Fractional discretization: The African’s tortoise walk
Introduction
The motivation of this work is based on an African story of wild animals. It’s a story of a one of a kind 20 km race among animals. All the racers speed up and rushed to the finish line as quicker as they could. Before a winner could be announced, the presiding officer the Lion king, together with all the other racing animals had to wait for the turtle who arrived only on the second day. The awarding procedure to choose the victor could only then begin. The Lion then requested that each racer gives a precise description of the racing track, and a detailed account of the moving obstacles he encountered. Every animal’s narration of the track and the race lacked one essential element as they mostly focused on speeding to the finish line in big leaps. As the turtle walked nonchalantly to the finish line its description of the track and the recount of the race was perfect. The lion awarded the victory to the turtle.
The moral of the story is that your focus should be kept at all time on the real objective. To do so one should always be mindful of the distractions and lures in order to avoid them. Finishing first in the story of the race should not have been the focus, rather describing the track and the obstacles on the race.
The concept of discretization has been based on the difference between two consecutive point gives Δt, also called the step size of the discretization. This concept is used to approximate differential operators with integer and non-integer orders. In addition, the same concept is used to discretize integral operators with integer and non-integer orders [1], [2], [3], [4], [5]. Such concept is fundamental the base of almost all numerical schemes suggested in the literature so far [6], [7], [8], [9]. Based on the concept of fractional differentiation and integration, where the order is considered to be 0 < α < 1, it comes to our mind to ask the following question, what happen if we instead considered λ/Δt where 0 < λ ≤ 1. If yes, could perhaps capture while discretizing real world problem capture complexities linked to interval using this tortoise walk approach? Will such concept be useful and accurate to enhance the stability of numerical method? In this paper, we present such numerical approach, and leave other researchers to provide their opinions. We shall already inform that such approach will appear to be scaling case where one will think that, the size step can just be replaced by . Yes perhaps in the case of classical differential operator, however when dealing with integral we obtain very clearly the effect of such approximation. The approach will therefore be opened to amelioration and criticisms.
Section snippets
Fractional discretization
We introduce a new way to discretize a differential or integral operation from classical to fractional. The new way will be achieved using the idea of fractional step. In the last past years, the discretization was made within such that and Here, we suggest thatwhere 0 < α ≤ 1. Then within such that and ThusHere αΔt will be the fractional step. If then we recover the classical discretization. We
Fractional discretization for classical derivative
We present here the fractional discretization for classical derivative.Alsoand backward EulerSecond approximation forwardand backwardCrank–NicholsonIllustration to solving heat equationTo show the difference between the classical discretization and the fractional counterpart,
Mean value theorem
If the function f is continuous on and differentiable on then there exists a point γ such thatSo with the Taylor series for where here
If we truncate the Taylor series with the framework of fractional discretization after the first derivativewhereThusfor approximation of nth order derivative by forward
Fractional discretization applied to fractal operators
We start first with the fractal derivative of a differentiable function without loss of generality since the conformable derivative is proportional to the fractal derivativeWith u differentiable, we have thatThereforeandHere AlsoTo show the second approximation of the fractal derivative, we show
Fractional discretization applied to fractional differential and integral operators
We present here the numerical approximation of fractional differential and integral operators. We start with the common used one namely the Riemann-Liouville and Caputo derivative with power-law kernel. We start with the Caputo power-law fractional derivativeAt we have
Error analysis
We present the error analysis of the new numerical discretization, but we will only show the case for fractal-fractional version as when we can recover the error for the classical differentiation. We start with at was approximated as
Fractional discretization applied to fractional integral
In this section, fractional integral operators are discretized using the African tortoise walk approach. Three cases are presented including the Riemann-Liouville, Caputo-Fabrizio and the Atangana-Baleanu fractional integral. We present first the Riemann-Liouville case.
At the point the Riemann-Liouville fractional integral can be approximated as:
Error analysis
We start with the Caputo senseFor simplicity, we take asand we writeThen
Application
In this section, we present example of some classical equations with exact solutions. Example 1 We consider here first the delay equationwith the exact solutionWe used the following discretizationWe compare the exact solution for different value of λ. The results are depicted in Fig. 5, Fig. 6, Fig. 7, Fig. 8 below. Example 2 We consider the following equationwhere with exact solutionWe apply the walk
Conclusion
New fields of science, technology and engineering are introduced when only when someone asked question out of the normal framework. The concept of fractional differentiation and integration was introduced due to a question by L’hopital. The concept of non-commutative algebra was introduced when a question was asked what if the set is non-commutative? It was discovered that the earth has attractive force called gravity when, the question was asked why when an able falls from the tree the fruit
Declaration of Competing Interest
None.
Acknowledgement
The author like to thank Prof. Dr. Toufik Mekkaoui for his advice toward enhancing this paper.
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