Fractional discretization: The African’s tortoise walk

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Abstract

We proposed a new way to discretizing a differential or integral equation using a fractional step. The new way has improved the stability and accuracy of numerical methods. We presented some examples with classical and fractional differential and integral equations.

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 λt 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 h=λΔt. 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 [tn,tn+1] such that tn+1=(n+1)Δt and tn=nΔt. Here, we suggest thattn+α=(n+α)Δtwhere 0 < α ≤ 1. Then within [tn,tn+α]=Iαn, such that tn+α=(n+α)Δt and tn=nΔt. Thustn+αtn=αΔt.Here αΔt will be the fractional step. If α=1, then we recover the classical discretization. We

Fractional discretization for classical derivative

We present here the fractional discretization for classical derivative.utuij+αuijαΔt,utuijuijααΔt.Alsouxui+αjuiαj2αΔxand backward Euleruxui+αj+αuiαj+α2αΔx.Second approximation forward2ux2ui+αj2uij+uiαj(αΔx)2and backward2ux2ui+αj+α2uij+α+uiαj+α(αΔx)2.Crank–Nicholson2ux212[ui+αj+α2uij+α+uiαj+α(αΔx)2+ui+αj2uij+uiαj(αΔx)2].Illustration to solving heat equationut=D2ux2.To show the difference between the classical discretization and the fractional counterpart,

Mean value theorem

If the function f is continuous on [tj,tj+α] and differentiable on (tj,tj+α], then there exists a point γ such thatf(γ)=f(tj+α)f(tj)tj+αtj.So with the Taylor series for n=0,f(tj+α)=f(tj)+R0where R0=f(γ)h here h= tj+αtj=αΔt.

If we truncate the Taylor series with the framework of fractional discretization after the first derivativef(tj+α)=f(tj)+f(tj)h+R1whereR1=f(γ)2h2=f(γ)2(αΔt)2orR1h=f(γ)h2=f(γ)αΔt2.Thusf(tj)=f(tj+α)f(tj)αΔt+O(αΔt)for 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 α proportional to the fractal derivative0FDxαu(x)=limxx1u(x)u(x1)xαx1α.With u differentiable, we have that0FDxαu(x)=u(x)x1αα.Therefore0FDtαu(x,t)uij+βuijβΔttj1ααand0FDtαu(x,t)uijuijββΔttj1αα.Here tj+βtj=βΔt. Also0FDxαu(x,t)ui+βjuiβj2βΔxxi1αα0FDxαu(x,t)ui+βj+βuiβj+β2βΔxxi1αα.To 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 derivative0CDtαu(x,t)=1Γ(1α)0tτu(x,τ)(tτ)αdτ.At (xi,tn+β), we have=1Γ(1α)0tn+βτu(xi,τ)(tn+βτ)αdτ=1Γ(1α)0tnτu(xi,τ)(tn+βτ)αdτ+tntn+βτu(xi,τ)(tn+βτ)αdτ=1Γ(1α)j=0n1tjtj+1(tn+βτ)αuij+1uijΔtdτ+1Γ(1α)tntn+β(tn+βτ)αuin+βuin

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 λ=1, we can recover the error for the classical differentiation. We start with 0FFPDtα,βu(x,t) at (xi,tn+λ) was approximated as=[(n+λ)Δt]1ββΓ(1α)tn+λαu(xi,0)+[(n+λ)Δt]1ββΓ(1α)0tn+λτu(xi,τ)(tn+λτ)αdτ+[(n+λ)Δt]1ββΓ(1α)tntn+λτu(xi,τ)(tn+λτ)αdτ=[(n+λ)Δt]1ββΓ(1α)[(n+λ)Δt]αu(xi,0)+[(n+λ)Δt]1ββΓ(1α)j=0n1tjtj+1[uij+1uijΔt+(τtjλ)uττ(xi,τ)](tn

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 (xi,tn+λ), the Riemann-Liouville fractional integral can be approximated as:1Γ(α)0tn+λ(tn+λτ)α1u(xi,τ)dτ=1Γ(α)[0tn(tn+λτ)α1u(xi,τ)dτ+tntn+λ(tn+λτ)α1u(xi,τ)dτ]=1Γ(α)[j=0n1tjtj+1(tn+λτ)α1u(xi,τ)dτ+tntn+λ(

Error analysis

We start with the Caputo sense0FFPDtα,β(t)u(x,t)=1Γ(1α)ddtβ(t)0tn+λu(xi,τ)(tn+λτ)αdτ=1Γ(1α)ddt0tn+λu(xi,τ)(tn+λτ)αdτ[1tn+λβ(tn+λ)(β(tn+λ)β(tn)λΔtlntn+λβ(tn+λ)tn+λ)].For simplicity, we take asγnα=1Γ(1α)1tn+λβ(tn+λ)(β(tn+λ)β(tn)λΔtlntn+λβ(tn+λ)tn+λ)and we write=γnα[u(xi,0)tn+λα+0tnτu(xi,τ)(tn+λτ)αdτ+tntn+λτu(xi,τ)(tn+λτ)αdτ]=γnα[u(xi,0)tn+λα+j=0n1tjtj+1uij+1uijΔt(tn+λτ)αdτEα,2(α1αtα)+tntn+λuin+λuinλΔt(tn+λτ)αdτ+R].ThenR=γnα[j=0n1tjtj+1uττ(xi,τ)(tn+λτ)α(τtj

Application

In this section, we present example of some classical equations with exact solutions.

Example 1

We consider here first the delay equationddty(t)=ay(t)with the exact solutiony(t)=y(0)eat.We used the following discretizationyn+λynλΔt=aynyn+λ=(λΔta+1)yn.We 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 equationu(x,t)t=Du(x,t)xwhere D=0.01, with exact solutionu(x,t)=sin(xDt).We apply the walkuin+λuinλΔt

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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