Fractional Differential Equations

New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gomez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. A new numerical scheme has been developed, in this paper, for the newly established fractional differentiation. We present in general the error analysis. The new numerical scheme was applied to solve linear and non-linear fractional differential equations. We do not need a predictor-corrector to have an efficient algorithm, in this method. The comparison of approximate and exact solutions leaves no doubt believing that, the new numerical scheme is very efficient and converges toward exact solution very rapidly.

New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models

We presented an analysis of evolutions equations generated by three fractional derivatives namely the Riemann–Liouville, Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives. For each evolution equation, we presented the exact solution for time variable and studied the semigroup principle. The Riemann–Liouville fractional operator verifies the semigroup principle but the associate evolution equation does not. The Caputo–Fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly, the exact solution satisfies very well all the principle of semigroup. However, the Atangana–Baleanu for small time is the stretched exponential derivative, which does not satisfy the semigroup as operators. For a large time the Atangana–Baleanu derivative is the same with Riemann–Liouville fractional derivative, thus satisfies semigroup principle as an operator. The solution of the associated evolution equation does not satisfy the semigroup principle as Riemann–Liouville. With the connection between semigroup theory and the Markovian processes, we found out that the Atangana–Baleanu fractional derivative has at the same time Markovian and non-Markovian processes. We concluded that, the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects, which are not always Markovian. We presented the exact solutions of some evolutions equation using the Laplace transform. In addition to this, we presented the numerical solution of a nonlinear equation and show that, the model with the Atangana–Baleanu fractional derivative has random walk for small time. We also observed that, the Mittag-Leffler function is a good filter than the exponential and power law functions, which makes the Atangana–Baleanu fractional derivatives powerful mathematical tools to model complex real world problems.

Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties

Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative.

Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order

We presented the nonlinear Baggs and Freedman model with new fractional derivative. We derived the special solution using an iterative method. The stability of the iterative method was presented using the fixed point theory. The uniqueness of the special solution was presented in detail using some properties of the inner product and the Hilbert space. We presented some numerical simulations to underpin the effectiveness of the used derivative and semi-analytical method. c ©2016 All rights reserved.

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On the new fractional derivative and application to nonlinear Baggs and Freedman model

The model of Ebola spread within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced by Atangana and Baleanu. It is expected that the proposed model will show better approximation than the models established before. The existence and uniqueness of solutions for the spread of Ebola disease model is given via the Picard-Lindelof method. Finally, numerical solutions for the model are given by using different parameter values.

Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

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Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

In this paper we investigate a possible applicability of the newly established fractional differentiation in the field of epidemiology. To do this we extend the model describing the Rubella spread by replacing the time derivative with the time fractional derivative for the inclusion of memory. Detailed analysis of existence and uniqueness of exact solution is presented using the Banach fixed point theorem. Finally some numerical simulations are showed to underpin the effectiveness of the used derivative.

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Ilknur Koca

Papers

Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order

On the new fractional derivative and application to nonlinear Baggs and Freedman model

Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators

Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

Analysis of rubella disease model with non-local and non-singular fractional derivatives