever
References