Abdon Atangana

Limitations of Groundwater Models With Local Derivative

This chapter presents the limitation of the application of classical differential operators to describe accurately the movement of groundwater pollution within subsurface water.

Aquifers and Their Properties

This chapter presents the general definition of the geological formation called aquifers together with their mathematical properties. Some reported examples of these aquifers around the world are presented.

Applications of Fractional Operators to Groundwater Models

This chapter presents models of groundwater flow for steady and unsteady state within a confined, unconfined and leaky aquifers within the scope of fractional differentiation and integration. For each model using fixed-theorem, the analysis of existence and uniqueness solution is presented in detail. The numerical and analytical solutions are derived and finally the limitation of fractional differentiation and integration to groundwater flow problems are listed.

Fractional Operators and Their Applications

This chapter presents different types of fractional differentiation and integration. It starts with a brief genesis of the concept of fractional calculus, then presents the concept of fractional differentiation with power law kernel known as Riemann–Liouville and Caputo derivatives with fractional order and their properties. Thus, the concept of local derivative with a fractional order is presented with its different properties. The concept of fractional differentiation and integration introduced by Caputo and Fabrizio is discussed in detail. Fractional differential operators with Mittag–Leffler kernel that was introduced by Atangana and Baleanu are analysed together with their respective properties. Finally physical interpretation, limitations and advantages of the concept of fractional differentiation and integration are presented.

Regularity of a General Parabolic Equation With Fractional Differentiation

This chapter presents a detailed regularity study for a general parabolic equation with fractional derivative with singular and non-singular kernels. The regularity of these general parabolic equation includes that of groundwater flow models as they are classified under parabolic equations.

Principle of Groundwater Flow

The principle of groundwater flowing within a geological formation called aquifers is presented. In particular, the derivation of the mathematical equations describing the movement of subsurface water within the geological formation is presented with their solutions.

Models of Groundwater Pollution With Fractional Operators

The groundwater transport model was extended to the concept of fractional differentiation with singular and non-singular kernel. For each model, a detailed study of existence and uniqueness solution was presented. Numerical and analytical methods were used to solve each models.

Groundwater Recharge Model With Fractional Differentiation

The groundwater recharge model is examined in this chapter within the scope of fractional differentiation with power law, exponential decay law and Mittag–Leffler law kernels. Numerical and exact solutions are derived using forward and Laplace transform methods respectively. Some numerical simulations are presented for different values of fractional orders.

Fractional Variable Order Derivatives

New operators of differentiation with variable orders were presented in this chapter. They are suggested in Caputo and Riemann–Liouville sense and have great advantages over existing differential operators with variable orders. Several properties of these new operators are presented including their relation with integral transforms. Numerical approximations of these new differential operators are presented.

Modeling Groundwater Pollution With Variable Order Derivatives

Abstract The concept of differential operators with variable orders has been used to model groundwater pollution problems. Two concepts of variable orders were used including: The convolution type, constructed from the fractional differentiation with constant order and the approximate Riemann–Liouville type. A fixed-point theorem was used to insure that conditions under which the exact solution exists and is unique. Analytical, semi-analytical and numerical methods where used to solve each model. Some examples are presented to illustrate the method.