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Research article

Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

  • Received: 11 April 2022 Revised: 11 June 2022 Accepted: 16 June 2022 Published: 23 June 2022
  • MSC : 65N35, 34A08

  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.

    Citation: Yumei Chen, Jiajie Zhang, Chao Pan. Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials[J]. AIMS Mathematics, 2022, 7(8): 15612-15632. doi: 10.3934/math.2022855

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  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.



    This paper is devoted to study the expressions forms of the solutions and periodic nature of the following third-order rational systems of difference equations

    xn+1=yn1znzn±xn2,yn+1=zn1xnxn±yn2, zn+1=xn1ynyn±zn2,

    with initial conditions are non-zero real numbers.

    In the recent years, there has been great concern in studying the systems of difference equations. One of the most important reasons for this is a exigency for some mechanization which can be used in discussing equations emerge in mathematical models characterizing real life situations in economic, genetics, probability theory, psychology, population biology and so on.

    Difference equations display naturally as discrete peer and as numerical solutions of differential equations having more applications in ecology, biology, physics, economy, and so forth. For all that the difference equations are quite simple in expressions, it is frequently difficult to realize completely the dynamics of their solutions see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the related references therein.

    There are some papers dealed with the difference equations systems, for example, The periodic nature of the solutions of the nonlinear difference equations system

    An+1=1Cn,Bn+1=BnAn1Bn1,Cn+1=1An1,

    has been studied by Cinar in [7].

    Almatrafi [3] determined the analytical solutions of the following systems of rational recursive equations

    xn+1=xn1yn3yn1(±1±xn1yn3),yn+1=yn1xn3xn1(±1±yn1xn3).

    In [20], Khaliq and Shoaib studied the local and global asymptotic behavior of non-negative equilibrium points of a three-dimensional system of two order rational difference equations

    xn+1=xn1ε+xn1yn1zn1,yn+1=yn1ζ+xn1yn1zn1, zn+1=zn1η+xn1yn1zn1.

    In [9], Elabbasy et al. obtained the form of the solutions of some cases of the following system of difference equations

    xn+1=a1+a2yna3zn+a4xn1zn, yn+1=b1zn1+b2znb3xnyn+b4xnyn1,zn+1=c1zn1+c2znc3xn1yn1+c4xn1yn+c5xnyn.

    In [12], Elsayed et al. have got the solutions of the systems of rational higher order difference equations

    An+1=1AnpBnp,Bn+1=AnpBnpAnqBnq,

    and

    An+1=1AnpBnpCnp,Bn+1=AnpBnpCnpAnqBnqCnq,Cn+1=AnqBnqCnqAnrBnrCnr.

    Kurbanli [25,26] investigated the behavior of the solutions of the following systems

    An+1=An1An1Bn1,Bn+1=Bn1Bn1An1,  Cn+1=1CnBn,An+1=An1An1Bn1,Bn+1=Bn1Bn1An1,  Cn+1=Cn1Cn1Bn1.

    In [32], Yalçınkaya has obtained the conditions for the global asymptotically stable of the system

    An+1=BnAn1+aBn+An1,Bn+1=AnBn1+aAn+Bn1.

    Zhang et al. [39] investigated the persistence, boundedness and the global asymptotically stable of the solutions of the following system

    Rn=A+1Qnp,   Qn=A+Qn1RnrQns.

    Similar to difference equations and systems were studied see [21,22,23,24,27,28,29,30,31,32,33,34,35,36,37,38].

    In this section, we obtain the expressions form of the solutions of the following three dimension system of difference equations

    xn+1=yn1znzn+xn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynyn+zn2, (1)

    where nN0 and the initial conditions are non-zero real numbers.

    Theorem 1. We assume that {xn,yn,zn} are solutions of system (1).Then

    x6n2=ak3nn1i=0(a+(6i)k)(a+(6i+2)k)(a+(6i+4)k),x6n1=bf3nn1i=0(g+(6i+1)f)(g+(6i+3)f)(g+(6i+5)f),x6n=c3n+1n1i=0(d+(6i+2)c)(d+(6i+4)c)(d+(6i+6)c),x6n+1=ek3n+1(a+k)n1i=0(a+(6i+3)k)(a+(6i+5)k)(a+(6i+7)k),
    x6n+2=f3n+2(g+2f)n1i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f),x6n+3=hc3n+2(d+c)(d+3c)n1i=0(d+(6i+5)c)(d+(6i+7)c)(d+(6i+9)c),
    y6n2=dc3nn1i=0(d+(6i)c)(d+(6i+2)c)(d+(6i+4)c),y6n1=ek3nn1i=0(a+(6i+1)k)(a+(6i+3)k)(a+(6i+5)k),y6n=f3n+1n1i=0(g+(6i+2)f)(g+(6i+4)f)(g+(6i+6)f),y6n+1=hc3n+1(d+c)n1i=0(d+(6i+3)c)(d+(6i+5)c)(d+(6i+7)c),y6n+2=k3n+2(a+2k)n1i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k),y6n+3=bf3n+2(g+f)(g+3f)n1i=0(g+(6i+5)f)(g+(6i+7)f)(g+(6i+9)f),

    and

    z6n2=gf3nn1i=0(g+(6i)f)(g+(6i+2)f)(g+(6i+4)f),z6n1=hc3nn1i=0(d+(6i+1)c)(d+(6i+3)c)(d+(6i+5)c),z6n=k3n+1n1i=0(a+(6i+2)k)(a+(6i+4)k)(a+(6i+6)k),z6n+1=bf3n+1(g+f)n1i=0(g+(6i+3)f)(g+(6i+5)f)(g+(6i+7)f),
    z6n+2=c3n+2(d+2c)n1i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c),z6n+3=ek3n+2(a+k)(a+3k)n1i=0(a+(6i+5)k)(a+(6i+7)k)(a+(6i+9)k),

    where x2=a, x1=b, x0=c, y2=d, y1=e, y0=f, z2=g, z1=h and z0=k.

    Proof. For n=0 the result holds. Now assume that n>1 and that our assumption holds for n1, that is,

    x6n8=ak3n3n2i=0(a+(6i)k)(a+(6i+2)k)(a+(6i+4)k),x6n7=bf3n3n2i=0(g+(6i+1)f)(g+(6i+3)f)(g+(6i+5)f),x6n6=c3n2n2i=0(d+(6i+2)c)(d+(6i+4)c)(d+(6i+6)c),x6n5=ek3n2(a+k)n2i=0(a+(6i+3)k)(a+(6i+5)k)(a+(6i+7)k),x6n4=f3n1(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f),x6n3=hc3n1(d+c)(d+3c)n2i=0(d+(6i+5)c)(d+(6i+7)c)(d+(6i+9)c),
    y6n8=dc3n3n2i=0(d+(6i)c)(d+(6i+2)c)(d+(6i+4)c),y6n7=ek3n3n2i=0(a+(6i+1)k)(a+(6i+3)k)(a+(6i+5)k),y6n6=f3n2n2i=0(g+(6i+2)f)(g+(6i+4)f)(g+(6i+6)f),
    y6n5=hc3n2(d+c)n2i=0(d+(6i+3)c)(d+(6i+5)c)(d+(6i+7)c),y6n4=k3n1(a+2k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k),y6n3=bf3n1(g+f)(g+3f)n2i=0(g+(6i+5)f)(g+(6i+7)f)(g+(6i+9)f),

    and

    z6n8=gf3n3n2i=0(g+(6i)f)(g+(6i+2)f)(g+(6i+4)f),z6n7=hc3n3n2i=0(d+(6i+1)c)(d+(6i+3)c)(d+(6i+5)c),z6n6=k3n2n2i=0(a+(6i+2)k)(a+(6i+4)k)(a+(6i+6)k),z6n5=bf3n2(g+f)n2i=0(g+(6i+3)f)(g+(6i+5)f)(g+(6i+7)f),z6n4=c3n1(d+2c)n2i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c),z6n3=ek3n1(a+k)(a+3k)n2i=0(a+(6i+5)k)(a+(6i+7)k)(a+(6i+9)k).

    It follows from Eq (1) that

    x6n2=y6n4z6n3z6n3+x6n5=(k3n1(a+2k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k)                            )(ek3n1(a+k)(a+3k)n2i=0(a+(6i+5)k)(a+(6i+7)k)(a+(6i+9)k)                            )(ek3n1(a+k)(a+3k)n2i=0(a+(6i+5)k)(a+(6i+7)k)(a+(6i+9)k)                            )+(ek3n2(a+k)n2i=0(a+(6i+3)k)(a+(6i+5)k)(a+(6i+7)k)                            )=(k3n(a+2k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k))(a+3k)n2i=0(a+(6i+9)k)[(k(a+3k)n2i=0(a+(6i+9)k))+(1n2i=0(a+(6i+3)k))]=(k3n(a+2k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k))[k+((a+3k)n2i=0(a+(6i+9)k)n2i=0(a+(6i+3)k))]=(k3n(a+2k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k))[k+(a+(6n3)k)]=ak3na(a+2k)(a+(6n2)k)n2i=0(a+(6i+4)k)(a+(6i+6)k)(a+(6i+8)k).

    Then we see that

    x6n2=k3nn1i=0(a+(6i)k)(a+(6i+2)k)(a+(6i+4)k).

    Also, we see from Eq (1) that

    y6n2=z6n4x6n3x6n3+y6n5=(c3n1(d+2c)n2i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c)                         )(hc3n1(d+c)(d+3c)n2i=0(d+(6i+5)c)(d+(6i+7)c)(d+(6i+9)c)                         )(hc3n1(d+c)(d+3c)n2i=0(d+(6i+5)c)(d+(6i+7)c)(d+(6i+9)c)                         )+(hc3n2(d+c)n2i=0(d+(6i+3)c)(d+(6i+5)c)(d+(6i+7)c)                         )=(c3n(d+2c)n2i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c))(d+3c)n2i=0(d+(6i+9)c)[(c(d+3c)n2i=0(d+(6i+9)c))+(1n2i=0(d+(6i+3)c))]=(c3n(d+2c)n2i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c))[c+d+(6n3)c]=c3n[d+(6n2)c](d+2c)n2i=0(d+(6i+4)c)(d+(6i+6)c)(d+(6i+8)c).

    Then

    y6n2=dc3nn1i=0(d+(6i)c)(d+(6i+2)c)(d+(6i+4)c).

    Finally from Eq (1), we see that

    z6n2=x6n4y6n3y6n3+z6n5=(f3n1(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f)                            )(bf3n1(g+f)(g+3f)n2i=0(g+(6i+5)f)(g+(6i+7)f)(g+(6i+9)f)                            )(bf3n1(g+f)(g+3f)n2i=0(g+(6i+5)f)(g+(6i+7)f)(g+(6i+9)f)                            )+(bf3n2(g+f)n2i=0(g+(6i+3)f)(g+(6i+5)f)(g+(6i+7)f)                            )=(f3n(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f))(g+3f)n2i=0(g+(6i+9)f)[(f(g+3f)n2i=0(g+(6i+9)f))+(1n2i=0(g+(6i+3)f))]=(f3n(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f))[f+((g+3f)n2i=0(g+(6i+9)f)n2i=0(g+(6i+3)f))]=(f3n(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f))[f+(g+(6n3)f)]=f3n(g+(6n2)f)(g+2f)n2i=0(g+(6i+4)f)(g+(6i+6)f)(g+(6i+8)f).

    Thus

    z3n2=gf3nn1i=0(g+(6i)f)(g+(6i+2)f)(g+(6i+4)f).

    By similar way, one can show the other relations. This completes the proof.

    Lemma 1. Let {xn,yn,zn} be a positive solution of system (1), then all solution of (1) is bounded and approaching to zero.

    Proof. It follows from Eq (1) that

    xn+1=yn1znzn+xn2yn1,     yn+1=zn1xnxn+yn2zn1,zn+1=xn1ynyn+zn2xn1,

    we see that

    xn+4yn+2,     yn+2zn,  znxn2,    xn+4<xn2,yn+4zn+2,   zn+2xn,   xnyn2,      yn+4<yn2,zn+4xn+2,   xn+2yn,   ynzn2,      zn+4<zn2,

    Then all subsequences of {xn,yn,zn} (i.e., for {xn} are {x6n2}, {x6n1}, {x6n}, {x6n+1}, {x6n+2}, {x6n+3}  are decreasing and at that time are bounded from above by K,L and M since K=max{x2,x1,x0,x1,x2,x3}, L=max{y2,y1,y0,y1,y2,y3} and M=max{z2,z1,z0,z1,z2,z3}.

    Example 1. We assume an interesting numerical example for the system (1) with x2=.22,x1=.4, x0=.12,y2=.62, y1=4, y0=.3,z2=.4,z1=.53 andz0=2. (See Figure 1).

    Figure 1.  This figure shows the behavior of the solutions of the system (1) with the initial conditions x2=.22,x1=.4, x0=.12,y2=.62, y1=4, y0=.3,z2=.4,z1=.53 andz0=2. (We see from this figure that all solutions converges to zero).

    In this section, we get the solution's form of the following system of difference equations

    xn+1=yn1znzn+xn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynynzn2, (2)

    where nN0 and the initial values are non-zero real numbers with x2±z0,2z0, z2y0,2y0,3y0 and y22x0,±x0.

    Theorem 2. Assume that {xn,yn,zn} are solutions of (2). Then for n=0,1,2,...,

    x6n2=(1)nk3na2n1(a+2k)n, x6n1=(1)nbf3n(fg)2n(3fg)n, x6n=(1)nc3n+1d2n(2cd)n,x6n+1=ek3n+1(ak)n(a+k)2n+1, x6n+2=(1)nf3n+2gn(2fg)2n+1, x6n+3=(1)nhc3n+2(cd)2n+1(c+d)n+1,
    y6n2=(1)nc3nd2n1(2cd)n, y6n1=ek3n(ak)n(a+k)2n, y6n=(1)nf3n+1gn(2fg)2n,y6n+1=(1)nhc3n+1(cd)2n(c+d)n+1, y6n+2=(1)nk3n+2a2n(a+2k)n+1, y6n+3=(1)nbf3n+2(fg)2n+1(3fg)n+1,

    and

    z6n2=(1)nf3ngn1(2fg)2n, z6n1=(1)nhc3n(cd)2n(c+d)n, z6n=(1)nk3n+1a2n(a+2k)n,z6n+1=(1)nbf3n+1(fg)2n+1(3fg)n, z6n+2=(1)n+1c3n+2d2n+1(2cd)n, z6n+3=ek3n+2(ak)n(a+k)2n+2,

    where x2=a, x1=b, x0=c, y2=d, y1=e, y0=f, z2=g, z1=h and z0=k.

    Proof. The result is true for n=0. Now suppose that n>0 and that our claim verified for n1. That is,

    x6n8=(1)n1k3n3a2n3(a+2k)n1, x6n7=(1)n1bf3n3(fg)2n2(3fg)n1, x6n6=(1)n1c3n2d2n2(2cd)n1,x6n5=ek3n2(ak)n1(a+k)2n1, x6n4=(1)n1f3n1gn1(2fg)2n1, x6n3=(1)n1hc3n1(cd)2n1(c+d)n,
    y6n8=(1)n1c3n3d2n3(2cd)n1, y6n7=ek3n3(ak)n1(a+k)2n2, y6n6=(1)n1f3n2gn1(2fg)2n2,y6n5=(1)n1hc3n2(cd)2n2(c+d)n, y6n4=(1)n1k3n1a2n2(a+2k)n, y6n3=(1)n1bf3n1(fg)2n1(3fg)n,

    and

    z6n8=(1)n1f3n3gn2(2fg)2n2, z6n7=(1)n1hc3n3(cd)2n2(c+d)n1, z6n6=(1)n1k3n2a2n2(a+2k)n1,z6n5=(1)n1bf3n2(fg)2n1(3fg)n1, z6n4=(1)nc3n1d2n1(2cd)n1, z6n3=ek3n1(ak)n1(a+k)2n.

    Now from Eq (2), it follows that

    x6n2=y6n4z6n3z6n3+x6n5=((1)n1k3n1a2n2(a+2k)n)(ek3n1(ak)n1(a+k)2n)(ek3n1(ak)n1(a+k)2n)+(ek3n2(ak)n1(a+k)2n1)=((1)nk3na2n2(a+2k)n)(k+a+k)=(1)nk3na2n1(a+2k)n,y6n2=z6n4x6n3x6n3+y6n5=((1)nc3n1d2n1(2cd)n1)((1)n1hc3n1(cd)2n1(c+d)n)((1)n1hc3n1(cd)2n1(c+d)n)+((1)n1hc3n2(cd)2n2(c+d)n)=((1)nc3nd2n1(2cd)n1)c+cd=(1)nc3nd2n1(2cd)n,z6n2=x6n4y6n3y6n3z6n5=((1)n1f3n1gn1(2fg)2n1)((1)n1bf3n1(fg)2n1(3fg)n)((1)n1bf3n1(fg)2n1(3fg)n)((1)n1bf3n2(fg)2n1(3fg)n1)=((1)n1f3ngn1(2fg)2n1)(f3f+g)=(1)nf3ngn1(2fg)2n.

    Also, we see from Eq (2) that

    x6n1=y6n3z6n2z6n2+x6n4=((1)n1bf3n1(fg)2n1(3fg)n)((1)nf3ngn1(2fg)2n)((1)nf3ngn1(2fg)2n)+((1)n1f3n1gn1(2fg)2n1)=((1)nbf3n(fg)2n1(3fg)n)(f+2fg)=(1)nbf3n(fg)2n(3fg)n,y6n1=z6n3x6n2x6n2+y6n4=(ek3n1(ak)n1(a+k)2n)((1)nk3na2n1(a+2k)n)((1)nk3na2n1(a+2k)n)+((1)n1k3n1a2n2(a+2k)n)=(ek3n(ak)n1(a+k)2n)k+a=ek3n(ak)n(a+k)2n,z6n1=x6n3y6n2y6n2z6n4=((1)n1hc3n1(cd)2n1(c+d)n)((1)nc3nd2n1(2cd)n)((1)nc3nd2n1(2cd)n)((1)nc3n1d2n1(2cd)n1)=((1)n1hc3n(cd)2n1(c+d)n)c(2cd)=(1)nhc3n(cd)2n(c+d)n.

    Also, we can prove the other relations.

    Example 2. See below Figure 2 for system (2) with the initial conditions x2=11,x1=5, x0=13,y2=6, y1=7, y0=3,z2=14, z1=9 andz0=2.

    Figure 2.  This figure shows the behavior of solutions of the systems of rational recursive sequence xn+1=yn1znzn+xn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynynzn2, when we take the initial conditions: x2=11,x1=5, x0=13,y2=6, y1=7, y0=3,z2=14, z1=9 andz0=2. (See the figure we can conclude that all the solutions unboundedness solutions).

    Here, we obtain the form of solutions of the system

    xn+1=yn1znzn+xn2,yn+1=zn1xnxnyn2, zn+1=xn1ynyn+zn2, (3)

    where nN0 and the initial values are non-zero real numbers with x2±z0,2z0, z2±y0,2y0 and y2x0,2x0,3x0.

    Theorem 3. If  {xn,yn,zn} are solutions of system (3) where x2=a, x1=b, x0=c, y2=d, y1=e, y0=f, z2=g, z1=h and z0=k. Then for n=0,1,2,...,

    x6n2=k3na2n1(a2k)n, x6n1=(1)nbf3n(fg)n(f+g)2n, x6n=(1)nc3n+1dn(d2c)2n,x6n+1=(1)nek3n+1(ak)2n(a+k)n+1, x6n+2=(1)nf3n+2g2n(2f+g)n+1, x6n+3=(1)nhc3n+2(cd)2n+1(3cd)n+1,
    y6n2=(1)nc3ndn1(d2c)2n, y6n1=(1)nek3n(ak)2n(a+k)n, y6n=(1)nf3n+1g2n(2f+g)n,y6n+1=(1)nhc3n+1(cd)2n+1(3cd)n, y6n+2=k3n+2a2n+1(a2k)n, y6n+3=(1)nbf3n+2(fg)n(f+g)2n+2,

    and

    z6n2=(1)nf3ng2n1(2f+g)n, z6n1=(1)nhc3n(cd)2n(3cd)n, z6n=k3n+1a2n(a2k)n,z6n+1=(1)nbf3n+1(fg)n(f+g)2n+1, z6n+2=(1)nc3n+2dn(2cd)2n+1, z6n+3=(1)n+1ek3n+2(ak)2n+1(a+k)n+1.

    Proof. As the proof of Theorem 2 and so will be left to the reader.

    Example 3. We put the initials x2=8,x1=15, x0=13,y2=6,y1=7, y0=3,z2=14,z1=19 andz0=2, for the system (3), see Figure 3.

    Figure 3.  This figure shows the unstable of the solutions of the difference equations system (3) with the initial values x2=8,x1=15, x0=13,y2=6,y1=7, y0=3,z2=14,z1=19 andz0=2.

    The following systems can be treated similarly.

    In this section, we deal with the solutions of the following system

    xn+1=yn1znznxn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynyn+zn2, (4)

    where nN0 and the initial values are non-zero real with x2z0,2z0,3z0, z2±y0,2y0 and y2±x0,2x0.

    Theorem 4. The solutions of system (4) are given by

    x6n2=(1)nk3nan1(a2k)2n, x6n1=(1)nbf3n(fg)2n(f+g)n, x6n=(1)nc3n+1d2n(d+2c)n,x6n+1=ek3n+1(ak)2n+1(a3k)n, x6n+2=(1)n+1f3n+2g2n+1(2fg)n, x6n+3=(1)n+1hc3n+2(cd)n(c+d)2n+2,
    y6n2=(1)nc3nd2n1(d+2c)n, y6n1=ek3n(ak)2n(a3k)n, y6n=(1)nf3n+1g2n(2fg)n,y6n+1=(1)nhc3n+1(c+d)2n+1(cd)n, y6n+2=k3n+2an(a2k)2n+1, y6n+3=(1)nbf3n+2(fg)2n+1(f+g)n+1,

    and

    z6n2=(1)nf3ng2n1(2fg)n, z6n1=(1)nhc3n(c+d)2n(cd)n, z6n=(1)nk3n+1an(a2k)2n,z6n+1=(1)nbf3n+1(fg)2n(f+g)n+1, z6n+2=(1)nc3n+2d2n(2c+d)n+1, z6n+3=ek3n+2(ak)2n+1(a3k)n+1,

    where x2=a, x1=b, x0=c, y2=d, y1=e, y0=f, z2=g, z1=h and z0=k.

    Example 4. Figure 4 shows the behavior of the solution of system (4) with x2=18,x1=15, x0=3,y2=6, y1=.7, y0=3, z2=4,z1=9 andz0=5.

    Figure 4.  This figure shows the behavior of the system xn+1=yn1znznxn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynyn+zn2 with the initial conditions:- x2=18,x1=15, x0=3, y2=6, y1=.7, y0=3, z2=4,z1=9 andz0=5.0.6, x1=0.2, x0=5. (From the figure, we see that all solutions goes to zero).

    In this section, we obtain the solutions of the difference system

    xn+1=yn1znznxn2,yn+1=zn1xnxnyn2, zn+1=xn1ynynzn2, (5)

    where the initials are arbitrary non-zero real numbers with x2z0, z2y0 and y2x0.

    Theorem 5. If  {xn,yn,zn} are solutions of system (5) where x2=a, x1=b, x0=c, y2=d, y1=e, y0=f, z2=g, z1=h and z0=k. Then

    x6n2=k3na3n1, x6n1=bf3n(fg)3n, x6n=c3n+1d3n,x6n+1=ek3n+1(ka)3n+1, x6n+2=f3n+2g3n+1, x6n+3=hc3n+2(cd)3n+2,
    y6n2=c3nd3n1, y6n1=ek3n(ka)3n, y6n=f3n+1g3n,y6n+1=hc3n+1(cd)3n+1, y6n+2=k3n+2a3n+1, y6n+3=bf3n+2(fg)3n+2,

    and

    z6n2=f3ng3n1, z6n1=hc3n(cd)3n, z6n=k3n+1a3n,z6n+1=bf3n+1(fg)3n+1, z6n+2=c3n+2d3n+1, z6n+3=ek3n+2(ka)3n+2.

    Example 5. Figure 5 shows the dynamics of the solution of system (5) with x2=18,x1=15,x0=3,y2=6,y1=.7, y0=3,z2=4,z1=9 andz0=5.

    Figure 5.  This figure shows the behavior of the system of nonlinear difference equations (5) with the initial conditions considered as follows:- x2=18,x1=15, x0=3,y2=6, y1=.7, y0=3,z2=4,z1=9 andz0=5.

    This paper discussed the expression's form and boundedness of some systems of rational third order difference equations. In Section 2, we studied the qualitative behavior of system xn+1=yn1znzn+xn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynyn+zn2, first we have got the form of the solutions of this system, studied the boundedness and gave numerical example and drew it by using Matlab. In Section 3, we have got the solution's of the system xn+1=yn1znzn+xn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynynzn2, and take a numerical example. In Sections 4–6, we obtained the solution of the following systems respectively, xn+1=yn1znzn+xn2,yn+1=zn1xnxnyn2, zn+1=xn1ynyn+zn2, xn+1=yn1znznxn2,yn+1=zn1xnxn+yn2, zn+1=xn1ynyn+zn2, and xn+1=yn1znznxn2,yn+1=zn1xnxnyn2, zn+1=xn1ynynzn2. Also, in each case we take a numerical example to illustrates the results.

    This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (G: 233–130–1441). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

    All authors declare no conflicts of interest in this paper.



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