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Fractional calculus is a generalization of classical calculus and many researchers have paid attention to this science as they encounter many of these issues in the real world. Most of these issues do not have analytical exact solution. Which made many researchers interest and search in numerical and approximate methods to obtain solutions using these methods. There are many of these methods, such as the homotopy analysis [1,2,3,4], He's variational iteration method [5,6], Adomians decomposition method [7,8,9], Fourier spectral methods [10], finite difference schemes [11], collocation methods [12,13,14]. To find out more about the fractal calculus, refer to the following references [15,16]. More recently, a new concept was introduced for the fractional operator, as this operator has two orders, the first representing the fractional order, and the second representing the fractal dimension. In our work we aim to applied the idea of fractal-fractional derivative of orders $ \beta, k $ to a reaction-diffusion equation with $ q $-th nonlinear. To this end [17], we replace the derivative with respect to t by the fractal-fractional derivatives power (FFP) law, the fractal-fractional exponential(FFE) law and the fractal-fractional Mittag-Leffler (FFM) law kernels which corresponds to the [18], Caputo-Fabrizio (CF) [19] and the Atangana-Baleanu (AB) [20] fractional derivatives, respectively. This topic has attracted many researchers and has been applied to research related to the real world, such as [21,22,23,24,25,26]. Some recent developments in the area of numerical techniques can be found in [27,28,29,30,31].
Merkin and Needham [32] considered the reaction-diffusion travelling waves that can develop in a coupled system involving simple isothermal autocatalysis kinetics. They assumed that reactions took place in two separate and parallel regions, with, in $ I $, the reaction being given by quadratic autocatalysis
$ F+G→2G(ratek1fg), $ | (1.1) |
together with a linear decay step
$ G→H(ratek2g) $ | (1.2) |
where $ f $ and $ g $ are the concentrations of reactant $ F $ and autocatalyst $ H $, the $ k_{i}(i = 1, 2) $ are the rate constants and $ H $ is some inert product of reaction. The reaction in region $ II $ was the quadratic autocatalytic step (1.1) only. The two regions were assumed to be coupled via a linear diffusive interchange of the autocatalytic species $ G $. We shall consider a similar system as I, but with cubic autocatalysis
$ F+2G→3G(ratek3fg2) $ | (1.3) |
together with a linear decay step
$ G→H(ratek4g). $ | (1.4) |
For $ q $-th autocatalytic, we have
$ F+qG→(q+1)G(ratek3fgq),1≤q≤2, $ | (1.5) |
together with a linear decay step
$ G→H(rate k4g). $ | (1.6) |
This yields to the following system
$ ∂η1∂t=∂2η1∂ξ2+ν(η2−η1)−η1ζq1, $ | (1.7) |
$ ∂ζ1∂t=∂2ζ1∂ξ2−κζ1+η1ζq1, $ | (1.8) |
$ ∂η2∂t=∂2η2∂ξ2+ν(η1−η2)−η2ζq2, $ | (1.9) |
$ ∂ζ2∂t=∂2ζ2∂ξ2+η2ζq2 $ | (1.10) |
where $ \nu $ represents the couple between (I) and (II) and $ \kappa $ represents the strength of the auto-catalyst decay. For more details see [32]. Omitting the diffusion terms in the system (1.7)-(1.10), one has the following ordinary differential equations
$ ∂η1∂t=ν(η2−η1)−η1ζq1, $ | (1.11) |
$ ∂ζ1∂t=−κζ1+η1ζq1, $ | (1.12) |
$ ∂η2∂t=ν(η1−η2)−η2ζq2, $ | (1.13) |
$ ∂ζ2∂t=η2ζq2. $ | (1.14) |
Now we provide some basic definitions that be needed in this work. As for the theorems and proofs related to the three fractal-fractional operators, they are found in details in [17]. Thus we suffice in this work by constructing the algorithms and making the numerical simulations of the set of Eqs (1.7)-(1.10) with the three fractal-fractional operators.
Definition 1. If $ \eta(t) $ is continuous and fractal differentiable on $ (a, b) $ of order $ k $, then the fractal-fractional derivative of $ \eta(t) $ of order $ \beta $ in Riemann Liouville sense with the power law is given by [17]:
$ {FFP}0Dβ,ktη(t)=1Γ(1−β)ddtk∫t0(t−τ)−βη(τ)dτ,(0<β,k≤1), $ | (1.15) |
and the fractal-fractional integral of $ \eta(t) $ is given by
$ FFP0Iβ,ktη(t)=kΓ(β)∫t0τk−1(t−τ)β−1η(τ)dτ. $ | (1.16) |
Definition 2. If $ \eta(t) $ is continuous in the $ (a, b) $ and fractal differentiable on $ (a, b) $ with order $ k $, then the fractal-fractional derivative of $ \eta(t) $ of order $ \beta $ in Riemann Liouville sense with the exponential decay kernel is given by [17]:
$ FFE0Dβ,ktη(t)=M(β)1−βddtk∫t0e−β1−β(t−τ)η(τ)dτ,(0<β,k≤1), $ | (1.17) |
and the fractal-fractional integral of $ \eta(t) $ is given by
$ FFE0Iβ,ktη(t)=(1−β)ktk−1M(β)η(t)+βkM(β)∫t0τk−1η(τ)dτ $ | (1.18) |
where $ M(\beta) $ is the normalization function such that $ M(0) = M(1) = 1. $
Definition 3. If $ \eta(t) $ is continuous in the $ (a, b) $ and fractal differentiable on $ (a, b) $ with order $ k $, then the fractal-fractional derivative of $ \eta(t) $ of order $ \beta $ in Riemann Liouville sense with the Mittag-Leffler type kernel is given by [17]:
$ FFE0Dβ,ktη(t)=A(β)1−βddtk∫t0Eβ(−β1−β(t−τ))η(τ)dτ,(0<β,k≤1), $ | (1.19) |
and the fractal-fractional integral of $ \eta(t) $ is given by
$ FFE0Iβ,ktη(t)=(1−β)ktk−1A(β)η(t)+βkA(β)Γ(β)∫t0τk−1(t−τ)β−1η(τ)dτ, $ | (1.20) |
$ dη(t)dtk=limτ→tη(τ)−η(t)τk−tk $ | (1.21) |
where where $ A(\beta) = 1-\beta+\dfrac{\beta}{\Gamma(\beta)} $ is a normalization function such that $ A(0) = A(1) = 1 $.
Our contribution to this paper is to construct the successive approximations and evaluate the numerical solutions of the FFRDE. These successive approximations allow us to study the behavior of numerical solutions based on power, exponential, and the Mittag-Leffler kernels. Also we can study the behavior of approximate solutions in the case of nonlinearity of the FFRDE in general. To our best knowledge, this is the first study of the FFRDE using fractal-fractional with these kernels. The importance of these results lies in the fact that they highlight the possibility of using these results for the benefit of chemical and physical researchers, by trying to link the numerical results of these mathematical models with the laboratory results. These results also contribute to the reliance on numerical results in the case of many models related to the real world, which often cannot find an analytical solution. The structure of this paper is summarized as follows: In sections, two, three and four, the FFRDE is presented with the three kernels that proposed in this work and construct the successive approximations. In section Five, numerical solutions for the FFRDE are discussed with a study of their behavior. Section Six the conclusion is presented.
The new model is obtained by replacing the ordinary derivative with the the fractal-fractional derivative the power law kernel as [17]
$ FFP0Dβtη1(t)=ν(η2(t)−η1(t))−η1(t)ζq1(t), $ | (2.1) |
$ FFP0Dβtζ1(t)=−κζ1(t)+η1(t)ζq1(t), $ | (2.2) |
$ FFP0Dβtη2(t)=ν(η1(t)−η2(t))−η2(t)ζq2(t), $ | (2.3) |
$ FFP0Dβtζ2(t)=η2(t)ζq2(t). $ | (2.4) |
By following the procedure in [17], we can obtain the following successive approximations:
$ η1(t)−η1(0)=kΓ(β)∫t0τk−1(t−τ)β−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (2.5) |
$ ζ1(t)−ζ2(0)=kΓ(β)∫t0τk−1(t−τ)β−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (2.6) |
$ η2(t)−η3(0)=kΓ(β)∫t0τk−1(t−τ)β−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (2.7) |
$ ζ2(t)−ζ2(0)=kΓ(β)∫t0τk−1(t−τ)β−1φ4(η1,ζ1,η2,ζ2,τ)dτ $ | (2.8) |
where
$ φ1(η1,ζ1,η2,ζ2,τ)=(ν(η2(τ)−η1(τ))−η1(τ)ζq1(τ)), $ | (2.9) |
$ φ2(η1,ζ1,η2,ζ2,τ)=(−κζ1(τ)+η1(τ)ζq1(τ)), $ | (2.10) |
$ φ3(η1,ζ1,η2,ζ2,τ)=(ν(η1(τ)−η2(τ))−η2(τ)ζq2(τ)), $ | (2.11) |
$ φ4(η1,ζ1,η2,ζ2,τ)=η2(τ)ζq2(τ). $ | (2.12) |
Equation (2.5)-(2.8) can be reformulated as
$ η1(t)−η1(0)=kΓ(β)n∑m=0∫tm+1tmτk−1(tn+1−τ)β−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (2.13) |
$ ζ1(t)−ζ1(0)=kΓ(β)n∑m=0∫tm+1tmτk−1(tn+1−τ)β−1φ2(η1(τ),ζ1(τ),η2(τ),ζ2(τ),τ)dτ, $ | (2.14) |
$ η2(t)−η2(0)=kΓ(β)n∑m=0∫tm+1tmτk−1(tn+1−τ)β−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (2.15) |
$ ζ2(t)−ζ2(0)=kΓ(β)n∑m=0∫tm+1tmτk−1(tn+1−τ)β−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (2.16) |
Using the two-step Lagrange polynomial interpolation, we obtain
$ η1(t)−η1(0)=kΓ(β)n∑m=0∫tm+1tm(tn+1−τ)β−1Q1,m(τ)dτ, $ | (2.17) |
$ ζ1(t)−ζ1(0)=kΓ(β)n∑m=0∫tm+1tm(tn+1−τ)β−1Q2,m(τ)dτ, $ | (2.18) |
$ η2(t)−η2(0)=kΓ(β)n∑m=0∫tm+1tm(tn+1−τ)β−1Q3,m(τ)dτ, $ | (2.19) |
$ ζ2(t)−ζ2(0)=kΓ(β)n∑m=0∫tm+1tm(tn+1−τ)β−1Q4,m(τ)dτ, $ | (2.20) |
where,
$ Q1,m(τ)=τ−tm−1tm−tm−1tk−1mφ1(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)−τ−tmtm−tm−1×tk−1m−1φ1(η1(τm−1),ζ1(τm−1),η2(τm−1),ζ2(τm−1),τm−1), $ | (2.21) |
$ Q2,m(τ)=τ−tm−1tm−tm−1tk−1mφ2(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)−τ−tmtm−tm−1×tk−1m−1φ2(η1(τm−1),ζ1(τm−1),η2(τm−1),ζ2(τm−1),τm−1), $ | (2.22) |
$ Q3,m(τ)=τ−tm−1tm−tm−1tk−1mφ3(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)−τ−tmtm−tm−1×tk−1m−1φ3(η1(τm−1),ζ1(τm−1),η2(τm−1),ζ2(τm−1),τm−1), $ | (2.23) |
$ Q4,m(τ)=τ−tm−1tm−tm−1tk−1mφ4(η4(τm),ζ1(τm),η2(τm),ζ2(τm),τm)−τ−tmtm−tm−1×tk−1m−1φ4(η1(τm−1),ζ1(τm−1),η2(τm−1),ζ2(τm−1),τm−1). $ | (2.24) |
These integrals are evaluated directly and the numerical solutions of (2.1)-(2.4) involving the FFP derivative are given by
$ η1(tn+1)=η1(0)+khβΓ(β+2)n∑m=0tk−1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)−tk−1m−1φ1(η1(τm−1),ζ1(tm−1),η2(tm−1),ζ2(tm−1),tm−1)Ξ2(n,m)), $ | (2.25) |
$ ζ1(tn+1)=ζ1(0)+khβΓ(β+2)n∑m=0tk−1mφ2(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)−tk−1m−1φ2(η1(τm−1),ζ1(tm−1),η2(tm−1),ζ2(tm−1),tm−1)Ξ2(n,m)), $ | (2.26) |
$ η2(tn+1)=η2(0)+khβΓ(β+2)n∑m=0tk−1mφ3(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)−tk−1m−1φ4(η1(τm−1),ζ1(tm−1),η2(tm−1),ζ2(tm−1),tm−1)Ξ2(n,m)), $ | (2.27) |
$ ζ2(tn+1)=ζ2(0)+khβΓ(β+2)n∑m=0tk−1mφ4(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)−tk−1m−1φ4(η1(τm−1),ζ1(tm−1),η2(tm−1),ζ2(tm−1),tm−1)Ξ2(n,m)), $ | (2.28) |
$ Ξ1(n,m)=((n+1−m)β(n−m+2+β)−(n−m)β×(n−m+2+2β)), $ | (2.29) |
$ Ξ2(n,m)=((n+1−m)β+1−(n−m)β(n−m+1+β)). $ | (2.30) |
Considering the FFE derivative, we have from [17]
$ FFE0Dβtη1(t)=ν(η2(t)−η1(t))−η1(t)ζq1(t), $ | (3.1) |
$ FFE0Dβtζ1(t)=−κζ1(t)+η1(t)ζq1(t), $ | (3.2) |
$ FFE0Dβtη2(t)=ν(η1(t)−η2(t))−η2(t)ζq2(t), $ | (3.3) |
$ FFE0Dβtζ2(t)=η2(t)ζq2(t). $ | (3.4) |
For the successive approximations of the system (3.1)-(3.4), we follow the same procedures as in [17], we obtain
$ η1(t)−η1(0)=ktk−1(1−β)M(β)φ1(η1,ζ1,η2,ζ2,t)+βM(β)∫t0kτk−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.5) |
$ ζ1(t)−ζ1(0)=ktk−1(1−β)M(β)φ2(η1,ζ1,η2,ζ2,t)+βM(β)∫t0kτk−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.6) |
$ η2(t)−η2(0)=ktk−1(1−β)M(β)φ3(η1,ζ1,η2,ζ2,t)+βM(β)∫t0kτk−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.7) |
$ ζ2(t)−ζ2(0)=ktk−1(1−β)M(β)φ4(η1,ζ1,η2,ζ2,t)+βM(β)∫t0kτk−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (3.8) |
Using $ t = t_{n+1} $ the following is established
$ η1(tn+1)−η1(0)=ktk−1(1−β)M(β)φ1(η1,ζ1,η2,ζ2,tn)+βM(β)∫tn+10kτk−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.9) |
$ ζ1(tn+1)−ζ1(0)=ktk−1(1−β)M(β)φ2(η1,ζ1,η2,ζ2,tn)+βM(β)∫tn+10kτk−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.10) |
$ η2(tn+1)−η2(0)=ktk−1(1−β)M(β)φ3(η1,ζ1,η2,ζ2,tn)+βM(β)∫tn+10kτk−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.11) |
$ ζ2(tn+1)−ζ2(0)=ktk−1(1−β)M(β)φ4(η1,ζ1,η2,ζ2,tn)+βM(β)∫tn+10kτk−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (3.12) |
Further, we have the following:
$ η1(tn+1)−η1(tn)=ktk−1n(1−β)M(β)φ1(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ1(η1,ζ1,η2,ζ2,tn−1)+βM(β)∫tn+1tnkτk−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.13) |
$ ζ1(tn+1)−ζ1(tn)=ktk−1n(1−β)M(β)φ2(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ2(η1,ζ1,η2,ζ2,tn−1)+βM(β)∫tn+1tnkτk−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.14) |
$ η2(tn+1)−η2(tn)=ktk−1n(1−β)M(β)φ3(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ3(η1,ζ1,η2,ζ2,tn−1)+βM(β)∫tn+1tnkτk−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (3.15) |
$ ζ2(tn+1)−ζ2(tn)=ktk−1n(1−β)M(β)φ4(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ4(η1,ζ1,η2,ζ2,tn−1)+βM(β)∫tn+1tnkτk−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (3.16) |
It follows from the Lagrange polynomial interpolation and integrating the following expressions:
$ η1(tn+1)−η1(tn)=ktk−1n(1−β)M(β)φ1(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ1(η1,ζ1,η2,ζ2,tn−1)+khβ2M(β)×(3tk−1nφ1(η1,ζ1,η2,ζ2,tn)−tk−1n−1φ1(η1,ζ1,η2,ζ2,tn−1), $ | (3.17) |
$ ζ1(tn+1)−ζ1(tn)=ktk−1n(1−β)M(β)φ2(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ2(η1,ζ1,η2,ζ2,tn−1)+khβ2M(β)×(3tk−1nφ2(η1,ζ1,η2,ζ2,tn)−tk−1n−1φ2(η1,ζ1,η2,ζ2,tn−1), $ | (3.18) |
$ η2(tn+1)−η2(tn)=ktk−1n(1−β)M(β)φ3(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ3(η1,ζ1,η2,ζ2,tn−1)+khβ2M(β)×(3tk−1nφ3(η1,ζ1,η2,ζ2,tn)−tk−1n−1φ3(η1,ζ1,η2,ζ2,tn−1), $ | (3.19) |
$ ζ2(tn+1)−ζ2(tn)=ktk−1n(1−β)M(β)φ4(η1,ζ1,η2,ζ2,tn)−ktk−1n−1(1−β)M(β)φ4(η1,ζ1,η2,ζ2,tn−1)+khβ2M(β)×(3tk−1nφ4(η1,ζ1,η2,ζ2,tn)−tk−1n−1φ4(η1,ζ1,η2,ζ2,tn−1). $ | (3.20) |
Finally, it is appropriate to write the successive approximations of the system (3.1)-(3.4) as follows:
$ η1(tn+1)−η1(tn)=ktk−1n((1−β)M(β)+3hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn)−ktk−1n−1((1−β)M(β)+hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn−1), $ | (3.21) |
$ ζ1(tn+1)−ζ1(tn)=ktk−1n((1−β)M(β)+3hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn)−ktk−1n−1((1−β)M(β)+hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn−1), $ | (3.22) |
$ η2(tn+1)−η2(tn)=ktk−1n((1−β)M(β)+3hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn)−ktk−1n−1((1−β)M(β)+hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn−1), $ | (3.23) |
$ ζ2(tn+1)−ζ2(tn)=ktk−1n((1−β)M(β)+3hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn)−ktk−1n−1((1−β)M(β)+hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn−1). $ | (3.24) |
Considering the FFM derivative, we have [18]
$ FFM0Dβtη1(t)=ν(η2(t)−η1(t))−η1(t)ζq1(t), $ | (4.1) |
$ FFM0Dβtζ1(t)=−κζ1(t)+η1(t)ζq1(t), $ | (4.2) |
$ FFM0Dβtη2(t)=ν(η1(t)−η2(t))−η2(t)ζq2(t), $ | (4.3) |
$ FFM0Dβtζ2(t)=η2(t)ζq2(t). $ | (4.4) |
Also, for this system (4.1)-(4.4), we follow the same treatment that was done in [17] to obtain the successive approximate solutions as follows:
$ η1(t)−η1(0)=ktk−1(1−β)A(β)φ1(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)∫t0kτk−1(t−τ)β−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.5) |
$ ζ1(t)−ζ1(0)=ktk−1(1−β)A(β)φ2(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)∫t0kτk−1(t−τ)β−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.6) |
$ η2(t)−η2(0)=ktk−1(1−β)A(β)φ3(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)∫t0kτk−1(t−τ)β−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.7) |
$ ζ2(t)−ζ2(0)=ktk−1(1−β)A(β)φ4(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)∫t0kτk−1(t−τ)β−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (4.8) |
At $ t_{n+1} $ we obtain the following
$ η1(tn+1)−η1(0)=ktk−1n(1−β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)∫tn+10kτk−1(tn+1−τ)β−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.9) |
$ ζ1(tn+1)−ζ1(0)=ktk−1n(1−β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)∫tn+10kτk−1(tn+1−τ)β−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.10) |
$ η2(tn+1)−η2(0)=ktk−1n(1−β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)∫tn+10kτk−1(tn+1−τ)β−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.11) |
$ ζ2(tn+1)−ζ2(0)=ktk−1n(1−β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)∫tn+10kτk−1(tn+1−τ)β−1φ4(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.12) |
The integrals involving in (4.9)-(4.12) can be approximated as:
$ η1(tn+1)−η1(0)=ktk−1n(1−β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)n∑m=0∫tm+1tmkτk−1(tn+1−τ)β−1φ1(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.13) |
$ ζ1(tn+1)−ζ1(0)=ktk−1n(1−β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)n∑m=0∫tm+1tmkτk−1(tn+1−τ)β−1φ2(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.14) |
$ η2(tn+1)−η2(0)=ktk−1n(1−β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)n∑m=0∫tm+1tmkτk−1(tn+1−τ)β−1φ3(η1,ζ1,η2,ζ2,τ)dτ, $ | (4.15) |
$ ζ2(tn+1)−ζ2(0)=ktk−1n(1−β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)n∑m=0∫tm+1tmkτk−1(tn+1−τ)β−1φ4(η1,ζ1,η2,ζ2,τ)dτ. $ | (4.16) |
The following numerical schemes after approximating the expressions $ \tau^{k-1} \varphi_{i}(\eta_{1}, \zeta_{1}, \eta_{2}, \zeta_{2}, \tau), \, \, i = 1, 2, 3, 4 $ in the interval $ [t_{m}, t_{m+1}] $ in (4.13)-(4.16) are given by
$ η1(tn+1)−η1(0)=ktk−1n(1−β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)n∑m=0[tk−1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)−tk−1m−1φ1(η1(tm−1),ζ1(tm−1),η2(tm−1),ζ2(tm−1),(tm−1))Ξ2(n,m)], $ | (4.17) |
$ \begin{eqnarray} \zeta_{1}(t_{n+1})- \zeta_{1}(0) & = & \frac{k t_{n}^{k-1}(1 - \beta)}{A(\beta)} \varphi_{2}(\eta_{1}(t_{n}), \zeta_{1}(t_{n}), \eta_{2}(t_{n}), \zeta_{2}(t_{n}), t_{n}) \\ &+& \frac{k h^{\beta}}{A(\beta)\Gamma(\beta+2)} \sum\limits_{m = 0}^{n}\Big[ t_{m}^{k-1} \varphi_{2}(\eta_{1}(t_{m}), \zeta_{1}(t_{m}), \eta_{2}(t_{m}), \zeta_{2}(t_{m}), (t_{m})) \Xi_{1}(n, m) \\ &-& t_{m-1}^{k-1} \varphi_{2}(\eta_{1}(t_{m-1}), \zeta_{1}(t_{m-1}), \eta_{2}(t_{m-1}), \zeta_{2}(t_{m-1}), (t_{m-1})) \Xi_{2}(n, m)\Big], \end{eqnarray} $ | (4.18) |
$ \begin{eqnarray} \eta_{2}(t_{n+1})- \eta_{2}(0) & = & \frac{k t_{n}^{k-1}(1 - \beta)}{A(\beta)} \varphi_{3}(\eta_{1}(t_{n}), \zeta_{1}(t_{n}), \eta_{2}(t_{n}), \zeta_{2}(t_{n}), t_{n}) \\ &+& \frac{k h^{\beta}}{A(\beta)\Gamma(\beta+2)} \sum\limits_{m = 0}^{n} \Big[ t_{m}^{k-1} \varphi_{3}(\eta_{1}(t_{m}), \zeta_{1}(t_{m}), \eta_{2}(t_{m}), \zeta_{2}(t_{m}), (t_{m})) \Xi_{1}(n, m) \\ &-& t_{m-1}^{k-1} \varphi_{3}(\eta_{1}(t_{m-1}), \zeta_{1}(t_{m-1}), \eta_{2}(t_{m-1}), \zeta_{2}(t_{m-1}), (t_{m-1})) \Xi_{2}(n, m)\Big], \end{eqnarray} $ | (4.19) |
$ \begin{eqnarray} \zeta_{2}(t_{n+1})- \zeta_{2}(0) & = & \frac{k t_{n}^{k-1}(1 - \beta)}{A(\beta)} \varphi_{4}(\eta_{1}(t_{n}), \zeta_{1}(t_{n}), \eta_{2}(t_{n}), \zeta_{2}(t_{n}), t_{n}) \\ &+& \frac{k h^{\beta}}{A(\beta)\Gamma(\alpha+2)} \sum\limits_{m = 0}^{n} \Big[ t_{m}^{k-1} \varphi_{4}(\eta_{1}(t_{m}), \zeta_{1}(t_{m}), \eta_{2}(t_{m}), \zeta_{2}(t_{m}), (t_{m})) \Xi_{1}(n, m) \\ &-& t_{m-1}^{k-1} \varphi_{4}(\eta_{1}(t_{m-1}), \zeta_{1}(t_{m-1}), \eta_{2}(t_{m-1}), \zeta_{2}(t_{m-1}), (t_{m-1})) \Xi_{2}(n, m)\Big]. \end{eqnarray} $ | (4.20) |
In this section, we study in detail the effect of the non-linear term in general, as well as the effect of the fractal-fractional order on the numerical solutions that we obtained by using successive approximations in the above sections. First we begin by satisfying the effective of the numerical solutions of the proposed system when $ \beta = 1 $ and $ k = 1. $
We compare only for the power kernel with a known numerical method which is the finite differences method. This is because all numerical solutions based on the three fractal-fractional operators that presented in this paper are very close each other when $ \beta = 1 $ and $ k = 1. $ Figure 1 illustrates the comparison between numerical solutions (2.25)-(2.28) and numerical solutions computed by using the finite differences method with $ k $ and $ \beta $. The parameters that used are $ \gamma = 0.4, \kappa = 0.004, h = 0.02. $ From this figure we note that an excellent agreement. And the accurate is increasing as we take small $ h $. From, Figure 1(a) and 1(c), we can see, that the profiles for $ \eta_{1} $ and $ \eta_{2} $ are very similar, but the profiles of $ \zeta_{1} $ and $ \zeta_{2} $ are more distinct with $ \zeta_{2} > \zeta_{2}. $ For Figure 1(b), the profiles of $ \zeta_{1} $ and $ \zeta_{2} $ are very close than in Figure 1(a) and 1(c), also for $ \zeta_{1} $ and $ \zeta_{2} $. Figures 2 and 3 show that the behavior of the approximate solutions based on FFP, FFE and FFM, when the degree of the non-linear term is cubic and for different values of $ k $ and $ \beta $. For the parameters $ \gamma $ and $ \kappa $, we fixed them in all computations. The remain parameters are the same as in Figure 1. Similarly, in Figures 4 and 5, the approximate solutions are plotted in the case of a non-linear with quadratic degree and for different values of $ k $ and $ \beta $. Finally in Figures 6 and 7, the approximate solutions are shown in the case of non-linear with fractional order and for different values for $ k $ and $ \beta $. For the Figures 2 and 3 which the nonlinear is cubic, all the profiles are distinct. Similarly with Figures 6 and 7 when the nonlinear is quadratic. From Figures 4 and 5, we can see in the case of fraction non-linear, the profiles of $ \eta_{1} $ and $ \eta_{2} $ are very close to each other than the profiles of $ \zeta_{1} $ and $ \zeta_{2}. $
In this paper, numerical solutions of the of the fractal-fractional reaction diffusion equations with general nonlinear have been studied. We introduced the FFRDE in three instances of fractional derivatives based on power, exponential, and Mittag-Leffler kernels. After that, we used the fundamental fractional calculus with the help of Lagrange polynomial functions. We obtained the iterative and approximate formulas in the three cases. We studied the effect of the non-linear term order, in the case of cubic, quadratic, and fractional for different values of the fractal-fractional derivative order. The accuracy of the numerical solutions in the classic case of the FFRDE was tested in the case of power kernel, where all the numerical solutions in the classic case of integer order coincide to each other, and the comparison result has excellent agreement. In all calculations was used the Mathematica Program Package.
The authors would like to express their Gratitudes to the ministry of education and the deanship of scientific research-Najran University-Kingdom of Saudi Arabia for their financial and Technical support under code number (NU/ESCI/17/025).
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] | Mazerolles P, Morancho R, Reynes A (1986) Silicon, Germanium, Tin, Lead Compd 9: 243-271. |
[2] | Seraphin BO (1976) Optical Properties of Solids. New Developments; North-Holland Publishing Co.: Amsterdam, NL. |
[3] | Drüsedau T, Andreas A, Schröder B, et al. (1994) Vibrational, optical and electronic properties of the hydrogenated amorphous germanium-carbon alloy system. Philos Mag 69: 1-20. |
[4] |
Saito N, Nakaaki I, Iwata H, et al. (2007) Optical and electrical properties of undoped and oxygen-doped a-GeC:H films prepared by magnetron sputtering. Thin Solid Film 515: 3766-3771. doi: 10.1016/j.tsf.2006.09.031
![]() |
[5] | Kumar S, Kashyap SC, Chopra KL (1998) Structure and transport properties of amorphous Ge1-xCx:H thin films obtained by activated reactive evaporation. J Non-Cryst Solids 101: 287. |
[6] |
Shinar J, Wu HS, Shinar R, et al. (1987). An IR, optical, and electron-spin-resonance study of as-deposit ed and annealed a-Ge1-xCx-H prepared by RF-sputtering in Ar/H2/C3H8. J Appl Phys 62:808-812. doi: 10.1063/1.339710
![]() |
[7] |
Saito N, Nakaaki I, Yamaguchi T, et al. (1995) Influence of deposition conditions on the properties of a-GeC:H and a-Ge:H films prepared by rf magnetron sputtering. Thin Solid Films 269: 69-74. doi: 10.1016/0040-6090(95)06671-3
![]() |
[8] |
Jacobsohn LG, Freire FL, Mariotto G (1998) Investigation on the chemical, structural and mechanical properties of carbon-germanium films deposited by dc-magnetron sputtering. Diam Relat Mater 7: 440-443. doi: 10.1016/S0925-9635(97)00171-4
![]() |
[9] | Vilcarromero J, Marques FC (1999) Hydrogen in amorphous germanium-carbon. Thin Solid Films343-344: 445-448. |
[10] |
Mariotto G, Vinegoni C, Jacobsohn LG, et al. (1999) Raman spectroscopy and scanning electron microscopy investigation of annealed amorphous carbon-germanium films deposited by d.c. magnetron sputtering. Diam Relat Mater 8: 668-672. doi: 10.1016/S0925-9635(98)00328-8
![]() |
[11] | Kumeda M, Masuda A, Shimizu T (1998) Structural studies on hydrogenated amorphous germanium-carbon films prepared by RF sputtering. Jpn J Appl Phys 36: 1754-1759. |
[12] |
Hu CQ, Zheng WT, Zheng B, et al. (2004) Chemical bonding of a-Ge1-xCx:H films grown by RF reactive sputtering. Vacuum 77: 63-68. doi: 10.1016/j.vacuum.2004.08.004
![]() |
[13] | Yuan H, Williams R (1993) Synthesis by laser ablation and characterization of pure germanium-carbon alloy thin-films. Chem Mater5: 479-485. |
[14] | Booth DC, Voss KJ (1981) The optical and structural properties of CVD germanium carbide. J Phys Colloques 42: C4-1033-C4-1036. |
[15] |
Gazicki M. (1999) Plasma deposition of thin carbon/germanium alloy films from organogermanium compounds. Caos Soliton Fract, 10: 1983-2017. doi: 10.1016/S0960-0779(98)00246-X
![]() |
[16] |
Kazimierski P, Tyczkowski J, Kozanecki M, et al. (2002) Transition from amorphous semiconductor to amorphous insulator in hydrogenated carbon-germanium films investigated by Raman spectroscopy. Chem Mater 14: 4694-4701. doi: 10.1021/cm020428s
![]() |
[17] |
Gazicki M, Ledzion R, Mazurczyk R, et al. (1998) Deposition and properties of germanium/carbon films deposited from tetramethylgermanium in a parallel plate RF discharge. Thin Solid Films 322:123-131. doi: 10.1016/S0040-6090(97)00908-5
![]() |
[18] | Kazimierski P, Tyczkowski J (2003) Deposition technology of a new nanostructured material for reversible charge storage. Surf Coat Tech 174-175: 770-773. |
[19] | Inagaki N, Mitsuuchi M (1984) Photoconductive films prepared by glow discharge polymerization. J Polym Sci 22: 301-305. |
[20] |
Szmidt J, Gazicki-Lipman M, Szymanowski H, et al. (2003) Electrophysical properties of thin germanium/carbon layers produced on silicon using organometallic radio frequency plasma enhanced chemical vapor deposition process. Thin Solid Films 441: 192-199. doi: 10.1016/S0040-6090(03)00884-8
![]() |
[21] | Sadhir RK, James WJ, Auerbach RA, et al. (1984) Synthesis of organogermanium by glow discharge polymerization. J Appl Polym Sci 38: 99-104. |
[22] |
Zhu JQ, Jiang CZ, Han JC, et al. (2012) Optical and electrical properties of nonstoichiometric a-Ge1-xCx films prepared by magnetron co-sputtering. Appl Surf Sci 258:3877-3881. doi: 10.1016/j.apsusc.2011.12.051
![]() |
[23] |
Mahmood A, Shah A, Castillon FF, et al. (2011) Surface analysis of GeC prepared by reactive pulsed laser deposition technique. Curr Appl Phys 11:547-550. doi: 10.1016/j.cap.2010.09.011
![]() |
[24] |
Schrader JS, Huguenin-Love JL, Soukup RJ, et al. (2006) Thin films of GeC deposited using a unique hollow cathode sputtering technique. Sol Energy Mater Sol Cells 90: 2338-2345. doi: 10.1016/j.solmat.2006.03.007
![]() |
[25] |
Yashiki Y, Miyajima S, Yamada A, et al. (2006) Deposition and characterization of mu c-Ge1-xCx thin films grown by hot-wire chemical vapor deposition using organo-germane. Thin Solid Films 501: 202-205. doi: 10.1016/j.tsf.2005.07.174
![]() |
[26] |
Hu CQ, Zheng WT, Tian HW, (2006) Effects of the chemical bonding on the optical and mechanical properties for germanium carbide films used as antireflection and protection coating of ZnS windows. J Phys: Condens Matter 18: 4231-4241. doi: 10.1088/0953-8984/18/17/011
![]() |
[27] |
Li YP, Li J, Wang N, et al. (2014) Optical and structural properties of co-sputtered Ge1 - xCx thin films as a function of the substrate temperature. Thin Solid Films 551: 74-78. doi: 10.1016/j.tsf.2013.11.110
![]() |
[28] |
Wu X, Zhang W, Y Lanqin, et al. (2008) The deposition and optical properties of Ge1-xCx thin film and infrared multilayer antireflection coatings. Thin Solid Films 516: 3189-3195. doi: 10.1016/j.tsf.2007.09.001
![]() |
[29] |
Li YP, Liu Z, Zhao H, et al. (2009) Infrared transmission properties of germanium carbon thin films deposited by reactive RF magnetron sputtering. Vacuum 83: 965-969. doi: 10.1016/j.vacuum.2008.11.005
![]() |
[30] |
Saito N, Iwata H, Nakaaki I, et al. (2009) Amorphous and microcrystalline GeC:H films prepared by magnetron sputtering. Physica Status Solidi (a) 206: 238-242. doi: 10.1002/pssa.200824228
![]() |
[31] | Mahmood A, Shah A, Castillon FF, et al. (2011)Surface analysis of GeC prepared by reactive pulsed laser deposition technique. Curr Appl Phys 11: 547-550 |
[32] |
Antoniotti P, Benzi P, Castiglioni M, et al. (1992) Studies on the solid obtained from radiolysis of germane methane mixtures. Chem Mater 4: 717-720. doi: 10.1021/cm00021a040
![]() |
[33] |
Benzi P, Castiglioni M, Volpe P (1994) α-GeC precursors obtained by radiolysis of GeH4-hydrocarbon mixtures. J Mater Chem 4: 1067-1070. doi: 10.1039/jm9940401067
![]() |
[34] |
Antoniotti P, Benzi P, Castiglioni M, et al. (1996) Radiolysis of binary systems containing germanium and carbon hydrides. Radiat Phys Chem 48: 457-462. doi: 10.1016/0969-806X(96)00011-4
![]() |
[35] |
Benzi P, Castiglioni M, Truffa E, et al. (1996) Thin film deposition of GexCyHz by radiolysis of GeH4-C3H8 mixtures. J Mater Chem 6: 1507-1509. doi: 10.1039/jm9960601507
![]() |
[36] | Antoniotti P, Benzi P, Castiglioni M, et al. (1999) An experimental and theoretical study of gaseous products in the radiolysis of germane/ethylene mixtures. Eur J Inorg Chem 323-332. |
[37] | Benzi P, Castiglioni M, Volpe P (2001) Characterisation and properties of amorphous nonstoichiometric Ge1-xCx:H compounds obtained from X-ray radiolysis of germane/ethylene mixtures. Eur J Inorg Chem 1235-1242. |
[38] |
Benzi P, Bottizzo E, Operti L, et al. (2002) Amorphous germanium carbides by radiolysis-CVD of germane/ethyne systems: Preparation and reaction mechanisms. Chem Mater 14: 2506-2513. doi: 10.1021/cm011261q
![]() |
[39] | Benzi P, Bottizzo E, Operti L, et al. (2004) Characterization and properties of amorphous nonstoichiometric Ge1-xCx:H compounds obtained by radiolysis-CVD of germane/ethyne systems. Chem Mater 216: 1068-1074. |
[40] |
Benzi P, Bottizzo E, Demaria C (2006) Characterization and properties of Ge1-xCx:H compounds obtained by X-ray CVD of germane/ethyne systems: Effect of the irradiation dose. Chem Vapor Depos 12: 25-32. doi: 10.1002/cvde.200506411
![]() |
[41] |
Benzi P, Bottizzo E, Demaria C, et al. (2007) Amorphous nonstoichiometric Ge1-xCx:H compounds obtained by radiolysis-chemical vapor deposition of germane/ethyne or germane/allene systems: A bonding and microstructure investigation performed by x-ray photoelectron spectroscopy and Raman spectroscopy. J Appl Phys 101: 124906. doi: 10.1063/1.2748710
![]() |
[42] |
Demaria C, Benzi P, Arrais A, et al. (2013) Growth and thermal annealing of amorphous germanium carbide obtained by X-ray chemical vapor deposition. J Mat Sci 48: 6357-6366. doi: 10.1007/s10853-013-7435-1
![]() |
[43] |
Arrais A, Benzi P, Bottizzo E, et al. (2007) Characterization of hydrogenated amorphous germanium compounds obtained by x-ray chemical vapor deposition of germane: Effect of the irradiation dose on optical parameters and structural order. J Appl Phys 102:104905. doi: 10.1063/1.2817464
![]() |
[44] |
Arrais A, Benzi P, Bottizzo E, et al. (2009) Correlations among hydrogen bonding configuration, structural order and optical coefficients in hydrogenated amorphous germanium obtained by x-ray-activated chemical vapour deposition. J Phys D Appl Phys 42:105406. doi: 10.1088/0022-3727/42/10/105406
![]() |
[45] |
Chew K, Rusli, Yoon SF, et al. (2002) Hydrogenated amorphous silicon carbide deposition using electron cyclotron40] resonance chemical vapor deposition under high microwave power and strong hydrogen dilution. J Appl Phys 92: 2937-2941. doi: 10.1063/1.1500418
![]() |
[46] | Gazicki M, Schalko J, Olcaytug F, et al. (1994) Study on electromagnetron for plasma polymerization. 2. Magnetic-field enhanced radio-frequency plasma deposition of organogermanium films from tetraethylgermanium. J Vac Sci Technol A12 345-353 |
[47] |
Taga K, Hamada S, Fukui H, et al. (2002) Vibrational spectra and density functional study of propylgermane. J Mol Struct 610: 85-97. doi: 10.1016/S0022-2860(02)00022-4
![]() |
[48] | Rübel H, Schröder B, Fuhs, W, et al. (1987) IR spectroscopy and structure of RF magnetron sputtered a-SiC-H films. Phys Stat Sol 139: 131-143 |
[49] |
Cardona M (1983) Vibrational Spectra of Hydrogen in Silicon and Germanium. Phys Stat Sol 118:463-481. doi: 10.1002/pssb.2221180202
![]() |
[50] | Bellamy LJ (1975) The Infrared Spectra of Complex Molecules, 3rd ed. Chapman and Hall, London 13-36. |
[51] | Schrader B (1995) Infrared and Raman Spectroscopy, VCH, Weinheim, 192-195. |
[52] |
Gharbi R, Fathallah M, Alzaied N, et al. (2008) Hydrogen and nitrogen effects on optical and structural properties of amorphous carbon. Mat Sci Eng C 28: 795-798. doi: 10.1016/j.msec.2007.10.022
![]() |
[53] |
Robertson J (1986) Amorphous-carbon. Adv Phys 35: 317-374. doi: 10.1080/00018738600101911
![]() |
[54] |
Akaoglu B, Sel K, Atilgan I, et al. (2008) Carbon content influence on the optical constants of hydrogenated amorphous silicon carbon alloys. Opt Mat 30: 1257-1267. doi: 10.1016/j.optmat.2007.06.005
![]() |
[55] | Lucovsky G (1985) Local bonding of hydrogen in a-Si H, a-Ge H and a-Si, Ge H alloy-films. J Non-Cryst Solids 76, 173-186. |
[56] | Ley L (1984) The Physic of Hydrogenated Amorphous Silicon II, Springer-Verlag, New York, Chap.3, 61-161. |
[57] | Tauc J (1974) Amorphous and Liquid Semiconductors, Plenum, London and New York Chap. 4,159-220. |
[58] | Mott NF, Davis EA (1979) Electronic Process in Non-Crystalline Materials, 2nd ed. Clarendon Press. Oxford, 287-318. |
[59] |
Masuda A, Kumeda M, Shimizu T (1991) Relationship between photodarkening and light-induced ESR in amorphous Ge-S films alloyed with lead. Jpn J Appl Phys 30: L1075. doi: 10.1143/JJAP.30.L1075
![]() |
[60] | Lide DR (1997-1998) CRC Handbook of Chemistry and Physics, 78th ed. CRC Press, Boca Raton and New York. |
[61] |
Robertson J (1992) The electronic and atomic structure of hydrogenated amorphous Si-C alloys. Phil Mag B 66: 615-638. doi: 10.1080/13642819208207664
![]() |
[62] |
Mihailova T, Toneva A (1995) Effect of the gas pressure during deposition on the optical properties of HOMOCVD a-Si:H thin films. Sol Energ Mat Sol C 36: 1-9. doi: 10.1016/0927-0248(94)00164-N
![]() |
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9. | Nauman Ahmed, Ali Raza, Ali Akgül, Zafar Iqbal, Muhammad Rafiq, Muhammad Ozair Ahmad, Fahd Jarad, New applications related to hepatitis C model, 2022, 7, 2473-6988, 11362, 10.3934/math.2022634 | |
10. | Esra Karatas Akgül, Wasim Jamshed, Kottakkaran Sooppy Nisar, S.K. Elagan, Nawal A. Alshehri, On solutions of gross domestic product model with different kernels, 2022, 61, 11100168, 1289, 10.1016/j.aej.2021.06.067 | |
11. | Rubayyi T. Alqahtani, Shabir Ahmad, Ali Akgül, On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels, 2022, 10, 2227-7390, 1102, 10.3390/math10071102 | |
12. | A. DLAMINI, EMILE F. DOUNGMO GOUFO, M. KHUMALO, CHAOTIC BEHAVIOR OF MODIFIED STRETCH–TWIST–FOLD FLOW UNDER FRACTAL-FRACTIONAL DERIVATIVES, 2022, 30, 0218-348X, 10.1142/S0218348X22402071 | |
13. | Anwar Zeb, Abdon Atangana, Zareen A. Khan, Salih Djillali, A robust study of a piecewise fractional order COVID-19 mathematical model, 2022, 61, 11100168, 5649, 10.1016/j.aej.2021.11.039 | |
14. | Shabir Ahmad, Aman Ullah, Ali Akgül, Manuel De la Sen, A study of fractional order Ambartsumian equation involving exponential decay kernel, 2021, 6, 2473-6988, 9981, 10.3934/math.2021580 | |
15. | Raheel Kamal, Gul Rahmat, Kamal Shah, Ricardo Escobar, On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations, 2021, 2021, 1563-5147, 1, 10.1155/2021/4640467 | |
16. | Kaihong Zhao, Shuang Ma, Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses, 2022, 7, 2473-6988, 3169, 10.3934/math.2022175 | |
17. | Kamsing Nonlaopon, Muhammad Naeem, Ahmed M. Zidan, Rasool Shah, Ahmed Alsanad, Abdu Gumaei, Muhammad Imran Asjad, Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators, 2021, 2021, 1099-0526, 1, 10.1155/2021/7979365 | |
18. | Saima Rashid, Rehana Ashraf, Ebenezer Bonyah, Azhar Hussain, On Analytical Solution of Time-Fractional Biological Population Model by means of Generalized Integral Transform with Their Uniqueness and Convergence Analysis, 2022, 2022, 2314-8888, 1, 10.1155/2022/7021288 | |
19. | Khadija Tul Kubra, Rooh Ali, Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan, 2023, 2363-6203, 10.1007/s40808-023-01747-w | |
20. | ZAREEN A. KHAN, KAMAL SHAH, BAHAAELDIN ABDALLA, THABET ABDELJAWAD, A NUMERICAL STUDY OF COMPLEX DYNAMICS OF A CHEMOSTAT MODEL UNDER FRACTAL-FRACTIONAL DERIVATIVE, 2023, 31, 0218-348X, 10.1142/S0218348X23401813 | |
21. | Kamal Shah, Thabet Abdeljawad, On complex fractal-fractional order mathematical modeling of CO 2 emanations from energy sector, 2024, 99, 0031-8949, 015226, 10.1088/1402-4896/ad1286 | |
22. | Samy A. Abdelhafeez, Anas A. M. Arafa, Yousef H. Zahran, Ibrahim S. I. Osman, Moutaz Ramadan, Adapting Laplace residual power series approach to the Caudrey Dodd Gibbon equation, 2024, 14, 2045-2322, 10.1038/s41598-024-57780-x | |
23. | Krunal B. Kachhia, Prit P. Parmar, A novel fractional mask for image denoising based on fractal–fractional integral, 2024, 11, 26668181, 100833, 10.1016/j.padiff.2024.100833 | |
24. | Harpreet Kaur, Amanpreet Kaur, Palwinder Singh, Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel, 2024, 13, 2192-8029, 10.1515/nleng-2022-0380 | |
25. | Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525 | |
26. | Ashish Rayal, System of fractal-fractional differential equations and Bernstein wavelets: a comprehensive study of environmental, epidemiological, and financial applications, 2025, 100, 0031-8949, 025236, 10.1088/1402-4896/ada592 | |
27. | Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan, Fahad Aljuaydi, An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator, 2025, 9, 2504-3110, 199, 10.3390/fractalfract9040199 |