A new approach for nuclear family model with fractional order Caputo derivative

  • 1 Mehmet Akif Ersoy University, 15100, Burdur, Turkey
  • 2 Mehmet Akif Ersoy University, 15100, Burdur, Turkey
Ilknur Koca
  • Corresponding author
  • Mehmet Akif Ersoy University, Department of Mathematics, Faculty of Sciences, 15100, Burdur, Turkey
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and Pelin Yaprakdal

Abstract

A work on a mathematical modeling is very popular in applied sciences. Nowadays many mathematical models have been considered and new methods have been used for approaching of these models. In this paper we are considering mathematical modeling of nuclear family model with fractional order Caputo derivative. Also the existence and uniqueness results and numerical scheme are given with Adams-Bashforth scheme via fractional order Caputo derivative.

1 Introduction

Recently, new efficient numerical methods have been developed for solutions of differential equations with different definitions of derivatives. For example the kernels including the power law for the Riemann-Liouville and Caputo type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu derivative [2,3,4,5,6, 11,12,13,14]. So these kernels history are beginning from the Leibniz’s letter to L’Hospital to Atangana-Baleanu derivative. In this work we are interesting in mathematical modeling of nuclear family. Model was introduced by Koca in 2015 with Caputo type fractional derivative [1]. In addition to previous paper, we reconsider model with searching the existence and uniqueness results of solutions and we give numerical approach for solutions of model with Caputo derivative. We believe that classical (ordinary) derivative is weak to explain the memory effect of the family dynamics. Because of this, we considered numerical solutions via fractional order Caputo derivative. Also the aim of the choose of the Caputo derivative is to give better meaning for modeling.

Adams-Bashforth is a powerful numerical method to solve linear and non-linear ordinary differential equations. Method was used only for ordinary differential equations generally with integer order. After that Atangana and Batogna have extended this method for partial differential equation with Caputo-Fabrizio derivative [10] in their thesis. Also Owolabi and Atangana formulated a new three-step fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative [7,8,9]. Method has been used for the solution of linear and nonlinear fractional differential equations.

In this paper we extend the applicability of the proposed scheme to solve system that is modeled by the Caputo derivative. The remainder of this paper is follows that in section one; some useful definitions of fractional order differentiation are given, in section two; we present in detail the existence and uniqueness results of solutions of our system. Finally in numerical part; we consider the solutions of system with two-step Adams-Bashforth scheme via fractional order Caputo derivative.

2 Preliminaries

Definition 1

Caputo fractional derivative of order α > 0 of a function f : (0, ∞) → R, according to Caputo, the fractional derivative of a continuous and differentiable function f is given as :

CDtα(f(t))=1Γ(1α)0t(tx)αddxf(x)dx,0<α1.

Definition 2

The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, ∞) → R, according to Riemann-Liouville, the fractional integral that is considered as anti-fractional derivative of a function f is :

Itα(f(t))=1Γ(α)0t(tx)α1f(x)dx,x>a.

Now we give two important properties for Caputo and Riemann-Liouville derivatives.

Property 1 : If f (t) is defined in the interval [a, b] and

1Γ(α)at(tx)α1f(x)dx=0
for α > 0 and for all t ∈ [a, b], then
f(t)0.

Property 2 : The following equation

CDtα(f(t))=g(x),α(0,1),xRf(0)=f0,
doesn’t have a periodic solution if f0 does not solve g(x) = 0, where g(x) is continuous.

3 Model derivation and existence and uniqueness of solutions for the nuclear family model

In this section, first we give integer order nuclear family model that is introduced by Koca in 2015 with four state variables [1]. The model describes baby’s emotions, in which baby (B) is involved in emotions with mother (M) and father (F). In model, the following notations for variables were used:

  • B(t): Baby’s love for the baby’s father,
  • F(t): Father’s love for the baby and his wife,
  • M(t): Mother’s love for the baby and her husband,
  • B1(t): Baby’s love for the baby’s mother.

The integer order nuclear family model is given as

dBdt=aB+b(FM)(c(FM))+γ1dFdt=eF+gB(hB)+jM+γ2,dMdt=kM+mB1(nB1)+pF+γ3,dB1dt=aB1+b(MF)(d(MF))+γ4,
with initial conditions
B(0)=B0,F(0)=F0,M(0)=M0,B1(0)=B10,
where e, g, h, j are specify father’s emotional style, k, m, n, p are specify mother’s emotional style and γ1, γ2, γ3, γ4 are attraction constants.

3.1 Existence of solution for the nuclear family model

In this part, we will present in detail the existence of the solutions of our system. The fixed-point theorem will help achieve this. Let P = K(q) × K(q) and K(q) be the Banach space of continuous RR valued function defined on the interval q with the norm

B,F,M,B1=B+F+M+B1.
Here
B=sup{|B(t)|:tq},F=sup{|F(t)|:tq},M=sup{|M(t)|:tq},B1=sup{|B1(t)|:tq}.

Let us redefine the nuclear family model spread by replacing the time derivative by Caputo fractional derivative:

aCDtαB(t)=F1(t,B(t)),aCDtαF(t)=F2(t,F(t)),aCDtαM(t)=F3(t,M(t)),aCDtαB1(t)=F4(t,B1(t)),
with initial conditions B(t0) = B0, F(t0) = F0, M(t0) = M0 and B1(t0) = B10.

Here,

F1(t,B(t))=aB(t)+b(F(t)M(t))(c(F(t)M(t))+γ1,F2(t,F(t))=eF(t)+gB(t)(hB(t))+jM(t)+γ2,F3(t,M(t))=kM(t)+mB1(t)(nB1(t))+pF(t)+γ3,F4(t,B1(t))=aB1(t)+b(M(t)F(t))(d(M(t)F(t))+γ4.

The above system (10) can be converted to the Caputo fractional integral. By definition (2), the model can be written as

B(t)=B0+1Γ(α)0t(tτ)α1F1(τ,B(τ))dτ,F(t)=F0+1Γ(α)0t(tτ)α1F2(τ,F(τ))dτ,M(t)=M0+1Γ(α)0t(tτ)α1F3(τ,M(τ))dτ,B1(t)=B10+1Γ(α)0t(tτ)α1F4(τ,B1(τ))dτ.

Theorem 1

The kernels F1, F2, F3 and F4 satisfy the Lipschitz condition if the following inequalities can be obtained :

0Li<1,fori=1,2,3,4.

Proof

Let us start the kernel F1. Let B and B1 be two function, so we have following:

F1(t,B(t))F1(t,B1(t))=aB(t)+b(F(t)M(t))(c(F(t)M(t))+γ1aB1(t)b(F(t)M(t))(c(F(t)M(t))γ1aB(t)B1(t)

Taking as L1 = a, then we get

F1(t,B(t))F1(t,B1(t))L1B(t)B1(t).

Hence, the Lipschitz condition is satisfied for F1, and if 0 ≤ L1 < 1, then it is also a contraction for F1. Similarly the other kernels have the Lipschitz condition as follows:

F2(t,F(t))F2(t,F1(t))L2F(t)F1(t),F3(t,M(t))F3(t,M1(t))L3M(t)M1(t),F4(t,B1(t))F4(t,B11(t))L4B1(t)B11(t).

When considering the kernels for the model, eq. (9) can be rewritten as follows:

B(t)=B0+1Γ(α)0t(tτ)α1F1(τ,B(τ))dτ,F(t)=F0+1Γ(α)0t(tτ)α1F2(τ,F(τ))dτ,M(t)=M0+1Γ(α)0t(tτ)α1F3(τ,M(τ))dτ,B1(t)=B10+1Γ(α)0t(tτ)α1F4(τ,B1(τ))dτ.

Now we can present the following recursive formula:

Bn(t)=B0+1Γ(α)0t(tτ)α1F1(τ,Bn1(τ))dτ,Fn(t)=F0+1Γ(α)0t(tτ)α1F2(τ,Fn1(τ))dτ,Mn(t)=M0+1Γ(α)0t(tτ)α1F3(τ,Mn1(τ))dτ,B1n(t)=B10+1Γ(α)0t(tτ)α1F4(τ,B1(n1)(τ))dτ.

Also the initial conditions are given as B(t0) = B0, F(t0) = F0, M(t0) = M0 and B1(t0) = B10. Now, we obtain the difference between the successive terms in the expression.

ϕn(t)=Bn(t)Bn1(t)=1Γ(α)0t(tτ)α1(F1(τ,Bn1(τ))F1(τ,Bn2(τ)))dτ,ψn(t)=Fn(t)Fn1(t)=1Γ(α)0t(tτ)α1(F2(τ,Fn1(τ))F2(τ,Fn2(τ)))dτ,μn(t)=Mn(t)Mn1(t)=1Γ(α)0t(tτ)α1(F3(τ,Mn1(τ))F3(τ,Mn2(τ)))dτ,εn(t)=B1n(t)B1(n1)(t)=1Γ(α)0t(tτ)α1(F4(τ,B1(n1)(τ))F4(τ,B1(n2)(τ)))dτ.

It is worth noticing that

Bn(t)=i=1nϕn(t),Fn(t)=i=1nψn(t),Mn(t)=i=1nμn(t),B1n(t)=i=1nεn(t).

Let us consider equality (18), applying the norm on both sides of the equation and considering triangular inequality and then the equation reduces to (20),

ϕn(t)=Bn(t)Bn1(t)1Γ(α)0t(tτ)α1(F1(τ,Bn1(τ))F1(τ,Bn2(τ)))dτ.

As the kernel satisfies the Lipschitz condition, we have

Bn(t)Bn1(t)L1Γ(α)0t(tτ)α1Bn1(τ)Bn2(τ)dτ,
then we get
ϕn(t)L1Γ(α)0t(tτ)α1ϕn1(t)dτ.

Similarly, we get the following results:

ψn(t)L2Γ(α)0t(tτ)α1ψn1(t)dτ,μn(t)L3Γ(α)0t(tτ)α1μn1(t)dτ,εn(t)L4Γ(α)0t(tτ)α1εn1(t)dτ,
after the above results, let us give a new theorem for solutions of model.

Theorem 2

The nuclear family model with the Caputo fractional derivative (9) has a unique solution under the conditions that we can find tmax satisfying

tmaxαΓ(α)Li<1,fori=1,2,3,4.

Proof

We know that the functions B(t), F(t), M(t) and B1(t) are bounded. Also we have shown that their kernels satisfy the Lipschitz condition. So from the equality (22)(23), we obtain the succeeding relations as follows:

ϕn(t)B0[tmaxαΓ(α)L1]n,ψn(t)F0[tmaxαΓ(α)L2]n,μn(t)M0[tmaxαΓ(α)L3]n,εn(t)B10[tmaxαΓ(α)L4]n.

Thus equality (19) exists and is a smooth function. To show that the above functions are the solutions of the model, let we assume

B(t)B0=Bn(t)bn(t),F(t)F0=Fn(t)cn(t),M(t)M0=Mn(t)dn(t),B1(t)B10=B1n(t)en(t).

Our aim here is to show that the term at infinity goes ‖b(t)‖ −→ 0. Therefore we have

bn(t)1Γ(α)0t(tτ)α1(F1(τ,B(τ))F1(τ,Bn1(τ)))dτ1Γ(α)0t(tτ)α1F1(τ,B(τ))F1(τ,Bn1(τ))dτtαL1Γ(α)BBn1.

Repeating this process recursively, we obtain

bn(t)B0[tαΓ(α)]n+1L1nM.

Then at tmax we have

bn(t)B0[tmaxαΓ(α)]n+1L1nM.

With applying the limit on both sides as n tends to infinity, we obtain ‖b(t)‖ −→ 0. This completes the proof.

3.2 Uniqueness of the special Solution

Another important application is to prove the uniqueness of the system of solutions. So we assume by contraction that there exists another system of solutions of (9), B2(t), F2(t), M2(t) and B12(t). Then

B(t)B2(t)1Γ(α)0t(tτ)α1(F1(τ,B(τ))F1(τ,B2(τ)))dτ.

Applying the norm to eq. (30), we get

B(t)B2(t)1Γ(α)0t(tτ)α1F1(τ,B(τ))F1(τ,B2(τ))dτ.

By using the Lipschitz condition properties of the kernel, we have

B(t)B2(t)tαL1Γ(α)B(t)B2(t).

This gives

B(t)B2(t)(1tαL1Γ(α))0,
B(t)B2(t)=0B(t)=B2(t).

So the equation has a unique solution. It is clear that we can show the same results for other solutions of F(t), M(t) and B1(t).

4 Two-step Adams-Bashforth scheme with fractional order Caputo derivative

In this section we consider the two-step Adams-Bashforth scheme with Caputo derivative which is given by Atangana and Owolabi in [9]. Let us give fractional differential equation with fractional order Caputo derivative as below:

0CDtαx(t)=F(t,x(t)),x(0)=x0.

The above fractional order Caputo equation is equal to integral equation as below:

x(t)=x(0)+1Γ(α)0tF(τ,x(τ))(tτ)α1dτ.

With using the fundamental theorem of calculus and taking t = tn+1, we have

x(tn+1)=x(0)+1Γ(α)0tn+1F(τ,x(τ))(tn+1τ)α1dτ.

When t = tn, we have

x(tn)=x(0)+1Γ(α)0tnF(τ,x(τ))(tnτ)α1dτ.

Then we can write follows that

x(tn+1)x(tn)=1Γ(α)0tn+1F(τ,x(τ))(tn+1τ)α1dτ1Γ(α)0tnF(τ,x(τ))(tnτ)α1dτ.

To get the value of integrals

0tn+1F(τ,x(τ))(tn+1τ)α1dτ
and
0tnF(τ,x(τ))(tnτ)α1dτ,
we can use the polynomial interpolation p(τ) as an approximation of F(τ, x (τ)). Then the interpolation is taking as with Lagrange polynomial
p(τ)=F(τ,x(τ))=ττn1τnτn1F(τn,x(τn))+ττnτn1τnF(τn1,x(τn1)).

If we integrate and simplify the right side of equality, then we get

0tn+1(ττn1τnτn1F(τn,x(τn))+ττnτn1τnF(τn1,x(τn1)))(tn+1τ)α1dτ=F(tn,x(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1)F(tn1,x(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1),
and
0tn(ττn1τnτn1F(τn,x(τn))+ττnτn1τnF(τn1,x(τn1)))(tnτ)α1dτ=F(tn,x(tn))hΓ(α)(hαtnαtnα+1α+1)+F(tn1,x(tn1))hΓ(α+1)tnα+1.

Here tn−1, tn and tn+1 are equally spaced then we take

tntn1=h,tn+1tn=h.

Therefore finally we get

x(tn+1)=x(tn)+F(tn,x(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1+hαtnαtnα+1α)+F(tn1,x(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1+tnαα+1)+Rnα(t).

Here Rnα(t) is error term for two step Adams-Bashforth scheme and calculated as below:

Rnα(t)=F(n+1)(t,x(t))(n+1)!i=0n(tti)<h3+αtmax12Γ(α+1)((n+1)α+n2).

Readers can be found detailed analysis of method in paper [9].

4.1 Application of the two-step fractional Adams-Bashforth method on fractional order nuclear family model via Caputo derivative

Let us consider the fractional order nuclear family model as below:

aCDtαB(t)=F1(t,B(t)),aCDtαF(t)=F2(t,F(t)),aCDtαM(t)=F3(t,M(t)),aCDtαB1(t)=F4(t,B1(t)),
with initial conditions B(t0) = B0, F(t0) = F0, M(t0) = M0 and B1(t0) = B10.

Here

F1(t,B(t))=aB(t)+b(F(t)M(t))(c(F(t)M(t))+γ1,F2(t,F(t))=eF(t)+gB(t)(hB(t))+jM(t)+γ2,F3(t,M(t))=kM(t)+mB1(t)(nB1(t))+pF(t)+γ3,F4(t,B1(t))=aB1(t)+b(M(t)F(t))(d(M(t)F(t))+γ4.

By using the numerical shcheme of above (46)(47) then we have

B(tn+1)=B(tn)+F1(tn,B(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1+hαtnαtnα+1α)+F1(tn1,B(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1+tnαα+1)+Rnα(t),F(tn+1)=F(tn)+F2(tn,F(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1+hαtnαtnα+1α)+F2(tn1,F(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1+tnαα+1)+Rnα(t),M(tn+1)=M(tn)+F3(tn,M(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1+hαtnαtnα+1α)+F3(tn1,M(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1+tnαα+1)+Rnα(t),B1(tn+1)=B1(tn)+F4(tn,B1(tn))hΓ(α)(2hαtn+1αtn+1α+1α+1+hαtnαtnα+1α)+F4(tn1,B1(tn1))hΓ(α)(hαtn+1αtn+1α+1α+1+tnαα+1)+Rnα(t).

5 Conclusion

In this paper fractional order nuclear family model is considered. Here, we generalize the previous model by considering the order as fractional order. As we saw that, the fractional order model is much more efficient in modeling than its integer order version. The detailed analysis such as existence and uniqueness results of the solution and efficient numerical scheme for model are presented.

References

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    I. Koca, Mathematical modeling of nuclear family and stability analysis, Appl. Math. Sci., 8 (2014), 3385–3392.

  • [2]

    I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).

  • [3]

    M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1, 73–85 (2015).

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    A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 1–7 (2016).

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    A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fract. : the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, 89, 447–454 (2016).

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    J.F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag–Leffler laws and applications, Eur Phys J Plus, 132, 13 (2017).

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    K.M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense. Chaos Solitons Fractals 99 (2017) 171–179.

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    K.M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffussive, diffusive, and superdiffusive scenarios. J. Comput. Nonlinear Dyn. 12 (2017) 031010.

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    A. Atangana and K.M. Owolabi, New numerical approach for fractional differential equations, Math Modell Nat Phenom, 13, 3 (2018).

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    R. Gnitchogna Batogna, A. Atangana, New two step Laplace Adam-Bashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34, 5, (2017).

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    J. Mishra, Analysis of the Fitzhugh Nagumo model with a new numerical scheme, Discrete & Continuous Dynamical Systems-S, 2019, 688–706.

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    J. Mishra, Fractional hyper-chaotic model with no equilibrium, Chaos, Solitons & Fractals 116, 43–53, 2018.

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    J. Mishra, A remark on fractional differential equation involving I-function, The European Physical Journal Plus 133 (2), 36, 2018.

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    J. Mishra, Modified Chua chaotic attractor with differential operators with non-singular kernels Chaos, Solitons & Fractals 125, 64–72

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  • [1]

    I. Koca, Mathematical modeling of nuclear family and stability analysis, Appl. Math. Sci., 8 (2014), 3385–3392.

  • [2]

    I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).

  • [3]

    M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1, 73–85 (2015).

  • [4]

    A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 1–7 (2016).

  • [5]

    A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fract. : the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, 89, 447–454 (2016).

  • [6]

    J.F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag–Leffler laws and applications, Eur Phys J Plus, 132, 13 (2017).

  • [7]

    K.M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense. Chaos Solitons Fractals 99 (2017) 171–179.

  • [8]

    K.M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffussive, diffusive, and superdiffusive scenarios. J. Comput. Nonlinear Dyn. 12 (2017) 031010.

  • [9]

    A. Atangana and K.M. Owolabi, New numerical approach for fractional differential equations, Math Modell Nat Phenom, 13, 3 (2018).

  • [10]

    R. Gnitchogna Batogna, A. Atangana, New two step Laplace Adam-Bashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34, 5, (2017).

  • [11]

    J. Mishra, Analysis of the Fitzhugh Nagumo model with a new numerical scheme, Discrete & Continuous Dynamical Systems-S, 2019, 688–706.

  • [12]

    J. Mishra, Fractional hyper-chaotic model with no equilibrium, Chaos, Solitons & Fractals 116, 43–53, 2018.

  • [13]

    J. Mishra, A remark on fractional differential equation involving I-function, The European Physical Journal Plus 133 (2), 36, 2018.

  • [14]

    J. Mishra, Modified Chua chaotic attractor with differential operators with non-singular kernels Chaos, Solitons & Fractals 125, 64–72