This work is licensed under the Creative Commons Attribution 4.0 International License.
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.
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 :
{\, ^C}D_t^\alpha \left( {f(t)} \right) = {1 \over {\Gamma (1 - \alpha )}}\int\limits_0^t {(t - x)^{ - \alpha }}{d \over {dx}}f(x)dx,\;\;0 < \alpha \le 1.
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 :
I_t^\alpha \left( {f(t)} \right) = {1 \over {\Gamma (\alpha )}}\int\limits_0^t {(t - x)^{\alpha - 1}}f(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
{1 \over {\Gamma (\alpha )}}\int\limits_a^t {(t - x)^{\alpha - 1}}f(x)dx = 0
for α > 0 and for all t ∈ [a, b], then
f(t) \equiv 0.
Property 2 : The following equation
\matrix{ {{\, ^C}D_t^\alpha \left( {f(t)} \right)} {\ = g(x),\;\;\alpha \in (0,1),\;\;x \in R} \cr {\kern 35pt}{f(0)} {\ = {f_{0,}}} \hfill}
doesn’t have a periodic solution if f0 does not solve g(x) = 0, where g(x) is continuous.
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
\matrix{ {{{dB} \over {dt}} = aB + b(F - M)(c - (F - M)) + {\gamma _1}} \hfill \cr {{{dF} \over {dt}} = eF + gB(h - B) + jM + {\gamma _2},} \hfill \cr {{{dM} \over {dt}} = kM + m{B_1}(n - {B_1}) + pF + {\gamma _3},} \hfill \cr {{{d{B_1}} \over {dt}} = a{B_1} + b(M - F)(d - (M - F)) + {\gamma _4},} \hfill}
with initial conditions
B(0) = {B_0},F(0) = {F_0},M(0) = {M_0},\,{B_1}(0) = {B_{10}},
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.
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 R → R valued function defined on the interval q with the norm
\left\| {B,F,M,{B_1}} \right\| = \left\| B \right\| + \left\| F \right\| + \left\| M \right\| + \left\| {{B_1}} \right\|.
Here
\matrix{ {\,\,\left\| B \right\| = \sup \left\{ {\left| {B(t)} \right|:t \in q} \right\},} \hfill \cr {\,\,\left\| F \right\| = \sup \left\{ {\left| {F(t)} \right|:t \in q} \right\},} \hfill \cr {\left\| M \right\| = \sup \left\{ {\left| {M(t)} \right|:t \in q} \right\},} \hfill \cr {\left\| {{B_1}} \right\| = \sup \left\{ {\left| {{B_1}(t)} \right|:t \in q} \right\}.} \hfill}
Let us redefine the nuclear family model spread by replacing the time derivative by Caputo fractional derivative:
\matrix{ {\,\,\,_a^CD_t^\alpha B(t) = {F_1}(t,B(t)),} \hfill \cr {\,\,\,_a^CD_t^\alpha F(t) = {F_2}(t,F(t)),} \hfill \cr {\, _a^CD_t^\alpha M(t) = {F_3}(t,M(t)),} \hfill \cr {\, _a^CD_t^\alpha {B_1}(t) = {F_4}(t,{B_1}(t)),} \hfill}
with initial conditions B(t0) = B0, F(t0) = F0, M(t0) = M0 and B1(t0) = B10.
The above system (10) can be converted to the Caputo fractional integral. By definition (2), the model can be written as
\matrix{\,\,{B(t) = {B_0} + {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}{F_1}(\tau ,B(\tau ))d\tau ,} \hfill \cr {\,\,F(t) = {F_0} + {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}{F_2}(\tau ,F(\tau ))d\tau ,} \hfill \cr {M(t) = {M_0} + {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}{F_3}(\tau ,M(\tau ))d\tau ,} \hfill \cr {{B_1}(t) = {B_{10}} + {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}{F_4}(\tau ,{B_1}(\tau ))d\tau .} \hfill}
Theorem 1
The kernels F1, F2, F3 and F4 satisfy the Lipschitz condition if the following inequalities can be obtained :
0 \le {L_i} < 1,\,\,\,{\rm{for}}\,\,i = 1,2,3,4.
Proof
Let us start the kernel F1. Let B and B1 be two function, so we have following:
\matrix{ {{\kern 15pt}\left\| {{F_1}(t,B(t)) - {F_1}(t,{B^1}(t))} \right\|} \hfill \cr { = \left\| {\matrix{ {aB(t) + b(F(t) - M(t))(c - (F(t) - M(t)) + {\gamma _1}} \cr { - a{B^1}(t) - b(F(t) - M(t))(c - (F(t) - M(t)) - {\gamma _1}} \cr } } \right\|} \hfill \cr {{\kern 10pt} \le a\left\| {B(t) - {B^1}(t)} \right\|} \hfill}
Taking as L1 = a, then we get
\left\| {{F_1}(t,B(t)) - {F_1}(t,{B^1}(t))} \right\| \le {L_1}\left\| {B(t) - {B^1}(t)} \right\|.
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:
\matrix{ {\,\,\,\left\| {{F_2}(t,F(t)) - {F_2}(t,{F^1}(t))} \right\| \le {L_2}\left\| {F(t) - {F^1}(t)} \right\|,} \hfill \cr {\left\| {{F_3}(t,M(t)) - {F_3}(t,{M^1}(t))} \right\| \le {L_3}\left\| {M(t) - {M^1}(t)} \right\|,} \hfill \cr {\left\| {\,{F_4}(t,{B_1}(t)) - {F_4}(t,B_1^1(t))} \right\| \le {L_4}\left\| {{B_1}(t) - B_1^1(t)} \right\|.} \hfill}
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),
\matrix{ {\left\| {{\phi _n}(t)} \right\|} {\ = \left\| {{B_n}(t) - {B_{n - 1}}(t)} \right\|} \hfill \cr {\kern 50pt} { \le {1 \over {\Gamma \left( \alpha \right)}}\left\| {\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}\left( {{F_1}(\tau ,{B_{n - 1}}(\tau )) - {F_1}(\tau ,{B_{n - 2}}(\tau ))} \right)d\tau } \right\|.} \hfill}
As the kernel satisfies the Lipschitz condition, we have
\left\| {{B_n}(t) - {B_{n - 1}}(t)} \right\| \le {{{L_1}} \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {(t - \tau )^{\alpha - 1}}\left\| {{B_{n - 1}}(\tau ) - {B_{n - 2}}(\tau )} \right\|d\tau ,
then we get
\left\| {{\phi _n}(t)} \right\| \le {{{L_1}} \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {(t - \tau )^{\alpha - 1}}\left\| {{\phi _{n - 1}}(t)} \right\|d\tau .
Similarly, we get the following results:
\matrix{ {\left\| {{\psi _n}(t)} \right\| \le {{{L_2}} \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}\left\| {{\psi _{n - 1}}(t)} \right\|d\tau ,} \hfill \cr {\left\| {{\mu _n}(t)} \right\| \le {{{L_3}} \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}\left\| {{\mu _{n - 1}}(t)} \right\|d\tau ,} \hfill \cr {\left\| {{\varepsilon _n}(t)} \right\| \le {{{L_4}} \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {{(t - \tau )}^{\alpha - 1}}\left\| {{\varepsilon _{n - 1}}(t)} \right\|d\tau ,} \hfill}
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
{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}{L_i} < 1,\,\,{\rm{for}}\,\,i = 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:
\matrix{ {\left\| {{\phi _n}(t)} \right\| \le \left\| {{B_0}} \right\|{{\left[ {{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}{L_1}} \right]}^n},} \hfill \cr {\left\| {{\psi _n}(t)} \right\| \le \left\| {{F_0}} \right\|{{\left[ {{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}{L_2}} \right]}^n},} \hfill \cr {\left\| {{\mu _n}(t)} \right\| \le \left\| {{M_0}} \right\|{{\left[ {{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}{L_3}} \right]}^n},} \hfill \cr {\left\| {{\varepsilon _n}(t)} \right\| \le \left\| {{B_{10}}} \right\|{{\left[ {{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}{L_4}} \right]}^n}.} \hfill}
Thus equality (19) exists and is a smooth function. To show that the above functions are the solutions of the model, let we assume
\matrix{\,\,\,\,{B(t) - {B_0}} \ = {B_n}(t) - {b_n}(t),\hfill \cr {\,\,\,\,\,F(t) - {F_0}} \ = {F_n}(t) - {c_n}(t), \hfill \cr {\,M(t) - {M_0}} \ = {M_n}(t) - {d_n}(t), \hfill \cr {{B_1}(t) - {B_{10}}} \ = {B_{1n}}(t) - {e_n}(t). \hfill}
Repeating this process recursively, we obtain
\left\| {{b_n}(t)} \right\| \le \left\| {{B_0}} \right\|{\left[ {{{{t^\alpha }} \over {\Gamma \left( \alpha \right)}}} \right]^{n + 1}}L_1^nM.
Then at tmax we have
\left\| {{b_n}(t)} \right\| \le \left\| {{B_0}} \right\|{\left[ {{{t_{\max }^\alpha } \over {\Gamma \left( \alpha \right)}}} \right]^{n + 1}}L_1^nM.
With applying the limit on both sides as n tends to infinity, we obtain ‖b∞(t)‖ −→ 0. This completes the proof.
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
\left\| {B(t) - {B_2}(t)} \right\| \le {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {(t - \tau )^{\alpha - 1}}\left( {{F_1}(\tau ,B(\tau )) - {F_1}(\tau ,{B_2}(\tau ))} \right)d\tau .
Applying the norm to eq. (30), we get
\left\| {B(t) - {B_2}(t)} \right\| \le {1 \over {\Gamma \left( \alpha \right)}}\int\limits_0^t {(t - \tau )^{\alpha - 1}}\left\| {{F_1}(\tau ,B(\tau )) - {F_1}(\tau ,{B_2}(\tau ))} \right\|d\tau .
By using the Lipschitz condition properties of the kernel, we have
\left\| {B(t) - {B_2}(t)} \right\| \le {{{t^\alpha }{L_1}} \over {\Gamma \left( \alpha \right)}}\left\| {B(t) - {B_2}(t)} \right\|.
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).
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:
\matrix{ {\, _0^CD_t^\alpha x(t)} \ = F(t,x\left( t \right)), \hfill \cr {\kern 25pt}{x(0)} \ = {x_{0.}} \hfill}
The above fractional order Caputo equation is equal to integral equation as below:
x(t) = x(0) + {1 \over {\Gamma (\alpha )}}\int\limits_0^t F(\tau ,x\left( \tau \right)){(t - \tau )^{\alpha - 1}}d\tau .
With using the fundamental theorem of calculus and taking t = tn+1, we have
x({t_{n + 1}}) = x(0) + {1 \over {\Gamma (\alpha )}}\int\limits_0^{{t_{n + 1}}} F(\tau ,x\left( \tau \right)){({t_{n + 1}} - \tau )^{\alpha - 1}}d\tau .
When t = tn, we have
x({t_n}) = x(0) + {1 \over {\Gamma (\alpha )}}\int\limits_0^{{t_n}} F(\tau ,x\left( \tau \right)){({t_n} - \tau )^{\alpha - 1}}d\tau .
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.
