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Study of multi term delay fractional order impulsive differential equation using fixed point approach

  • This manuscript is devoted to investigate a class of multi terms delay fractional order impulsive differential equations. Our investigation includes existence theory along with Ulam type stability. By using classical fixed point theorems, we establish sufficient conditions for existence and uniqueness of solution to the proposed problem. We develop some appropriate conditions for different kinds of Ulam-Hyers stability results by using tools of nonlinear functional analysis. We demonstrate our results by an example.

    Citation: Amjad Ali, Kamal Shah, Dildar Ahmad, Ghaus Ur Rahman, Nabil Mlaiki, Thabet Abdeljawad. Study of multi term delay fractional order impulsive differential equation using fixed point approach[J]. AIMS Mathematics, 2022, 7(7): 11551-11580. doi: 10.3934/math.2022644

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  • This manuscript is devoted to investigate a class of multi terms delay fractional order impulsive differential equations. Our investigation includes existence theory along with Ulam type stability. By using classical fixed point theorems, we establish sufficient conditions for existence and uniqueness of solution to the proposed problem. We develop some appropriate conditions for different kinds of Ulam-Hyers stability results by using tools of nonlinear functional analysis. We demonstrate our results by an example.



    Chemical kinetics deals with chemistry experiments and interprets them in terms of a mathematical model. The experiments are done on chemical reactions with the passage of time. The models are differential equations for the rates at which reactants are consumed and products are produced. Chemists are able to understand how chemical reactions take place at the molecular level by combining models with investigation. Molecules react in steps to lead to the overall stoichiometric reaction which is reaction mechanism for collection of reactions. The set of reactions specifies the path (or paths) that reactant molecules take to finally arrive at the product molecules. All species in the reaction appear in at least one step and the sum of the steps gives the overall reaction. The govern the rate of the reaction which leads directly to the mechanism of differential equations [1]. Many processes and phenomena in chemistry generally in sciences can be designated by first-order differential equations. These equations are the most important and most frequently used to describe natural laws. The following examples are discussed: the Bouguer-Lambert-Beer law in spectroscopy, time constants of sensors, chemical reaction kinetics, radioactive decay, relaxation in nuclear magnetic resonance, and the RC constant of an electrode [2]. The induced kinetic differential equations of a reaction network endowed with mass action type kinetics are a system of polynomial differential equations [3]. We review the basic ideas of fractional differential equations and their applications on non-linear biochemical reaction models. We apply this idea to a non-linear model of enzyme inhibitor reactions [4].

    The fractional-order, which involves integration and transect differentiation using fractional calculus is helping to better understand the explanation of real-world problems than ordinary integer order, as well as in the modeling of real phenomena due to a characterization of memory and hereditary properties in [5,6]. Riemann Liouville developed the concept of fractional derivative, which is based on power law, [7,8] offers a novel fractional derivative that makes use of the exponential kernel. Several issues include the non-singular kernel fractional derivative, which covers the trigonometric and exponential functions, and [9,10,11,12] illustrates some relevant techniques for epidemic models. This virus's suggested outbreak efficiently catches the timeline for the COVID-19 disease conceptual model [13]. In the literature, many fractional operators are employed to solve real-world issues [14,15].

    In this paper, section 1 is introduction and section 2 consists of some basic fractional order derivative which are helpful to solve the epidemiological model. Section 3 and 4 consists of generalized from of the model, uniqueness and stability of the model. Fractal Fractional techniques with exponential decay kernel and Mittag-Leffler kernel respectively in section 5. Results and conclusion are discussed in section 6, and 7 respectively.

    Following are the basic definitions [7,8,14,15] used for analysis and solution of the problem.

    Definition 1: Sumudu transform for any function ϕ(t) over a set is given as,

    A={ϕ(t):there exist Λ,τ1,τ2>0,|ϕ(t)|<Λexp(|t|τi),if t(1)j×[0,)}

    is defined by

    F(u)=ST[ϕ(t)]=0exp(t)ϕ(ut)dt,      u(τ1,τ2).

    Definition 2: For a function g(t)W12(0,1),b>a and σ(0,1], the definition of Atangana–Baleanu derivative in the Caputo sense is given by

    ABC0Dσtg(t)=AB(σ)1σt0ddτg(τ)Mσ[σ1σ(tτ)σ]dτ,      n1<σ<n

    where

    AB(σ)=1σ+σΓ(σ).

    By using Sumudu transform (ST) for (1), we obtain

    ST[ABC0Dσtg(t)](s)=q(σ)1σ{σΓ(σ+1)Mσ(11σVσ)}×[ST(g(t))g(0)].

    Definition 3: For a function g(t)W12(0,1),b>a and α1(0,1], the definition of Atangana–Baleanu derivative in the Caputo sense is given by

    ABC0Dα1tg(t)=AB(α1)1α1t0ddτg(τ)Eα1[α11α1(tτ)α1]dτ,

    where

    AB(α1)=1α1+α1Γ(α1).

    Definition 4: Suppose that g(t) is continuous on an open interval (a,b), then the fractal-fractional integral of g(t) of order α1 having Mittag-Leffler type kernel and given by

    FFMJα1,α20,t(g(t))=α1α2AB(α1)Γ(α1)t0sα21g(s)(ts)α1ds+α2(1α1)tα21g(t)AB(α1)

    Robertson introduces this chemical process in [19,20]. Schafer pioneered the following chemical reactions method in 1975 [19,20]. It represents the high irradiance response (HIRES) of photomorphogenesis based on phytochrome. A stiff system of eight non-linear ordinary differential equations is used to create the following mathematical model.

    y1=M1y1+M2y2+M3y3+M4,y2=M1y1M5y2,y3=M6y3+M2y4+M7y5,y4=M3y2+M8y3M9y4,y5=M10y5+M2y6+M2y7,y6=M11y6y8+M12y4+M8y5M2y6+M12y7,y7=M11y6y8M13y7,y8=M11y6y8+M13y7. (1)

    Here M1=1.7,M2=0.43,M3=8.32, M4=0.0007,M5=8.75, M6=10.03,M7=0.035, M8=1.71,M9=1.12,M10=1.745, M11=280,M12=0.69,M13=1.81. The initial values can be represented by y=(1,0,0,0,0,0,0,0.0057)T. By using Atangana-Baleanu in Caputo sense for system (1), we get

    ABC0Dαty1=M1y1+M2y2+M3y3+M4,ABC0Dαty2=M1y1M5y2,ABC0Dαty3=M6y3+M2y4+M7y5,ABC0Dαty4=M3y2+M8y3M9y4,ABC0Dαty5=M10y5+M2y6M2y7,ABC0Dαty6=M11y6y8+M12y4+M8y5M2y6+M12y7,ABC0Dαty7=M11y6y8M13y7,ABC0Dαty8=M11y6y8+M13y7. (2)

    Here OABCDαt is the Atanagana-Baleanue Caputo sense fractional derivative with 0<α1.

    With given initial conditions

    yi(0)0,i=1,2,3,,8 (3)

    Theorem 3.1: The solution of the proposed fractional-order model (1) along initial conditions is unique and bounded in R+8.

    Proof: In (1), we can get its existence and uniqueness on the time interval (0, ∞). Afterwards, we need to show that the non-negative region R+8 is a positively invariant region. For this

    ABC0Dαty1|y1=0=M2y2+M3y3+M40,
    ABC0Dαty1|y2=0=M1y10,
    ABC0Dαty1|y3=0=M2y4+M7y50,
    ABC0Dαty1|y4=0=M3y2+M8y30,
    ABC0Dαty1|y5=0=M2y6M2y70,
    ABC0Dαty1|y6=0=M12y4+M8y5+M12y70,
    ABC0Dαty1|y7=0=M11y6y80,
    ABC0Dαty1|y8=0=M13y70

    If (y1(0)), (y2(0)), (y3(0)), (y4(0)), (y5(0)), (y6(0)), (y7(0)), (y8(0)) ϵ R8+, then from above expression, the solution cannot escape from the hyperplane. Also on each hyperplane bounding the non-negative orthant, the vector field points into R8+, i.e., the domain R8+ is a positively invariant set.

    Now, with the help of Sumudu transform definition, we get

    QEα(11αPα)ST{y1(t)y1(0)}=ST[M1y1+M2y2+M3y3+M4],QEα(11αPα)ST{y2(t)y2(0)}=ST[M1y1M5y2],QEα(11αPα)ST{y3(t)y3(0)}=ST[M6y3+M2y4+M7y5],QEα(11αPα)ST{y4(t)y4(0)}=ST[M3y2+M8y3M9y4],QEα(11αPα)ST{y5(t)y5(0)}=ST[M10y5+M2y6M2y7],QEα(11αPα)ST{y6(t)y6(0)}=ST[M11y6y8+M12y4+M8y5M2y6+M12y7],QEα(11αPα)ST{y7(t)y7(0)}=ST[M11y6y8M13y7],QEα(11αPα)ST{y8(t)y8(0)}=ST[M11y6y8+M13y7]. (4)

    Where Q=M(α)αΓ(α+1)1α

    Rearranging, we get

    ST(y1(t))=y1(0)+H×ST[M1y1+M2y2+M3y3+M4],ST(y2(t))=y2(0)+H×ST[M1y1M5y2],ST(y3(t))=y3(0)+H×ST[M6y3+M2y4+M7y5],ST(y4(t))=y4(0)+H×ST[M3y2+M8y3M9y4],ST(y5(t))=y5(0)+H×ST[M10y5+M2y6M2y7],ST(y6(t))=y6(0)+H×ST[M11y6y8+M12y4+M8y5M2y6+M12y7],ST(y7(t))=y7(0)+H×ST[M11y6y8M13y7],ST(y8(t))=y8(0)+H×ST[M11y6y8+M13y7]. (5)

    Using the inverse Sumudu transform on both sides of the system (5), we obtain

    y1(t)=y1(0)+ST1[H×ST[M1y1+M2y2+M3y3+M4]],y2(t)=y2(0)+ST1[H×ST[M1y1M5y2]],y3(t)=y3(0)+ST1[H×ST[M6y3+M2y4+M7y5]],y4(t)=y4(0)+ST1[H×ST[M3y2+M8y3M9y4]],y5(t)=y5(0)+ST1[H×ST[M10y5+M2y6M2y7]],y6(t)=y6(0)+ST1[H×ST[M11y6y8+M12y4+M8y5M2y6+M12y7]],y7(t)=y7(0)+ST1[H×ST[M11y6y8M13y7]],y8(t)=y8(0)+ST1[H×ST[M11y6y8+M13y7]]. (6)

    We next obtain the following recursive formula.

    y1(n+1)(t)=y1(n)(0)+ST1[H×ST{M1y1(n)+M2y2(n)+M3y3(n)+M4}],y2(n+1)(t)=y2(n)(0)+ST1[H×ST{M1y1(n)M5y2(n)}],y3(n+1)(t)=y3(n)(0)+ST1[H×ST{M6y3(n)+M2y4(n)+M7y5(n)}],y4(n+1)(t)=y4(n)(0)+ST1[H×ST{M3y2(n)+M8y3(n)M9y4(n)}],y5(n+1)(t)=y5(n)(0)+ST1[H×ST{M10y5(n)+M2y6(n)M2y7(n)}],y6(n+1)(t)=y6(n)(0)+ST1[H×ST{M11y6(n)y8(n)+M12y4(n)+M8y5(n)M2y6(n)+M12y7(n)}],y7(n+1)(t)=y7(n)(0)+ST1[H×ST{M11y6(n)y8(n)M13y7(n)}],y8(n+1)(t)=y8(n)(0)+ST1[H×ST{M11y6(n)y8(n)+M13y7(n)}]. (7)

    Where H=1αM(α)αΓ(α+1)Eα(11αPα)

    And the solution of system is provided by

    y1(t)=limny1(n)(t),    y2(t)=limny2(n)(t),    y3(t)=limny3(n)(t),
    y4(t)=limny4(n)(t),    y5(t)=limny5(n)(t),    y6(t)=limny6(n)(t),
    y7(t)=limny7(n)(t),    y8(t)=limny8(n)(t).

    Theorem 4.1: Define K be a self-map is given by

    K[y1(n+1)(t)]=y1(n+1)(t)=y1(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M1y1(n)+M2y2(n)+M3y3(n)+M4}],K[y2(n+1)(t)]=y2(n+1)(t)=y2(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M1y1(n)M5y2(n)}],K[y3(n+1)(t)]=y3(n+1)(t)=y3(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M6y3(n)+M2y4(n)+M7y5(n)}],K[y4(n+1)(t)]=y4(n+1)(t)=y4(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M3y2(n)+M8y3(n)M9y4(n)}],K[y5(n+1)(t)]=y5(n+1)(t)=y5(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M10y5(n)+M2y6(n)M2y7(n)}],K[y6(n+1)(t)]=y6(n+1)(t)=y6(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11y6(n)y8(n)+M12y4(n)+M8y5(n)M2y6(n)+M12y7(n)}],K[y7(n+1)(t)]=y7(n+1)(t)=y7(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11y6(n)y8(n)M13y7(n)}],K[y8(n+1)(t)]=y8(n+1)(t)=y8(n)(0)+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11y6(n)y8(n)+M13y7(n)}]. (8)

    Proof: By using triangular inequality with the definition of norms, we get

    ||K[y1(n)(t)]K[y1(m)(t)]||||y1(n)(t)y1(m)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M1||y1(n)y1(m)||+M2||y2(n)y2(m)||+M3||y3(n)y3(m)||+M4}],||K[y2(n)(t)]K[y2(m)(t)]||||y2(n)(t)y2(m)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M1||y1(n)y1(m)||M5||y2(n)y2(m)||}],||K[y3(n)(t)]K[y3(m)(t)]||||y3(n)(t)y3(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M6||y3(n)y3(m)||+M2||y4(n)y4(m)||+M7||y5(n)y5(m)||}],||K[y4(n)(t)]K[y4(m)(t)]||||y4(n)(t)y4(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M3||y2(n)y2(m)||+M8||y3(n)y3(m)||M9||y4(n)y4(m)||}],||K[y5(n)(t)]K[y5(m)(t)]||||y5(n)(t)y5(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M10||y5(n)y5(m)||+M2||y6(n)y6(m)||M2||y7(n)y7(m)||}],||K[y6(n)(t)]K[y6(m)(t)]||||y6(n)(t)y6(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11||y6(n)y8(n)y6(m)y8(m)||+M12||y4(n)y4(m)||+M8||y5(n)y5(m)||M2||y6(n)y6(m)||+M12||y7(n)y7(m)||}],||K[y7(n)(t)]K[y7(m)(t)]||||y7(n)(t)y7(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11||y6(n)y8(n)y6(m)y8(m)||M13||y7(n)y7(m)||}],||K[y8(n)(t)]K[y8(m)(t)]||||y8(n)(t)y8(n)(t)||+ST1[1αM(α)αΓ(α+1)Eα(11αPα)×ST{M11||y6(n)y8(n)y6(m)y8(m)||+M13||y7(n)y7(m)||}].

    Hence satisfied given conditions.

    θ=(0,0,0,0,0,0,0,0),θ={||y1(n)(t)y1(m)(t)||×||(y1(n)(t)+y1(m)(t))||M1||y1(n)y1(m)||+M2||y2(n)y2(m)||+M3||y3(n)y3(m)||+M4||y2(n)(t)y2(m)(t)||×||(y2(n)(t)+y2(m)(t))||+M1||y1(n)y1(m)||M5||y2(n)y2(m)||||y3(n)(t)y3(m)(t)||×||(y3(n)(t)+y3(m)(t))||M6||y3(n)y3(m)||+M2||y4(n)y4(m)||+M7||y5(n)y5(m)||||y4(n)(t)y4(m)(t)||×||(y4(n)(t)+y4(m)(t))||+M3||y2(n)y2(m)||+M8||y3(n)y3(m)||M9||y4(n)y4(m)||||y5(n)(t)y5(m)(t)||×||(y5(n)(t)+y5(m)(t))||M10||y5(n)y5(m)||+M2||y6(n)y6(m)||M2||y7(n)y7(m)||||y6(n)(t)y6(m)(t)||×||(y6(n)(t)+y6(m)(t))||M11||y6(n)y8(n)y6(m)y8(m)||+M12||y4(n)y4(m)||+M8||y5(n)y5(m)||M2||y6(n)y6(m)||+M12||y7(n)y7(m)||||y7(n)(t)y7(m)(t)||×||(y7(n)(t)+y7(m)(t))||+M11||y6(n)y8(n)y6(m)y8(m)||M13||y7(n)y7(m)||||y8(n)(t)y8(m)(t)||×||(y8(n)(t)+y8(m)(t))||M11||y6(n)y8(n)y6(m)y8(m)||+M13||y7(n)y7(m)||

    Hence the system is stable.

    Theorem 4.2: Unique singular solution with the iterative method for the special solution of system (2).

    Proof: Considering the Hilbert space H=L2((p,q)×(0,T)) which can be defined as

    h:(p,q)×(0,T)R,ghdgdh<.

    For this purpose, we consider the following operator

    θ(0,0,0,0,0,0,0,0),θ={M1y1+M2y2+M3y3+M4,M1y1M5y2,M6y3+M2y4+M7y5,M3y2+M8y3M9y4,M10y5+M2y6M2y7,M11y6y8+M12y4+M8y5M2y6+M12y7,M11y6y8M13y7,M11y6y8+M13y7.

    By using inner product, we get

    T((y1(11)y1(12),y2(21)y2(22),y3(31)y3(32),y4(41)y4(42),y5(51)y5(52),y6(61)        y6(62),y7(71)y7(72),y8(81)y8(82)),(V1,V2,V3,V4,V5,V6,V7,V8)).

    Where

    (y1(11)y1(12),y2(21)y2(22),y3(31)y3(32), y4(41)y4(42),y5(51)y5(52), y6(61)y6(62), y7(71)y7(72), y8(81)y8(82)), are the special solutions of the system. Taking into account the inner function and the norm, we have

    {M1(y1(11)y1(12))+M2(y2(21)y2(22))+M3(y3(31)y3(32))+M4,V1}M1||y1(11)y1(12)||||V1||+M2||y2(21)y2(22)||||V1||+M3||y3(31)y3(32)||||V1||+M4||V1||,{M1(y1(11)y1(12))M5(y2(21)y2(22)),V2}M1||y1(11)y1(12)||||V2||+M5||y2(21)y2(22)||||V2||,{M6(y3(31)y3(32))+M2(y4(41)y4(42))+M7(y5(51)y5(52)),V3}M6||(y3(31)y3(32))||||V3||+M2||(y4(41)y4(42))||||V3||+M7||(y5(51)y5(52))||||V3||,{M3(y2(21)y2(22))+M8(y3(31)y3(32))M9(y4(41)y4(42)),V4}M3||(y2(21)y2(22))||||V4||+M8||(y3(31)y3(32))||||V4||+M9||(y4(41)y4(42))||||V4||,{M10(y5(51)y5(52))+M2(y6(61)y6(62))M2(y7(71)y7(72)),V5}M10||(y5(51)y5(52))||||V5||+M2||(y6(61)y6(62))||||V5||+M2||(y7(71)y7(72))||||V5||,{M11(y6(61)y6(62))(y8(81)y8(82))+M12(y4(41)y4(42))+M8(y5(51)y5(52))M2(y6(61)y6(62))+M12(y7(71)y7(72)),V6}M11||(y6(61)y6(62))||||(y8(81)y8(82))||||V6||+M12||(y4(41)y4(42))||||V6||+M8||(y5(51)y5(52))||||V6||+M2||(y6(61)y6(62))||||V6||+M12||(y7(71)y7(72))||||V6||,{M11(y6(61)y6(62))(y8(81)y8(82))M13(y7(71)y7(72)),V7}M11||(y6(61)y6(62))||||(y8(81)y8(82))||||V7||+M13||(y7(71)y7(72))||||V7||,{M11(y6(61)y6(62))(y8(81)y8(82))+M13(y7(71)y7(72)),V8}M11||(y6(61)y6(62))||||(y8(81)y8(82))||||V8||+M13||(y7(71)y7(72))||||V8||.

    In the case for large number e1,e2,e3,e4,e5,e6,e7ande8, both solutions happen to be converged to the exact solution. Employing the topology concept, we can obtain eight positive very small parameters (χe1,χe2,χe3,χe4,χe5,χe6,χe7andχe8).

    ||y1y1(11)||,||y1y1(12)||χe1ϖ,||y2y2(21)||,||y2y2(22)||χe2ς,
    ||y3y3(31)||,||y3y3(32)||χe3υ,||y4y4(41)||,||y4y4(42)||χe4κ,
    ||y5y5(51)||,||y5y5(52)||χe5ϱ,||y6y6(61)||,||y6y6(62)||χe6ζ,
    ||y7y7(71)||,||y7y7(72)||χe7ν,||y8y8(81)||,||y8y8(82)||χe8ε.

    Where

    ϖ=8(M1||y1(11)y1(12)||+M2||y2(21)y2(22)||+M3||y3(31)y3(32)||+M4)||V1||
    ς=8(M1||y1(11)y1(12)||+M5||y2(21)y2(22)||)||V2||
    υ=8(M6||(y3(31)y3(32))||+M2||(y4(41)y4(42))||+M7||(y5(51)y5(52))||)||V3||
    κ=8(M3||(y2(21)y2(22))||+M8||(y3(31)y3(32))||+M9||(y4(41)y4(42))||)||V4||
    ϱ=8(M10||(y5(51)y5(52))||+M2||(y6(61)y6(62))||+M2||(y7(71)y7(72))||)||V5||
    ζ=8(M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M12||(y4(41)y4(42))||+M8||(y5(51)y5(52))||+M2||(y6(61)y6(62))||+M12||(y7(71)y7(72))||)||V6||
    ν=8(M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M13||(y7(71)y7(72))||)||V7||
    ε=8(M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M13||(y7(71)y7(72))||)||V8||

    But, it is obvious that

    (M1||y1(11)y1(12)||+M2||y2(21)y2(22)||+M3||y3(31)y3(32)||+M4)0
    (M1||y1(11)y1(12)||+M5||y2(21)y2(22)||)0
    (M6||(y3(31)y3(32))||+M2||(y4(41)y4(42))||+M7||(y5(51)y5(52))||)0
    (M3||(y2(21)y2(22))||+M8||(y3(31)y3(32))||+M9||(y4(41)y4(42))||)0
    (M10||(y5(51)y5(52))||+M2||(y6(61)y6(62))||+M2||(y7(71)y7(72))||)0
    (M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M12||(y4(41)y4(42))||+M8||(y5(51)y5(52))||+M2||(y6(61)y6(62))||+M12||(y7(71)y7(72))||)0
    (M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M13||(y7(71)y7(72))||)0
    (M11||(y6(61)y6(62))||||(y8(81)y8(82))||+M13||(y7(71)y7(72))||)0

    where ||V1||,||V2||,||V3||,||V4||,||V5||,||V6||,||V7||,||V8||0.

    Therefore, we have

    ||y1(11)y1(12)||=0,||y2(21)y2(22)||=0,||y3(31)y3(32)||=0,
    ||(y4(41)y4(42))||=0,||(y5(51)y5(52))||=0,||(y6(61)y6(62))||=0,
    ||(y7(71)y7(72))||=0,||(y8(81)y8(82))||=0.

    Which yields that

    y1(11)=y1(12),y2(21)=y2(22),y3(31)=y3(32),y4(41)=y4(42),y5(51)=y5(52),y6(61)=y6(62),y7(71)=y7(72),y8(81)=y8(82)

    This completes the proof of uniqueness.

    An operator B:ZZ can be defined as:

    B(φ)(t)=φ(0)+μtμ1(1α1)AB(α1)£(t,φ(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(t,φ(t))dλ (10)

    If £(t,φ(t)) satisfies the Lipschitz condition and the following extension then

    ● For every φZ there exists constants L£>0 and M£ such that

    |£(t,φ(t))|L£|φ(t)|+M£ (11)

    ● For every φ,¯φZ, there exists a constant M£>0 such that

    |£(t,φ(t))£(t,¯φ(t))||M£|φ(t)¯φ(t)| (12)

    Theorem 4.2: If the condition of (11) holds then for the function £:[0,T]×ZR there exists at least one solution for the (1).

    Proof: Since £ in (10) is continuous function, so B is also a continuous. Assume M={φ||φ||R,R>0}, then for φZ, we have

    B(φ)(t)=maxt[0,T]|φ(0)+μtμ1(1α1)AB(α1)£(t,φ(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(t,φ(t))dλ
    |φ(0)+μTμ1(1α1)AB(α1)(L£||φ(t)||+M£)+maxt[0,T]μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(t,φ(t))dλ|
    φ(0)+μTμ1(1α1)AB(α1)(L£||φ(t)||+M£)+μα1AB(α1)Γ(α1)(L£||φ(t)||+M£)Tμ+α11M(μ,α1)R.

    Hence, B is uniformly bounded, and M(μ,α1) is a beta function. For equicontinuity of B, we take t1<t2T, then consider

    B(φ)(t2)B(φ)(t1)=|μt2μ1(1α1)AB(α1)£(t2,φ(t2))+μα1AB(α1)Γ(α1)
    t20λμ1(t2λ)μ1£(t,φ(t))dλμt1μ1(1α1)AB(α1)£(t1,φ(t1))            +μα1AB(α1)Γ(α1)t20λμ1(t1λ)μ1£(t,φ(t))dλ|
    μt2μ1(1α1)AB(α1)(L£|φ(t)|+M£)+μα1AB(α1)Γ(α1)(L£|φ(t)|+M£)t2μ+α11M(μ,α1)
    μt1μ1(1α1)AB(α1)(L£|φ(t)|+M£)μα1AB(α1)Γ(α1)(L£|φ(t)|+M£)t1μ+α11M(μ,α1)

    If t1t2 then ||B(φ)(t2)B(φ)(t1)0|| Consequently ||B(φ)(t2)B(φ)(t1)0||,ast1t2. Hence B is equicontinous. Thus, by Arzela-Ascoli theorem B is completely continuous. Consequently, by the result of Schauder's fixed point, it has at least one solution.

    Theorem 4.3: If η=μTμ1(1α1)AB(α1)+μα1AB(α1)Γ(α1)Tμ+α11M(μ,α1)M£<1 and the condition (12) holds, then η has a unique solution.

    Proof: For φ,¯φZ, we have

    |B(φ)B(¯φ)|=maxt[0,T]|μtμ1(1α1)AB(α1)[£(t,φ(t))|£(t,¯φ(t))]
    +μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1[£(t,φ(t))|£(t,¯φ(t))]dλ|
    [μTμ1(1α1)AB(α1)+μα1AB(α1)Γ(α1)Tμ+α11M(μ,α1)]||B(φ)B(¯φ)||
    η||B(φ)B(¯φ)||.

    Hence, B is a contraction. So, by the principle of Banach contraction, it has a unique solution.

    Ulam-Hyres stability

    The proposed model is Ulam-Hyres stable if there exists Bμ,α10 such that for every ε>0 and for every φ(L[0,T],R) satisfies the following inequality FFMJμ,α10,t(φ(t))£(t,φ(t))ε,t[0,T] such that |φ(t)£(t)|Bμ,α1ε,t[0,T].

    Suppose a perturbation ωL[0,T],R then ω(0)=0 and

    ● For every ε>0ω(t)ε|

    0FFMJμ,α1t(φ(t))=£(t,φ(t))+ω(t).

    Lemma 4.4: The solution of the perturbed model 0FFMJμ,α1t(φ(t))=£(t,φ(t))+ω(t),φ(0)=φ0 fulfills the relation

    B(t)[φ(0)+μtμ1(1α1)AB(α1)£(t,φ(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(λ,φ(λ))dλ        α1α1,με

    Where α1α1,με=μTμ1(1α1)AB(α1)+μα1AB(α1)Γ(α1)Tμ+α11M(μ,α1).

    Lemma 4.5: By using condition (12) with lemma (4.4), proposed model is Ulam-Hyres stable if η<1.

    Proof: Suppose α1Z be a solution and φZ be any solution of (1), then

    |φ(t)α1(t)|=|φ(t)[α1(0)+μtμ1(1α1)AB(α1)£(t,α1(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(λ,α1(λ))dλ]|
    |φ(t)[φ(0)+μtμ1(1α1)AB(α1)£(t,φ(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(λ,φ(λ))dλ]|            +|φ(0)+μtμ1(1α1)AB(α1)£(t,φ(t))            +μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(λ,φ(λ))dλ|
    |α1(0)+μtμ1(1α1)AB(α1)£(t,α1(t))+μα1AB(α1)Γ(α1)t0λμ1(1λ)μ1£(λ,α1(λ))dλ|
    α1α1,με+(μTμ1(1α1)AB(α1)+μα1AB(α1)Γ(α1)Tμ+α11)L£|φ(t)α1(t)|
    α1α1,με+η|φ(t)α1(t)|.

    Consequently,

    ||φα1||α1α1,με+η||φ(t)α1(t)||.

    So, we can write it as

    ||φα1||Bα1,με,

    Where Bα1,με=α1α1,μ1η. Hence the solution is Ulam-Hyres stable.

    In this section, we present the Hires problem model (1) using fractal-fractional Atangana-Baleanu derivative. We have

    FFDα1,α20,ty1=M1y1+M2y2+M3y3+M4,FFDα1,α20,ty2=M1y1M5y2,FFDα1,α20,ty3=M6y3+M2y4+M7y5,FFDα1,α20,ty4=M3y2+M8y3M9y4,FFDα1,α20,ty5=M10y5+M2y6M2y7,FFDα1,α20,ty6=M11y6y8+M12y4+M8y5M2y6+M12y7,FFDα1,α20,ty7=M11y6y8M13y7,FFDα1,α20,ty8=M11y6y8+M13y7. (13)

    With initial conditions

    y1(0)=y1(0),y2(0)=y2(0),y3(0)=y3(0),y4(0)=y4(0),y5(0)=y5(0),y6(0)=y6(0),y7(0)=y7(0),y8(0)=y8(0).

    We present the numerical algorithm for the fractal-fractional Hires problem model (13). The following is obtained by integrating the system (13).

    y1(t)y1(0)=(1α1)C(α1)α2tα21{M1y1(t)+M2y2(t)+M3y3(t)+M4}+α1α2C(α1)Γ(α1)t0τα21{M1y1(τ)+M2y2(τ)+M3y3(τ)+M4}(tτ)α11dτ,y2(t)y2(0)=(1α1)C(α1)α2tα21{M1y1(t)M5y2(t)}+α1α2C(α1)Γ(α1)t0τα21{{M1y1(τ)M5y2(τ)}}(tτ)α11dτ,y3(t)y3(0)=(1α1)C(α1)α2tα21{M6y3(t)+M2y4(t)+M7y5(t)}+α1α2C(α1)Γ(α1)t0τα21{M6y3(τ)+M2y4(τ)+M7y5(τ)}(tτ)α11dτ,y4(t)y4(0)=(1α1)C(α1)α2tα21{M3y2(t)+M8y3(t)M9y4(t)}+α1α2C(α1)Γ(α1)t0τα21{M3y2(τ)+M8y3(τ)M9y4(τ)}(tτ)α11dτ,y5(t)y5(0)=(1α1)C(α1)α2tα21{M10y5(t)+M2y6(t)M2y7(t)}+α1α2C(α1)Γ(α1)t0τα21{{M10y5(τ)+M2y6(τ)M2y7(τ)}}(tτ)α11dτ,y6(t)y6(0)=(1α1)C(α1)α2tα21{M11y6(t)y8(t)+M12y4(t)+M8y5(t)M2y6(t)+M12y7(t)}+α1α2C(α1)Γ(α1)t0τα21{M11y6(τ)y8(τ)+M12y4(τ)+M8y5(τ)M2y6(τ)+M12y7(τ)}(tτ)α11dτ,y7(t)y7(0)=(1α1)C(α1)α2tα21{M11y6(t)y8(t)M13y7(t)}+α1α2C(α1)Γ(α1)t0τα21{M11y6(τ)y8(τ)M13y7(τ)}(tτ)α11dτ,y8(t)y8(0)=(1α1)C(α1)α2tα21{M11y6(t)y8(t)+M13y7(t)}+α1α2C(α1)Γ(α1)t0τα21{{M11y6(τ)y8(τ)+M13y7(τ)}}(tτ)α11dτ, (14)

    Let

    k(t,y1(t))=α2tα21{M1y1(t)+M2y2(t)+M3y3(t)+M4},
    k(t,y2(t))=α2tα21{M1y1(t)M5y2(t)},
    k(t,y3(t))=α2tα21{M6y3(t)+M2y4(t)+M7y5(t)},
    k(t,y4(t))=α2tα21{M3y2(t)+M8y3(t)M9y4(t)},
    k(t,y5(t))=α2tα21{M10y5(t)+M2y6(t)M2y7(t)},
    k(t,y6(t))=α2tα21{M11y6(t)y8(t)+M12y4(t)+M8y5(t)M2y6(t)+M12y7(t)},
    k(t,y7(t))=α2tα21{M11y6(t)y8(t)M13y7(t)},
    k(t,y8(t))=α2tα21{M11y6(t)y8(t)+M13y7(t)}.

    Then system (14) becomes

    y1(t)y1(0)=(1α1)C(α1)k(t,y1(t))+α1C(α1)Γ(α1)t0k(τ,y1(τ))(tτ)α11dτ,y2(t)y2(0)=(1α1)C(α1)k(t,y2(t))+α1C(α1)Γ(α1)t0k(τ,y2(τ))(tτ)α11dτ,y3(t)y3(0)=(1α1)C(α1)k(t,y3(t))+α1C(α1)Γ(α1)t0k(τ,y3(τ))(tτ)α11dτ,y4(t)y4(0)=(1α1)C(α1)k(t,y4(t))+α1C(α1)Γ(α1)t0k(τ,y4(τ))(tτ)α11dτ,y5(t)y5(0)=(1α1)C(α1)k(t,y5(t))+α1C(α1)Γ(α1)t0k(τ,y5(τ))(tτ)α11dτ,y6(t)y6(0)=(1α1)C(α1)k(t,y6(t))+α1C(α1)Γ(α1)t0k(τ,y6(τ))(tτ)α11dτ,y7(t)y7(0)=(1α1)C(α1)k(t,y7(t))+α1C(α1)Γ(α1)t0k(τ,y7(τ))(tτ)α11dτ,y8(t)y8(0)=(1α1)C(α1)k(t,y8(t))+α1C(α1)Γ(α1)t0k(τ,y8(τ))(tτ)α11dτ, (15)

    At tn+1=(n+1)Δt, we have

    y1(tn+1)y1(0)=(1α1)C(α1)k(tn,y1(tn))+α1C(α1)Γ(α1)tn+10k(τ,y1(τ))(tn+1τ)α11dτ,y2(tn+1)y2(0)=(1α1)C(α1)k(tn,y2(tn))+α1C(α1)Γ(α1)tn+10k(τ,y2(τ))(tn+1τ)α11dτ,y3(tn+1)y3(0)=(1α1)C(α1)k(tn,y3(tn))+α1C(α1)Γ(α1)tn+10k(τ,y3(τ))(tn+1τ)α11dτ,y4(tn+1)y4(0)=(1α1)C(α1)k(tn,y4(tn))+α1C(α1)Γ(α1)tn+10k(τ,y4(τ))(tn+1τ)α11dτ,y5(tn+1)y5(0)=(1α1)C(α1)k(tn,y5(tn))+α1C(α1)Γ(α1)tn+10k(τ,y5(τ))(tn+1τ)α11dτ,y6(tn+1)y6(0)=(1α1)C(α1)k(tn,y6(tn))+α1C(α1)Γ(α1)tn+10k(τ,y6(τ))(tn+1τ)α11dτ,y7(tn+1)y7(0)=(1α1)C(α1)k(tn,y7(tn))+α1C(α1)Γ(α1)tn+10k(τ,y7(τ))(tn+1τ)α11dτ,y8(tn+1)y8(0)=(1α1)C(α1)k(tn,y8(tn))+α1C(α1)Γ(α1)tn+10k(τ,y8(τ))(tn+1τ)α11dτ. (16)

    Also, we have

    y1(tn+1)=y1(0)+(1α1)C(α1)k(tn,y1(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y1(τ))(tn+1τ)α11dτ,y2(tn+1)=y2(0)+(1α1)C(α1)k(tn,y2(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y2(τ))(tn+1τ)α11dτ,y3(tn+1)=y3(0)+(1α1)C(α1)k(tn,y3(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y3(τ))(tn+1τ)α11dτ,y4(tn+1)=y4(0)+(1α1)C(α1)k(tn,y4(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y4(τ))(tn+1τ)α11dτ,y5(tn+1)=y5(0)+(1α1)C(α1)k(tn,y5(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y5(τ))(tn+1τ)α11dτ,y6(tn+1)=y6(0)+(1α1)C(α1)k(tn,y6(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y6(τ))(tn+1τ)α11dτ,y7(tn+1)=y7(0)+(1α1)C(α1)k(tn,y7(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y7(τ))(tn+1τ)α11dτ,y8(tn+1)=y8(0)+(1α1)C(α1)k(tn,y8(tn))+α1C(α1)Γ(α1)nj=0tj+1tjk(τ,y8(τ))(tn+1τ)α11dτ. (17)

    In general, approximating the function k\left(\tau, y\left(\tau \right)\right), using the Newton polynomial, we have

    \begin{array}{l} {P}_{n}\left(\tau \right) = \frac{k\left({t}_{n}, y\left({t}_{n}\right)\right)}{{t}_{n}-{t}_{n-1}}\left(\tau -{t}_{n-1}\right)+\frac{k\left({t}_{n-1}, y\left({t}_{n-1}\right)\right)}{{t}_{n}-{t}_{n-1}}\left(\tau -{t}_{n}\right) = \frac{k\left({t}_{n}, y\left({t}_{n}\right)\right)}{h}\left(\tau -{t}_{n-1}\right)\frac{k\left({t}_{n-1}, y\left({t}_{n-1}\right)\right)}{h}\left(\tau -\\ {t}_{n}\right). \end{array} (18)

    Using Eq (18) into system (17) we have

    \begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}\int_{{t}_{j}}^{{t}_{j+1}} \{k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)+\\ \frac{k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{\Delta t}\left(\tau -{t}_{j-2}\right)+\frac{k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{{2\left(\Delta t\right)}^{2}}\left(\tau -{t}_{j-2}\right)(\tau -\\ {t}_{j-1})\}{\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \end{array} (19)

    Rearranging the above equation, we have

    \begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}[\int_{{t}_{j}}^{{t}_{j+1}} k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \int_{{t}_{j}}^{{t}_{j+1}} \frac{k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{\Delta t}\left(\tau -{t}_{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \int_{{t}_{j}}^{{t}_{j+1}} \frac{k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{{2\left(\Delta t\right)}^{2}}\left(\tau -{t}_{j-2}\right)\left(\tau -{t}_{j-1}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau ], \end{array} (20)

    Writing further system (20) we have

    \begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\int_{{t}_{j}}^{{t}_{j+1}} {\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}\frac{k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{\Delta t}\int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}\frac{k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{{2\left(\Delta t\right)}^{2}}\int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right)\left(\tau -{t}_{j-1}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \end{array} (21)

    Now, calculating the integrals in system (21) we get

    \int_{{t}_{j}}^{{t}_{j+1}} {\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}}}{{\alpha }_{1}}\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right],
    \begin{array}{l} \int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}+1}}{{\alpha }_{1}\left({\alpha }_{1}+1\right)}[{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-{\left(n-j+1\right)}^{{\alpha }_{1}}(n-\\ j+3+3{\alpha }_{1})], \end{array}
    \begin{array}{l} \int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right)\left(\tau -{t}_{j-1}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}+2}}{{\alpha }_{1}\left({\alpha }_{1}+1\right)\left({\alpha }_{1}+2\right)}[{\left(n-j+1\right)}^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\\ \left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\right\}-{\left(n-j\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+\left(5{\alpha }_{1}+10\right)\left(n-j\right)+6{{\alpha }_{1}}^{2}+\\ 18{\alpha }_{1}+12\}]. \end{array}

    Inserting them into system (21) we get

    \begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+1\right)}\sum _{j = 2}^{n}k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right]+\\ \frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+2\right)}\sum _{j = 2}^{n}\left[k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\right][{(n-j+1)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-(n-j+\\ 1)^{{\alpha }_{1}}\left(n-j+3+3{\alpha }_{1}\right)]+\frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{2\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+3\right)}\sum _{j = 2}^{n}\left[k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\right][(n-\\ j+1)^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\right\}-{\left(n-j\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+(5{\alpha }_{1}+\\ 10)\left(n-j\right)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}] \end{array} (22)

    Finally, we have the following approximation:

    \begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{1}{y}_{1}(t)+{M}_{2}{y}_{2}(t)+{M}_{3}{y}_{3}(t)+{M}_{4}\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+{M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-1}+{M}_{2}{{y}_{2}}^{j-1}+{M}_{3}{{y}_{3}}^{j-1}+{M}_{4}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+\\ {M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j}+{M}_{2}{{y}_{2}}^{j}+{M}_{3}{{y}_{3}}^{j}+{M}_{4}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-1}+{M}_{2}{{y}_{2}}^{j-1}+\\ {M}_{3}{{y}_{3}}^{j-1}+{M}_{4}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+{M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+\\ (3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+\\ 18{\alpha }_{1}+12\}], \\ {{y}_{2}}^{n+1} = {{y}_{2}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{1}{y}_{1}(t)-{M}_{5}{y}_{2}(t)\}+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-\\ {M}_{5}{{y}_{2}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-1}-{M}_{5}{{y}_{2}}^{j-1}\}-\\ {t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-{M}_{5}{{y}_{2}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+\\ 3{\alpha }_{1})]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j}-{M}_{5}{{y}_{2}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-1}-{M}_{5}{{y}_{2}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-{M}_{5}{{y}_{2}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-\\ {(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{3}}^{n+1} = {{y}_{3}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{6}{y}_{3}(t)+{M}_{2}{y}_{4}(t)+{M}_{7}{y}_{5}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+{M}_{7}{{y}_{5}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-1}+{M}_{2}{{y}_{4}}^{j-1}+{M}_{7}{{y}_{5}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+\\ {M}_{7}{{y}_{5}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j}+{M}_{2}{{y}_{4}}^{j}+\\ {M}_{7}{{y}_{5}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-1}+{M}_{2}{{y}_{4}}^{j-1}+{M}_{7}{{y}_{5}}^{j-1}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+\\ {M}_{7}{{y}_{5}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-\\ j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{4}}^{n+1} = {{y}_{4}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{3}{y}_{2}(t)+{M}_{8}{y}_{3}(t)-{M}_{9}{y}_{4}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-{M}_{9}{{y}_{4}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-1}+{M}_{8}{{y}_{3}}^{j-1}-{M}_{9}{{y}_{4}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-\\ {M}_{9}{{y}_{4}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j}+{M}_{8}{{y}_{3}}^{j}-{M}_{9}{{y}_{4}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-1}+{M}_{8}{{y}_{3}}^{j-1}-{M}_{9}{{y}_{4}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-{M}_{9}{{y}_{4}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{5}}^{n+1} = {{y}_{5}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{10}{y}_{5}(t)+{M}_{2}{y}_{6}(t)-{M}_{2}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+{M}_{2}{{y}_{6}}^{j-2}-{M}_{2}{{y}_{7}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-1}+{M}_{2}{{y}_{6}}^{j-1}-{M}_{2}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+{M}_{2}{{y}_{6}}^{j-2}-\\ {M}_{2}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j}+{M}_{2}{{y}_{6}}^{j}-\\ {M}_{2}{{y}_{7}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-1}+{M}_{2}{{y}_{6}}^{j-1}-{M}_{2}{{y}_{7}}^{j-1}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+\\ {M}_{2}{{y}_{6}}^{j-2}-{M}_{2}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+\\ 10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+\\ 10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+\\ 12\}], \end{array} (23)
    \begin{array}{l} {{y}_{6}}^{n+1} = {{y}_{6}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{11}{y}_{6}(t){y}_{8}(t)+{M}_{12}{y}_{4}(t)+{M}_{8}{y}_{5}(t)-{M}_{2}{y}_{6}(t)+{M}_{12}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+{M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}[(n-\\ j+1)^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{12}{{y}_{4}}^{j-1}+{M}_{8}{{y}_{5}}^{j-1}-\\ {M}_{2}{{y}_{6}}^{j-1}+{M}_{12}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+\\ {M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}+{M}_{12}{{y}_{4}}^{j}+{M}_{8}{{y}_{5}}^{j}-{M}_{2}{{y}_{6}}^{j}+{M}_{12}{{y}_{7}}^{j}\}-\\ 2{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{12}{{y}_{4}}^{j-1}+{M}_{8}{{y}_{5}}^{j-1}-{M}_{2}{{y}_{6}}^{j-1}+{M}_{12}{{y}_{7}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+{M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2(n-\\ j)^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+\\ 6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \end{array}
    \begin{array}{l} {{y}_{7}}^{n+1} = {{y}_{7}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{11}{y}_{6}(t){y}_{8}(t)-{M}_{13}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}-{M}_{13}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}][(n-\\ j+1)^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}-{M}_{13}{{y}_{7}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}-{M}_{13}{{y}_{7}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \end{array}
    \begin{array}{l} {{y}_{8}}^{n+1} = {{y}_{8}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{13}{y}_{7}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+1\right)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{13}{{y}_{7}}^{j-2}\right\}\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+2\right)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{13}{{y}_{7}}^{j-1}\right\}-{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+\\ {M}_{13}{{y}_{7}}^{j-2}\}]\left[{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+3{\alpha }_{1}\right)\right]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{2\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+3\right)}\sum _{j = 2}^{n}\left[{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}+{M}_{13}{{y}_{7}}^{j}\right\}-2{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{13}{{y}_{7}}^{j-1}\right\}+\\ {t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{13}{{y}_{7}}^{j-2}\right\}\right][{\left(n-j+1\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+\left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{\left(n-j\right)}^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\left(5{\alpha }_{1}+10\right)\left(n-j\right)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\right\}]. \end{array}

    A fractional-order model is proposed for analysis and simulation, to observe the concentration of chemicals in chemistry kinematics problems with a stiff differential equation. For this purpose, we used ABC with Mittage-Lefffier law, Atangana-Tufik scheme, and fractal fractional derivative for hires problem with given initial conditions. Details of parameters values of real data are also given in [18,19] which will consider for simulation analysis for the proposed study. Solution of compartment shows in Figures 1 to 8 with fractional fractal operator at different order. Effect of fraction order can easily be observed in simulation of the compartments having a concentration of chemical reaction with stiff differential equations. The concentration {y}_{1} and {y}_{8} of the chemical species start decreasing by decreasing fractional values respectively while concentration {y}_{2}, {y}_{3} , {y}_{4} , {y}_{5} , {y}_{6} and {y}_{7} of the chemical species start increasing by decreasing fractional values. These concentrations of chemical species converge to our desired value according to steady state by decreasing the fractional values which shows that it provides us appropriate results at non integer value. We can get better concentration of the components by using the fractional derivative which are very important for chemical problem to check the actual behavior of the concentration of the chemical with smallest changes in derivative with respect to time. It is also very important for solutions of nonlinear problems which are commonly used researcher and scientist in kinetics chemistry.

    Figure 1.  Simulation of {y}_{1}\left(t\right) with fractal fractional derivative.
    Figure 2.  Simulation of {y}_{2}\left(t\right) with fractal fractional derivative.
    Figure 3.  Simulation of {y}_{3}\left(t\right) with fractal fractional derivative.
    Figure 4.  Simulation of {y}_{4}\left(t\right) with fractal fractional derivative.
    Figure 5.  Simulation of {y}_{5}\left(t\right) with fractal fractional derivative.
    Figure 6.  Simulation of {y}_{6}\left(t\right) with fractal fractional derivative.
    Figure 7.  Simulation of {y}_{7}\left(t\right) with fractal fractional derivative.
    Figure 8.  Simulation of {y}_{8}\left(t\right) with fractal fractional derivative.

    We examine the hires problems with stiff systems of nonlinear ordinary equations that rely on the concentration of chemical reaction of components in this study. The advanced techniques of fractional operator have been implemented for initial value problem arising from chemical reactions composed of large systems of stiff ordinary differential equations. The arbitrary derivative of fractional order has been taken with Atangana-Toufik scheme and fractal fractional derivative. Solutions have been obtained efficiently within limited time which shows the actual behavior of kinetic chemical reactions. Existence and uniqueness of results have been verified by fixed point theorem. Simulations are carried out for different fractional values. New chemical reactions can be done with the help of these analyses. These concepts are very important to use for real life problems like Brine tank cascade, Recycled Brine tank cascade, pond pollution, home heating and biomass transfer problem.

    Research Supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.

    No conflict of interest.



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