Modeling the potential energy field caused by mass density distribution with Eton approach

Badr Saad T. Alkahtani; Abdon Atangana

doi:10.1515/phys-2016-0008

Article Open Access

Modeling the potential energy field caused by mass density distribution with Eton approach

Badr Saad T. Alkahtani and Abdon Atangana

Published/Copyright: April 20, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

A new approach for modeling real world problems called the “Eton Approach” was presented in this paper. The "Eton approach" combines both the concept of the variable order derivative together with Atangana derivative with memory derivative. The Atangana derivative with memory is used to account for the memory and fractional derivative for its filter effect. The approach was used to describe the potential energy field that is caused by a given charge or mass density distribution.We solve the modified model numerically and present supporting numerical simulations.

Keywords: Eton approach; Atangana derivative with memory; fractional derivative; Poisson equation; numerical analysis

PACS: 02.30.Jr; 02.60.Cb; 02.60.Lj

1 Introduction

The concept of memory effect has long been a problem within the modeling community. There are many physical problems that requested to be modeled taking into account the memory effect, however, the big challenge was to find a suitable operator that can efficiently describe the physically observed fact [1–6]. The approach was used to describe the potential energy field that is caused by a given charge or mass density distribution. Many researchers suggest that the memory effect could fully be described via fractional derivatives. Nonetheless, the idea of memory requests time and space components; for instance with the movement of pollution within a geological formations, one would like to know at what time and space the pollution was retarded in the aquifer. Indeed this cannot be described with fractional derivative. Therefore a new operator able to take care of this problem is needed. On the other hand, as shown in the literature, the local derivative could not explain many physical problems as compared with derivative with fractional order. It was recently suggested that since the derivative with fractional order are based upon the convolution design, the fractional derivative could therefore be considered as filter operators. These filters are able to get rid of all impurities introduced by the local derivative while modeling real world problems [7,8]. Thus in order to introduce the memory effect into the mathematical formulation of a given real world problem and also to filter all impurities introduced by local derivative, we propose in this paper a new approach that will combine the two derivatives.

2 New trends on fractional calculus

In this section we will present some useful information about the new derivative with fractional order. However, more information around this derivative can be found in the following works [7,8]

Definition 1. Let f ∈ H¹ (a, b), b > a, a ∈ [0, 1] then, the new Caputo derivative of fractional derivative is defined as:

Dta(f(t))=M(a)1−a∫atf′(x)exp[−at−x1−a]dx.(1)

Where M(a) is a normalization function such that M(1) = 1[1]. However, if the function does not belongs to H¹ (a, b) then, the derivative can be reformulated as

Dta(f(t))=aM(a)1−a∫at(f(t)−f(x))exp[−at−x1−a]dx.(2)

Remark 2. The authors remarked that, ifs=1−aa∈[0,∞), a=11+s∈[0,1], thenequation (2)assumes the form

Dta(f(t))=N(s)s∫atf′(x)exp[−t−xs]dx, N(0)=N(∞)=1 .(3)

In addition,

lims→01s exp[−t−xs]=d(x−t).(4)

Now after the introduction of a new derivative, the associate anti-derivative becomes important, the associated integral of the new Caputo derivative with fractional order was proposed by Nieto and Losada [2].

Definition 3. [2] Let 0 < a < 1. The fractional integral of order a of a function f is defined by

Iatf(t)=21−a2−aMaft+2a2−aMa∫0tfsds,t≥0(5)

Remark [2]. Note that, according to the above definition, the fractional integral of Caputo type of function of order 0 < a < 1 is an average between function f and its integral of order one. This therefore imposes

2(1−a)(2−a)M(a)+2a(2−a)M(a)=1.(6)

The above expression yields an explicit formula for

M(a)=22−a, 0≤a≤1

Due to this, Nieto and Losada suggested that the new Caputo derivative of order 0 < α < 1 can be reformulated as

Dta(f(t))=11−a∫atf′(x)exp[−at−x1−a]dx.(7)

2.1 Some properties of the Caputo Fabrizio derivative

– For any natural number, n > 0, the following is obtained

Dtα(tn+1n+1)=M(α)1−α∫0ttnexp−αt−x1−αdx=M(α)1−αexp−αt1−αntntαα−1−nΓ(n)−Γn,tαα−1.(8)

– The Sumudu transform of a function f is given as:

S(f(t))=∫0∞1uexp−tuf(t)dt.(9)

Let f(t) be a function for which the Caputo-Fabrizio exists, then, the Sumudu transform of the Caputo-Frabrizio fractional derivative of f(t) is given as:

S(Dtα(f(t)))=M(α)S(f(t))−f(0)1−α−αu.(10)

– If a function, f, is infinitely differentiable, the following result is obtained

Dtα∫0tf(x)dx=M(α)1−α×∑n=0∞M(α)1−αexp−αt1−αntntαα−1−nΓ(n)−Γn,tαα−1n!.(11)

2.2 Caputo-Fabrzio Derivative with fractional order in Riemann-Liouville(RL) sense

One problem faced by the Caputo-Fabrizio derivative with fractional order is that, when alpha goes to zero we do not recover the initial function but the function and addition term. To solve this problem Atangana and Goufo proposed [15] an alternative definition of fractional derivative with no singular kernel, the result was also obtained in the work done by Caputo and Fabrizio [16].

Definition 4. Let f be a function not necessary differential, let alpha be a real number such that 0 ≤ α ≤ 1, then the Caputo-Fabrizio derivative with order alpha in RL sense is given as:

Dta(f(t))=11−addt∫atf(x)exp[−at−x1−a]dx.(12)

If α = 0 is zero we have the following:

Dt0(f(t))=ddt∫atf(x)dx=f(x).(13)

Using the argument by Caputo and Fabrizio, we also have that when α = 1 also goes to 1 we recover the first derivative.

3 Atangana derivative with memory

In mathematics, a dynamic scheme is a tuple (D, h, B) with D a manifold which can be a locally Banach space or Euclidean space, B the domain for time which is a set of non-negative real, and h is an evolution rule t → f^t the range is of course a diffeomorphism of a manifold to itself [10].

Definition 5. Let D be a dynamic system with domain B (time or space), let u a positively defined function called uncertainty function of D, within the domain B, then if h ∈ D, the Atangana derivative with memory of a functionh denoted by U^u(f)is defined as:

Uu(t)(h(t))=(1−u(t))h(t)+u(t)h′(t).(14)

Remark 6. If u = 1 we recover the first derivative (the the local derivative), if u = 0, we recover the initial function, this is conforms with the primary law of derivative.

3.1 Properties of Atangana derivative with memory

– Addition:

Uu(t)(ah(t)+bf(t))=aUu(t)(h(t))+bUu(t)(f(t)).(15)

– Multiplication:

Uu(t)(fg(t))=f(t)Uu(t)(g(t))+u(t)g(t)f′(t).(16)

Proof.Using the definition of Atangana derivative with memory, we obtain the following:

Uu(t)(fg(t))=(1−u(t))f(t)g(t)+u(t)(f(t)g(t))′Uu(t)(fg(t))=(1−u(t))f(t)g(t)+u(t)[f(t)g′(t)+g(t)f′(t)]Uu(t)(fg(t))=f(t)Uu(t)(g(t))+u(t)g(t)f′(t).(17)

This completes the proof. □

– Division:

Uu(t)(f(t)g(t))=g(t)Uu(t)f(t)−u(t)f(t)g′(t)g2(t).(18)

– Lipchitz condition: Let f(t) and g(t) be two functions then

||Uu(t)(g(t))−Uu(t)(f(t))|| ≤ ||(1−u(t))g(t)+u(t)g′(t) −(1−u(t))f(t)−u(t)f′(t)|| ≤|1−u(t)|||g(t)−f(t)||+|u(t)|||g′(t)−f′(t)|| ≤a||g(t)−f(t)||+b||g′(t)−f′(t)|| ≤a||g(t)−f(t)||+bα||g(t)−f(t)|| ≤H||g(t)−f(t)||.(19)

This proves that the Atangana derivative with memory possess the Lipchitz condition.

4 Poisson equation with Eton approach

In mathematics notably within applied mathematics, a partial differential equation is classified under elliptic type of equation and called Poisson’s equation is used in many fields of science for instance electrostatics, theoretical physics, and mechanical engineering. This equation is employed to describe the potential energy field caused by a given charge or mass density distribution. The aim of this section is to apply the new approach to the well-known Poisson equation, and present the difference between both models. We shall first recall that the one dimensional Poisson equation is given by:

∂x2ρ(x)=f(x).(20)

Now, to apply the Eton approach of modeling real world problems, we apply first the Atangana derivative with memory as follows:

Uxu(x)(Uxu(x)(ρ(x)))=f(x).(21)

The above will first produce

Uxu(x)((1−u(x))ρ(x)+u(x)ρ′(x))=f(x).(22)

Then using again the definition, we obtain:

ρ(x)[(1−u(x))2−u(x)u′(x)]+ρ′(x)2u(x)(1−u(x))+u′(x)+u(x)ρ″(x)=f(x).(23)

For simplicity we have the following version

ρ(x)a(x)+ρ'(x)b(x)+u(x)ρ″(x)=f(x).(24)

The second step is to replace the first order derivative with the variable order derivative to obtain

ρ(x)a(x)+Dxαρ(x)b(x)+u(x)ρ″(x)=f(x).(25)

The above equation is the result of Poisson equation with Eton approach. In the next section, we shall present the existence and uniqueness of the above equation.

5 Existence and uniqueness of exact solution

In this section, we present in detail the analysis of existence of the solution of above equation. We start this investigation by converting Equation (25) into a Volterra equation by applying on both sides the anti-derivative of the C.F. derivative obtaining

ρ(x)a1(x)+u1(x)ρ″(x)−f1(x)=Dxαρ(x),(26)

ρ(x)−ρ(0)=2(1−α)(2−α)M(α)(ρ(x)a1(x)+u1(x)ρ′′(x)−f1(x))+2α(2−α)M(α)∫0x(ρ(s)a1(s)+u1(s)ρ′′(s)−f1(s))ds.(27)

The above equation can be reduced for simplicity to:

ρ(x)−ρ(0)=2(1−α)(2−α)M(α)K(x,ρ(x))+2α(2−α)M(α)∫0xK(s,ρ(s))ds.(28)

Since K is a linear operator, then we can find a positive constant, σ, such that

||K(x,ρ1(x))−K(x,ρ0(x))||≤σ||ρ1(x)−ρ0(x)||.(29)

The above relation means the operator K satisfies Lipschitz condition. However assuming that, we can find two different solutions for Equation (27) then

||ρ1(x)−ρ0(0)||=||2(1−α)(2−α)M(α)K(x,ρ1(x)−K(x,ρ0(x)+2α(2−α)M(α)∫0x(K(s,ρ1(s)−K(s,ρ0(s))||.(30)

Nonetheless using the Lipschitz property of K we obtain

||ρ1(x)−ρ0(0)||≤2(1−α)(2−α)M(α)σ||ρ1(x)−ρ0(x)||+σ2α(2−α)M(α)∫0x||(ρ1(s)−ρ0(s))||.(31)

Thus we have the following result

||ρ1(x)−ρ0(0)||≤2(1−α)(2−α)M(α)σ+x0σ2α(2−α)M(α)×||(ρ1(s)−ρ0(s)||.(32)

1−2(1−α)(2−α)M(α)σ+x0σ2α(2−α)M(α)×||(ρ1(s)−ρ0(s)||≤0.(33)

Nevertheless, we chose a specific, x₀ > 0, such that

(2(1−α)(2−α)M(α)σ+x0σ2α(2−α)M(α))>0.(34)

Therefore if

||ρ1(x)−ρ0(0)||≤2(1−α)(2−α)M(α)σ+x0σ2α(2−α)M(α)×||(ρ1(s)−ρ0(s)||(35)

which implies that

∥ρ1(x)−ρ0(x)∥≤0→ρ1(x)=ρ0(x).(36)

This proves the uniqueness of the solution of the modified equation. We next present the existence of the solution, to do this we propose the following recursive formula

ρ0(x)=ρ(0)ρn+1(x)=2(1−α)(2−α)M(α)K(x,ρn(x)+2α(2−α)M(α)∫0xK(s,ρn(s))ds.(37)

The solution of the time fractional with modified equation by the the "Eton approach" is given as

ρ=limn→∞ρn+1(x).(38)

We shall now show that such solution. To achieve this, we shall first present the following preliminaries

Preliminaries: see([7,8,9,10]) Let us consider the following Banach space (X,||.|| ) be a Banach space and h a self-map of X. Let y_n+1 = g(H, y_n) be some iterative technique. Assuming that, F(H), the fixed point set of h has at least one element and that y_n converges to a point p ∈ F(H). Let x_n ⊆; X and define e_n = || x_n+1 – g(H, x_n)||. If limn→∞en=0, this implies that limn→∞xn=p, and as such the iteration method, y_n+1 = g(H, y_n), is said to be H – Stable. Without any loss of generality, we must assume that our sequence x_n has an upper boundary; otherwise we cannot expect the possibility of convergence. If all these conditions are satisfied for y_n+1 = Hy_n which is known as Picard’s iteration, consequently the iteration will is H – Stable.

Theorem 7. (see [7,9]). Let (X,||.|| ) be a Banach space and h a self-map of X satisfying

|| Hx – Hy|| ≤ C||x – Hx|| + c||x – y||, for all x, y in X where 0 ≤ C, 0 ≤ α < 1. Suppose that H has fixed point p. Then, H is Picard H – Stable.

Theorem 8. Let h be a self-map that we chose to defined as follows

ρn+1(x)=2(1−α)(2−α)M(α)Kx,ρn(x)+2α(2−α)M(α)∫0xK(s,ρn(s))ds(39)

then the self-map is H-stable in L₂( α, β) providing that the following inequality is satisfied,

(2(1−α)(2−α)M(α)σ+x0σ2α(2−α)M(α))<0.(40)

Proof. For all n, m ∈ ℕ we have

∥H(ρn(x))−H(ρm(0))∥=∥2(1−α)(2−α)M(α)K(x,ρn(x))−K(x,ρm(x))+2α(2−α)M(α)∫0x(K(s,ρn(s)−K(s,ρm(s))∥.(41)

The above can be converted using the triangular inequality property of the norm

∥H(ρn(x))−H(ρm(0))∥≤2(1−α)(2−α)M(α)∥K(x,ρn(x)−K(x,ρm(x)∥+2α(2−α)M(α)∫0x∥(K(s,ρn(s)−K(s,ρm(s))∥.(42)

Using the Lipschitz condition of K, we obtain the following

‖H(ρn(x))−H(ρm(x))‖≤2(1−α)(2−α)M(α)σ‖ρn(x)−ρm(x)‖+2α(2−α)M(α)σx0‖ρn(x)−ρm(x)‖.(43)

Rearranging, we obtain

‖H(ρn(x))−H(ρm(x))‖≤(2(1−α)(2−α)M(α)σ+2α(2−α)M(α)σx0)‖ρn(x)−ρm(x)‖.(44)

Or we have

||H(ρn(x))−H(ρm(x))||≤θ||ρn(x)−ρm(x)||.(45)

This shows that H-self-mapping has a fixed point and the proof is complete. □

In addition to this if we consider

c=0, C=θ.(46)

We have the conditions of a hypothesis which holds for H. Thus, since all conditions of the hypothesis hold, then, H is Picard’s H-stable.

6 Numerical analysis

In this section, the new equation will be solved numerically using the well-known finite element method associate to the Caputo-Fabrizio derivative with fractional order. We will present the detail of stability and convergence of the numerical scheme. To accommodate readers that are not use to this new derivative, we present the first and second approximation of the Caputo-Fabrizio derivative [10]. The first approximation in space is given as:

Dtα(f(tn))=M(α)α∑j=1n(fj+1−fjΔx+O(Δ))dj,k,(47)

where

dj,k=exp[−αk1−α(n−j+1)]−exp[−αk1−α(n−j)].(48)

With the second order approximation we derive the following:

Dxα(f(x))=α(1−α)π∫0xd2f(y)dy2exp[−α2(x−y)2(1−α)2]dy.(49)

At a particular point x_m we have the following

Dxα(f(xm))=α(1−α)π∫0xmd2f(y)dy2exp[−α2(xm−y)2(1−α)2]dy.(50)

Using the second order approximation of the ordinary derivative, we obtain

Dxα(f(xm))=α(1−α)π×∑k=1m(fk+1−2fk+fk−12(Δ)2)∫(k−1)ikiexp[−α2(im−y)2(1−α)2]dy.(51)

After integrating we obtained finally:

Dxα(f(xm))=12∑k=1m(fk+1−2fk+fk−12(Δ)2)[erf((m−k)iα1−α)−erf((m−k+1)iα1−α)](52)

or in more simpler way:

Dxα(f(xm))=12∑k=1m(fk+1−2fk+fk−12(Δ)2)di,km(53)

di,km=erf(m−k)iα1−α−erf(m−k+1)iα1−α.(54)

Nevertheless replacing this in the modified equation obtained from the "Eton approach", we obtain

(ρm+1−ρm2)a1m+u1m(ρm+1−2ρm+ρm−12(Δ)2)−f1m=12∑k=1m(ρk+1−2ρk+ρk−12(Δ)2)di,km−1−α2ρm−1(55)

above recursive formula can therefore be used to generate numerical approximation of the new model of potential energy field caused by mass density distribution.

7 Numerical simulations

In this section we present some numerical simulations of the modified model for different function of uncertain derivative. For each function, we present the numerical representation for different values of alpha, the fractional order derivative. The uncertain order derivatives chosen here are u(x)=sin[x+π4] andu(x)=3−2cos[x+π4]. Figure 1 shows the numerical simulation of the model as a function of fractional order and space. The numerical simulations of the modified model with the first function are depicted in Figures 2 and 3 for the first uncertain function. Figure 4 shows the numerical simulation of the model as function of fractional order and space for the second uncertain function. The numerical simulations of the modified model with the second function are depicted in Figures 5 and 6.The function used in the below simulations is given as:

$Figure 1 Solution as function of fractional order and space.$

Figure 1

Solution as function of fractional order and space.

Figure 2

Numerical simulation for different values of alpha.

Figure 3

Parametric plot of the 3 models.

$Figure 4 Solution as function of fractional order and space.$

Figure 4

Solution as function of fractional order and space.

Figure 5

Numerical simulation for different values of alpha.

Figure 6

Parametric plot of the 3 models.

f(x)=erfc(x)((cos(x+π4)−2)2−14sin(x+π4)(3−cos(x+π4)))−(56)

2e−αx1−α−x2(14sin(x+π4)+2(3−cos(x+π4))(cos(x+π4)−2))π+4e−x2x(3−cos(x+π4))π.(57)

Additionally

f(x)=Ci(19.x)((cos(x+π4)−2)2 −14sin(x+π4)(3−cos(x+π4))) +(14sin(x+π4)+2(3−cos(x+π4)) ×(cos(x+π4)−2))(∫0x1.e−19.(x−v)cos(19.v)v dv) +(3−cos(x+π4))(−1.cos(19.x)x2−19.sin(19.x)x).(58)

8 Conclusion

The aim of this paper was to introduce a new approach of modeling real world problems. The presented approach combines the filter effect that is obtained by using the Caputo-Fabrizio derivative with fractional order and the memory effect, which is introduced by using the novel operator called uncertain derivative, this operator was proposed by Atangana. This approach is referred to as UFM or simply Eton approach. This novel approach was used to describe the potential energy field that is caused by a given charge or mass density distribution. The new approach help describe physical problems with memory.

Acknowledgement

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this group No RG-1437-017.

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Received: 2015-12-3

Accepted: 2016-1-4

Published Online: 2016-4-20

Published in Print: 2016-1-1

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