
Citation: Ranbir Kumar, Sunil Kumar, Jagdev Singh, Zeyad Al-Zhour. A comparative study for fractional chemical kinetics and carbon dioxide CO2 absorbed into phenyl glycidyl ether problems[J]. AIMS Mathematics, 2020, 5(4): 3201-3222. doi: 10.3934/math.2020206
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Mathematical modelling is the best way to formulating problems from an application area and it is well known that several mathematical characterization of numerous growth in chemical and physical sciences is described by differential equations (DEs). In chemistry, chemical kinetics problem and CO2 with PGE problems are described by system of nonlinear (DEs) with different kind of Neumann boundary and Drichlet type conditions in different published work such as chemistry problem by Jawary and Raham [1], chemistry problem by Abbasbandy and Shirzadi [2], CO2 absorbed into PGE problem by Jawary et al. [3], Choe et al. [4], Singha et al. [5], CO2 absorbed into PGE problem by Robertson [6], chemistry problem by Matinfar et al. [7], chemistry problem by Ganji et al. [8], Dokoumetzidis et al. [9]. In the past few years, fractional calculus (FC) has found many diverse and robust applications in various research areas such as fluid dynamics, image processing, viscoelasticity and other physical phenomena. Many definitions of fractional derivatives are discovered by several mathematicians but two most famous definitions of fractional derivative are Riemann − Liouville and Caputo. Some interesting and fundamental works on various direction of the FC is given in several famous books such as by Mainardi [10], fractional differential equations by Podlubny [11], Diethelm [12], Kilbas et al. [13] and Das [14].
In the past few years, wavelets have become an increasingly newly developed famous mechanism in the several research areas of physical, chemical, computational sciences, Image manipulation, signal analysis, data compression, numerical analysis and several others research areas such as a primer on wavelets and their scinentific applications by Walker [15], wavelet: mathematics and applications by Benedetto [16], a mathematical tool for signal analysis by Chui [17], wavelet methods for dynamical problems by Gopalakrishnan and Mitra [18], Wang [19] and a wavelet operational matrix method by Wu [20]. Due to this reason, wavelets have been applied for the solution of differential equations (DEs) since the 1980s. The interesting features in this method are possibility to find-out singularities, irregular structure and transient phenomena exhibited by the analysed equations such as by Heydari et al. [21], Wang and Fan [19], Balaji [22], Rehman and Khan [23], Hosseininia [24], Pirmohabbati et al. [25], Hosseininia [26], Heydari [27] and Kumar et al. [28].
Among the several wavelet families most simple are the Haar wavelets and it has been successfully applied to several linear and nonlinear problems of physical science and other research areas such as fractional order stationary neutron transport equation, neutron point kinetics equation, fractional order nonlinear oscillatory van der pol system and fractional bagley torvik equation by Ray and Patra [29,30,31,32], a comparative study on haar wavelet and hybrid functions, nonlinear integral and integro −differential equation of first and higher order and parabolic differential equatons by Aziz et al. [33,34,35], burgers equation by Jiwari [36], fractional integral equations by Lepik [37], Poisson and biharmonic equations by Shi and Cao [38], delamination detection in composite beams by Hein and Feklistova [39], fractional order integral equations by Gao and Liao [40], lumped and distributed parameters systems by Chen and Hsiao [41], FDEs by Chen et al. [42], free vibration analysis by Xie et al. [43], fractional nonlinear differential equations by Saeed and Rehman [44], magnetohydrodynamic flow equations by Celik and Brahin [45,46], fishers equations by Hariharan et al. [47], FPDEs by Wang et al. [48], nonlinear oscillators equations Kaur et al. [49], poisson and biharmonic equations by Shi et al. [50] and free vibration analysis of functionally graded cylindrical shells by Jin et al. [51].
It is compulsory to note that the fractional chemical kinetics and condensations of CO2 and PGE problems is the first one to be solved by the Haar wavelet and generalization of Adams– Bashforth−Moulton method by us. It is also noted that there are no similar works with these methods for fractional chemical kinetics and condensations of CO2 and PGE problems available in any present published literature. It is well known by the several published research papers that the Caputo and Riemann-Liouville is most popular definition of fractional calculus.
The complete work is systematized in the following sections: Overview of basic FC are provided in section 2. Fractional Model of both Chemical Kinetics and CO2 absorbed into PHE problems are provided in section 3. In section 4, a haar wavelet and Adam Bashforth's-Moulton methods are discussed and presented for both chemistry problems. The proposed methods for solutions of both chemistry problem are provided in section 5. Numerical result and discussions are provided in section 6. Conclusion and future scope are given in sections 7.
There are numerous definition of derivative and integration are available in literature [52,53,54,55,56,57,58,59,60,61].
Definition 1. The (left sided) Riemann−Liouville fractional integral of order α>0 of a function Θ(t)∈Cα,α≥−1 is defined as,
IαtΘ(t)=1Γ(α)t∫0(t−ξ)α−1Θ(ξ)dξ,α>0,t>0; | (2.1) |
where Γ(.) is well known Gamma function.
Definition 2. The next two equations define Riemann – Liouville and Caputo fractional derivatives of order a, respectively,
RLDαtΘ(t)=dmdtm(Im−αtΘ(t))={dmΘ(t)dtm,α=m∈N,1Γ(m−α)dmdtmt∫0Θ(ξ)(t−ξ)α−m+1dξ,0≤m−1<α<m,
and,
CDαtΘ(t)=Im−αt(dmdtmΘ(t))={dmΘ(t)dtm,α=m∈N,1Γ(m−α)t∫0Θm(ξ)(t−ξ)α−m+1dξ,0≤m−1<α<m,
where t>0 and m is an integer. Two basic properties for m−1<α≤m and Θ∈L1[a,b] are given as
{(CDαtIαΘ)(t)=Θ(t),(IαCDαtΘ)(t)=Θ(t)−∑m−1k=0Θk(0+)(t−a)kk!. | (2.2) |
Let D,E and H are different location of a model of chemical process then the reactions are presented as
D⟶E, | (3.1) |
E+H⟶D+H, | (3.2) |
E+E⟶H, | (3.3) |
The concetrations of all three spaces of D,E and H are denoted by Θ1,Θ2 and Θ3 respectively. Let r1,r2 and r3 denotes the reaction rate of Eqs (3.1), (3.2) and (3.3) respectively. We consider an integer order model of chemical kinetics problem as [1,2,6,7,8]
{dΘ1(t)dt=−r1Θ1(t)+r2Θ2(t)Θ3(t),dΘ2(t)dt=r1Θ1(t)−r2Θ2(t)Θ3(t)−r3Θ22(t),dΘ3(t)dt=r3Θ22(t), | (3.4) |
with the initial conditions, Θ1(0)=1, Θ2(0)=0, Θ3(0)=0. The main target of this section is converted above inter order CK problem into fractional order CK problem. The fractional model of CK problem is presented as
{CDαtΘ1(t)=−r1Θ1(t)+r2Θ2(t)Θ3(t),0<α≤1,CDβtΘ2(t)=r1Θ1(t)−r2Θ2(t)Θ3(t)−r3Θ22(t),0<β≤1,CDγtΘ3(t)=r3Θ22(t),0<γ≤1, | (3.5) |
with the initial conditions, Θ1(0)=1, Θ2(0)=0, Θ3(0)=0 where, Dαt=dαdtα,Dβt=dβdtβ,Dγt=dγdtγ are fractional derivative with 0<α,β,γ≤1. If r1=1,r2=0, and r3=1 then
{CDαtΘ1(t)=−Θ1(t),0<α≤1,CDβtΘ2(t)=Θ1(t)−Θ22(t),0<β≤1,CDγtΘ3(t)=Θ22(t),0<γ≤1, | (3.6) |
with the initial conditions, Θ1(0)=1, Θ2(0)=0, Θ3(0)=0. The above system is representing a nonlnear reaction which was taken from litrature [2,7,8,62].
The CO2 causes in ocean acidification because it dissolves in water to form carbonic acid [63].The mathematical formulation of the concentration of CO2 and PGE is shown in Muthukaruppan et al. [64]. Now, the two nonlinear reactions equations in normalized form is presented as
{d2Υ1dt2=α1Υ1Υ21+β1Υ1+β2Υ2,d2Υ2dt2=α2Υ1Υ21+β1Υ1+β2Υ2, | (3.7) |
with boundary conditions, Υ1(0)=0, Υ1(1)=1m, Υ′2(0)=1m, Υ2(1)=1m. The whole chemistry of the above problem is given in several litratures [1,3,4]. The fractional model of the condensation of CO2 and PGE in operator form is given as,
{CDαtΥ1(t)=α1Υ1Υ21+β1Υ1+β2Υ2,1<α≤2,CDβtΥ2(t)=α2Υ1Υ21+β1Υ1+β2Υ2,1<β≤2, | (3.8) |
with the same boundary conditions Υ1(0)=0, Υ1(1)=1m and Υ′2(0)=1m,Υ2(1)=1m; where m≥3 and fractional operator is taken in Caputo sence.
The Haar functions have been discovered by Alfred Haar in 1910 and Haar wavelets are the simplest wavelet among all wavelet. The Haar sequence was also introduced by itself Alfred Haar in 1909 which is recognised as wavelet basis. The Haar wavelets are the mathematical operations which are known as Haar transform. These wavelets are build up by piecewise constant function on the real line. We used Haar wavelet operational matrix method because of its flexibility, simplicity and require very less effort of computation. Usually Haar wavelet is defined for [0, 1) but in general case we extend it up to certain interval. Haar functions are very useful in many applications as image coding, extraction of edge, binary logic design etc [20,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. The Haar scaling function is defined as
ϕ(x)={10≤x<1,0otherwise. | (4.1) |
The Haar wavelet mother function is defined as
ψ(x)={10≤x<12,−112≤x<1,0otherwise. | (4.2) |
The orthogonal set of Haar wavelet functions for t∈[0,1] are defined as
hi(t)=1√m{2j/2,k−12j≤t<k−0.52j,−2j/2,k−0.52j≤t<k2j,0,otherwise, | (4.3) |
where i=0,1,2,...,m−1, m=2r+1 and r is positive integer known as resolution of Harr wavelet. Also j and k represent integer decomposition of i=2j+k−1.
Any function Θ(t)∈L2([0,1)) can be expanded in terms of Haar wavelet by
Θ(t)=∞∑i=0cihi(t);Whereci=1∫0Θ(t)hi(t)dt. | (4.4) |
If we approximated as piecewise constant during each interval, Eq. 4.4 will terminated at finite terms as [65]:
Θ(t)≈m−1∑i=0cihi(t)=CTmHm(t), | (4.5) |
where Cm=[c0,c1,c2,...,cm−1]T and Hm(t)=[h0(t),h1(t),h2(t),...,hm−1(t)]T,
Using collocation points tl=(l−0.5)m, where l=0,1,...,m−1, we obtained the discrete form as
H=[h0(t0)h0(t1)⋯h0(tm−1)h1(t0)h1(t1)⋯h1(tm−1)⋮⋮⋱⋮hm−1(t0)hm−1(t1)⋯hm−1(tm−1)]. | (4.6) |
The HWOM of fractional order integration without using block pulse functions we integrate Hm(t) using Reimann-Liouville integration operator [41,66]. Then the HWOM of fractional order integration Qα is given by
QαHm(t)=IαHm(t)=[Iαh0(t),Iαh1(t),Iαh2(t),...,Iαhm−1(t)]T =[Qh0(t),Qh1(t),Qh2(t),...,Qhm−1(t)]T, | (4.7) |
where
Qh0(t)=1√mtαΓ(1+α),0≤t≤1,
Qhi(t)=1√m{0,0≤t<k−12j,2j/2ζ1(t)k−12j≤t<k−0.52j,2j/2ζ2(t)k−0.52j≤t<k2j,2j/2ζ3(t)k2j≤t<1,
where
ζ1(t)=1Γ(1+α)(t−k−12j)α,
ζ2(t)=1Γ(1+α)(t−k−12j)α−2Γ(1+α)(t−k−0.52j)α,
ζ3(t)=1Γ(1+α)(t−k−12j)α−2Γ(1+α)(t−k−0.52j)α+1Γ(1+α)(t−k2j)α.
If we take, α=1/2,m=8, then we have the operational matrix as given below:
Q1/2H8 = [0.09970.17270.22300.26390.29920.33080.35960.38630.09970.17270.22300.26390.0997−0.0147−0.0864−0.14150.24430.0333−0.1154−0.0666−0.0343−0.0223−0.0193−0.013200000.14100.24430.033−0.11540.1995−0.0534−0.0455−0.0188−0.0111−0.0075−0.0055−0.0043000.1995−0.0534−0.0455−0.0188−0.0111−0.007500000.1995−0.0534−0.0455−0.01880000000.1995−0.0534].
The above matrix is the operational matrix of Haar wavelets.
In this section we discuss about Predictor-Corrector scheme (PECE), which is the genralization of (ABM) mehod [67,68]. We obtain the numerical solution of nonlinear FDES as
DαΘ(t)=f(t,Θ(t)),0<t≤T,Θ(k)(0)=Θ(k)0, | (4.8) |
where derivative in Caputo's sense. which is equivalent to the Volterra integral equation
Θ(t)=α−1∑k=0Θk0tkk!+1Γ(α)∫t0(t−τ)α−1f(t,Θ(τ))dτ. | (4.9) |
Assume h=T/N, tn=nh, n=0,1,2,...,N ∈Z+ then the discrete form for the above equation will be
Θh(tn+1)=α−1∑k=0Θ(k)0tkn+1k!+hαΓ(α+2)f(tn+1,Θph(tn+1))+hαΓ(α+2)n∑j=0aj,n+1f(th,Θh(tj)), | (4.10) |
aj,n+1={nα+1−(n−α)(n+1)α,ifj=0,(n−j+2)α+1+(n−j)α+1−2(n−j+1)α+1,if0≤j≤n,1,ifj=1, | (4.11) |
Θph(tn+1)=α−1∑k=0Θ(k)0tkn+1k!+1Γ(α)n∑j=0bj,n+1f(tj,Θh(tj)), | (4.12) |
bj,n+1=hαα((n+1−j)α−(n−j)α). | (4.13) |
The corrector values for chemistry problem is
Θ1(n+1)=Θ1(0)+hαΓ(α+2)(−r1Θp1(n+1)+r2Θp2(n+1)Θp3(n+1))+hαΓ(α+2)n∑j=0αj,n+1(−r1Θ1(j)+r2Θ2(j)Θ3(j)),Θ2(n+1)=Θ2(0)+hβΓ(β+2)(r1Θp1(n+1)−r2Θp2(n+1)Θp3(n+1)−r3Θp2(n+1)2)+hβΓ(β+2)n∑j=0βj,n+1(r1Θ1(j)−r2Θ2(j)Θ3(j)−r3Θ22(j)),Θ3(n+1)=Θ3(0)+hγΓ(γ+2)(r3Θp2(n+1)2)+hγΓ(γ+2)n∑j=0γj,n+1r3Θ22(j). |
The corresponding predictor values are,
Θp1(n+1)=Θ1(0)+1Γ(α)n∑j=0Bj,n+1(−r1Θ1(j)+r2Θ2(j)Θ3(j)),Θp2(n+1)=Θ2(0)+1Γ(β)n∑j=0Cj,n+1(r1Θ1(j)−r2Θ2(j)Θ3(j)−r3Θ22(j)),Θp3(n+1)=Θ3(0)+1Γ(γ)n∑j=0Dj,n+1(r3Θ22(j)). |
From Eqs (4.12) and (4.14) we can calculate {αj,n+1}, {βj,n+1}, {γj,n+1}, and Bj,n+1, Cj,n+1, Dj,n+1.
Example: 1 We assume a fractional model of chemical kinetics problem is given as
{CDαtΘ1=−r1Θ1+c2Θ2Θ3,0<α≤1,CDβtΘ2=r1Θ1−r2Θ2Θ3−r3Θ22,0<β≤1,CDγtΘ3=r3Θ22,0<γ≤1, | (5.1) |
with the initial conditions, Θ1(0)=1, Θ2(0)=0 Θ3(0)=0, where r1, r2 and r3 are reaction rates. Let us assume higher derivatives in the terms of haar wavelet series.
{CDαtΘ1=CTHm(t),CDβtΘ2=GTHm(t),CDγtΘ3=KTHm(t), | (5.2) |
where C=[c0,c1,c2,...,cm−1]T, G=[g0,g1,g2,...,gm−1]T and K=[k0,k1,k2,...,km−1]T are unknown vectors. Applying Riemann-Liouville fractional integral in Eq. (5.2) and using initial conditions, we obtained
{Θ1=CTQαHm(t)+1,Θ2=GTQβHm(t),Θ3=KTQγHm(t). | (5.3) |
Now substituting the values of Θ1, Θ2 and Θ3 into the Eq. (5.1), we obtained.
{CTHm(t)=−r1(CTQαHm(t)+1)+r2(GTQβ)(KTQγ),GTHm(t)=r1(CTQαHm(t)+1)−r2(GTQβ)(KTQγ)−r3(KTQγ)(KTQγ),KTHm(t)=r3(GTQβ)2. | (5.4) |
Let r1=0.1, r2=0.02 and r3=0.009 as given in Aminikhah [69]. Now disperse the Eq. (5.4) at the collocation points tl=(l−0.5)m, where l=1,2,3,...,m. We obtained 3m nonlinear algebraic equations which can be solved by Newton iteration method, after solving we obtained the coefficients ci, gi and ki. Substitute these coefficients into the Eq. (5.3) we get desired solutions Θ1, Θ2 and Θ3.
Example 2: Consider the system of condensations of CO2 and PGE problem of arbitrary order.
{Υα1(t)=α1Υ1(t)Υ2(t)−Υα1(t)(β1Υ1(t)+β2Υ2(t)),1<α≤2,Υβ1(t)=α2Υ1(x)Υ2(x)−Υβ2(t)(β1Υ1(t)+β2Υ2(t)),1<β≤2, | (5.5) |
with boundary conditions Υ1(0)=0, Υ1(1)=1m, and Υ′2(0)=1m, Υ2(1)=1m where Υα1(t)= CDαtΥ1(t). Here for simplicity we have taken m=3 and we will take the value of α1=1, α2=2, β1=1 and β2=3 as given in Duan et al. [70], AL-jawary ad Radhi [71]. Further, we assume the higher derivative in terms of Haar wavelet series.
{Υα1(t)=CTHm(t),Υβ1(t)=KTHm(t), | (5.6) |
applying Riemann-Liouville integral operator on the above equation and using boundary conditions, we obtained
Υ1(t)−Υ′1(0)t=CTQαHm(t), | (5.7) |
substituting t=1 into Eq. (5.7) we obtained
Υ1(1)−Υ′1(0)=CTQαHm(1)
Υ′1(0)=13−CTQαHm(1), | (5.8) |
and
Υ2(0)=−KTQβHm(1). | (5.9) |
Therefore,
Υ1(t)=(13−CTQαHm(1))t+CTQαHm(t), | (5.10) |
similarly
Υ2(t)=t3−KTQβHm(1)+KTQβHm(t). | (5.11) |
Substituting the values of Υ1, Υ2 into the Eq. (5.5) and using Eq. (5.6) we obtained
CTHm(t)=(t3−CTQαHm(1)t+CTQαHm(t))(t3−KTQβHm(1)+KTQβHm(t))−CTHm(t)((t3−CTQαHm(1)t+CTQαHm(t))+3(t3−KTQβHm(1)+KTQβHm(t))).} | (5.12) |
KTHm(t)=2(t3−CTQαHm(1)t+CTQαHm(t))(t3−KTQβHm(1)+KTQβHm(t))−KTHm(t)((t3−CTQαHm(1)t+CTQαHm(t))+3(t3−KTQβHm(1)+KTQβHm(t))).} | (5.13) |
Now disperse the Eqs (5.12) and (5.13) at the collocation points tl=(l−0.5)m, where l=0,1,...,m−1. We obtained a system of nonlinear algebraic equations which can be easily solved by Newton-Iteration method using mathematical softwares, after solving we obtained the unknowns coefficients ci and ki. Substituting these coefficients into the Eqs (5.10) and (5.11) we get desired solutions Υ1 and Υ2.
All numerical simulation and graphical results of both examples are depicted through the Figures 1–14 where Figures 1–6 and Figures 7–14 are depicted for examples 1 and 2 respectively. We have depicted a comparison between numerical obtained solutions using by Haar wavelet and Adam's-Bashforth-Moulton predictor-corrector schemes through the Figures 1–3 and these figures are depicted for the values of m=64. It is clear from all figures that both obtained solutions by HWM and ABM are identical. The obtained solutions Θ1, Θ2 and Θ3 are plotted through the Figures 3–6 where the nature of solution Θ1 is of decreasing nature while other solutions Θ2 and Θ3 is of increasing nature. We plotted the resolutions Figures 7–14 for better understanding the nature of obtained solution of example 2. We plotted resolutions figures due to non-availability of its exact solution.
t | Θ1(HWM) | Θ1(ABM) | Θ2(HWM) | Θ2(ABM) |
0.1 | 0.9901 | 0.9893 | 0.0100 | 0.0107 |
0.2 | 0.9802 | 0.9794 | 0.0198 | 0.0206 |
0.3 | 0.9704 | 0.9697 | 0.0296 | 0.0303 |
0.4 | 0.9608 | 0.9600 | 0.0393 | 0.0400 |
0.5 | 0.9512 | 0.9505 | 0.0489 | 0.0495 |
0.6 | 0.9418 | 0.9410 | 0.0585 | 0.0590 |
0.7 | 0.9324 | 0.9317 | 0.0680 | 0.0683 |
0.8 | 0.9231 | 0.9224 | 0.0775 | 0.0776 |
0.9 | 0.9139 | 0.9132 | 0.0869 | 0.0868 |
1.0 | 0.9048 | 0.9041 | 0.0963 | 0.0962 |
t | Θ3(HWM) | Θ3(ABM) |
0.1 | 3.0×10−8 | 4.0×10−8 |
0.2 | 2.4×10−7 | 2.7×10−7 |
0.3 | 7.9×10−7 | 8.6×10−7 |
0.4 | 1.8×10−6 | 1.9×10−6 |
0.5 | 3.6×10−6 | 3.8×10−6 |
0.6 | 6.2×10−6 | 6.4×10−6 |
0.7 | 9.8×10−6 | 1.0×10−5 |
0.8 | 1.5×10−5 | 1.5×10−5 |
0.9 | 2.0×10−5 | 2.0×10−5 |
1.0 | 2.8×10−5 | 2.8×10−5 |
In this work, Haar wavelet operational matrix and Adam Bashforth's Moulton scheme are proposed to solve fractional chemical kinetics and another problem that relates the condensations of carbon dioxide CO2 numerically. A comparative study between fractional chemical kinetics and another problem that relates the conden sations of carbon dioxide CO2 has been done for m=64 in this work. Our tabulated and graphical results indicate that the solution will ameliorate if we will take more collocation points, i.e greater values of m. The essential advantage of HWM is that it converts problems into the system of linear or nonlinear algebraic equations so that the computation is facile and computer-oriented. Furthermore, wavelet method is much easier than other numerical methods for system of FDEs. Again, we have solved the chemistry problems at different resolutions, which produced the same results at each resolution. The precision of the solution will ameliorate if we increase the resolution. This new comparative study between the Haar wavelet operational matrix and Adam Bashforth's Moulton scheme for fractional chemical kinetics and another problem that relates the condensations of carbon dioxide CO2 indicates that both approaches can be applied successfully to the chemistry problems of chemistry science.
The first author Dr. Sunil Kumar would like to acknowledge the financial support received from the National Board for Higher Mathematics, Department of Atomic Energy, Government of India (Approval No. 2/48(20)/2016/ NBHM(R.P.)/R and D II/1014). The authors are also grateful to the editor and anonymous reviewers for their constructive comments and valuable suggestions to improve the quality of article.
The authors declare no conflict of interest in this manuscript.
[1] |
M. A. Al-Jawary, R. K. Raham, A semi-analytical iterative technique for solving chemistry problems, J. King Saud Univ. Sci., 29 (2017), 320-332. doi: 10.1016/j.jksus.2016.08.002
![]() |
[2] | S. Abbasbandy, A. Shirzadi, Homotopy analysis method for a nonlinear chemistry problem, Studies in Nonlinear Sciences, 1 (2010), 127-132. |
[3] | M. AL-Jawary, R. Raham, G. Radhi, An iterative method for calculating carbon dioxide absorbed into phenyl glycidyl ether, Journal of Mathematical and Computational Science, 6 (2016), 620-632. |
[4] |
Y. S. Choe, S. W. Park, D. W. Park, et al. Reaction kinetics of carbon dioxide with phenyl glycidyl ether by TEA-CP-MS41 catalyst, J. JPN Petrol. Inst., 53 (2010), 160-166. doi: 10.1627/jpi.53.160
![]() |
[5] | R. Singha, A. M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether: An optimal homotopy analysis method, Match-Commun Math. Co., 81 (2019), 800-812. |
[6] | H. Robertson, The solution of a set of reaction rate equations, Numerical analysis: an introduction, 178182. |
[7] |
M. Matinfar, M. Saeidy, B. Gharahsuflu, et al. Solutions of nonlinear chemistry problems by homotopy analysis, Comput. Math. Model., 25 (2014), 103-114. doi: 10.1007/s10598-013-9211-0
![]() |
[8] |
D. Ganji, M. Nourollahi, E. Mohseni, Application of he's methods to nonlinear chemistry problems, Comput. Math. Appl., 54 (2007), 1122-1132. doi: 10.1016/j.camwa.2006.12.078
![]() |
[9] |
A. Dokoumetzidis, R. Magin, P. Macheras, Fractional kinetics in multi-compartmental systems, J. Pharmacokinet. Phar., 37 (2010), 507-524. doi: 10.1007/s10928-010-9170-4
![]() |
[10] | F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, 2010. |
[11] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Elsevier, 1998. |
[12] | K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, 2010. |
[13] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, 2006. |
[14] | S. Das, Functional fractional calculus for system identification and controls, Springer-Verlag, 2008. |
[15] | J. S. Walker, A primer on wavelets and their scientific applications, CRC press, 2002. |
[16] | J. J. Benedetto, Wavelets: mathematics and applications, Vol. 13, CRC press, 1993. |
[17] | C. K. Chui, Wavelets: a mathematical tool for signal analysis, Vol. 1, Siam, 1997. |
[18] | S. Gopalakrishnan, M. Mitra, Wavelet methods for dynamical problems: with application to metallic, composite, and nano-composite structures, CRC Press, 2010. |
[19] | Y. Wang, Q. Fan, The second kind chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218 (2012), 8592-8601. |
[20] | J.-L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (2009), 31-40. |
[21] |
M. Heydari, M. Hooshmandasl, F. Mohammadi, et al. Wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations, Commun. Nonlinear Sci., 19 (2014), 37-48. doi: 10.1016/j.cnsns.2013.04.026
![]() |
[22] |
S. Balaji, Legendre wavelet operational matrix method for solution of fractional order riccati differential equation, Journal of the Egyptian Mathematical Society, 23 (2015), 263-270. doi: 10.1016/j.joems.2014.04.007
![]() |
[23] |
M. ur Rehman, R. A. Khan, The legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci., 16 (2011), 4163-4173. doi: 10.1016/j.cnsns.2011.01.014
![]() |
[24] |
M. Hosseininia, M. Heydari, F. M. Ghaini, et al. A wavelet method to solve nonlinear variableorder time fractional 2d klein-gordon equation, Comput. Math. Appl., 78 (2019), 3713-3730. doi: 10.1016/j.camwa.2019.06.008
![]() |
[25] |
P. Pirmohabbati, A. R. Sheikhani, H. S. Najafi, et al. Numerical solution of fractional mathieu equations by using block-pulse wavelets, Journal of Ocean Engineering and Science, 4 (2019), 299-307. doi: 10.1016/j.joes.2019.05.005
![]() |
[26] |
M. Hosseininia, M. Heydari, R. Roohi, et al. A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation, J. Comput. Phys., 395 (2019), 1-18. doi: 10.1016/j.jcp.2019.06.024
![]() |
[27] | M. H. Heydari, Z. Avazzadeh, Legendre wavelets optimization method for variable-order fractional poisson equation, Chaos, Solitons & Fractals, 112 (2018), 180-190. |
[28] | S. Kumar, R. Kumar, J. Singh, et al. An efficient numerical scheme for fractional model of hiv-1 infection of cd4+ t-cells with the effect of antiviral drug therapy, Alexandria Engineering Journal, 2020. |
[29] |
S. S. Ray, A. Patra, Numerical simulation for fractional order stationary neutron transport equation using haar wavelet collocation method, Nucl. Eng. Des., 278 (2014), 71-85. doi: 10.1016/j.nucengdes.2014.07.010
![]() |
[30] |
A. Patra, S. S. Ray, A numerical approach based on haar wavelet operational method to solve neutron point kinetics equation involving imposed reactivity insertions, Ann. Nucl. Energy, 68 (2014), 112-117. doi: 10.1016/j.anucene.2014.01.008
![]() |
[31] | S. S. Ray, A. Patra, Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system, Appl. Math. Comput., 220 (2013), 659-667. |
[32] | S. S. Ray, On haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation, Appl. Math. Comput., 218 (2012), 5239-5248. |
[33] |
I. Aziz, F. Haq, et al. A comparative study of numerical integration based on haar wavelets and hybrid functions, Comput. Math. Appl., 59 (2010), 2026-2036. doi: 10.1016/j.camwa.2009.12.005
![]() |
[34] |
I. Aziz, A. Al-Fhaid, et al. An improved method based on haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders, J. Comput. Appl. Math., 260 (2014), 449-469. doi: 10.1016/j.cam.2013.10.024
![]() |
[35] |
I. Aziz, A. Al-Fhaid, A. Shah, et al. A numerical assessment of parabolic partial differential equations using haar and legendre wavelets, Appl. Math. Model., 37 (2013), 9455-9481. doi: 10.1016/j.apm.2013.04.014
![]() |
[36] |
R. Jiwari, A haar wavelet quasilinearization approach for numerical simulation of burgers' equation, Comput. Phys. Commun., 183 (2012), 2413-2423. doi: 10.1016/j.cpc.2012.06.009
![]() |
[37] | Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Appl. Math. Comput., 214 (2009), 468-478. |
[38] |
Z. Shi, Y.-y. Cao, A spectral collocation method based on haar wavelets for poisson equations and biharmonic equations, Math. Comput. Model., 54 (2011), 2858-2868. doi: 10.1016/j.mcm.2011.07.006
![]() |
[39] |
H. Hein, L. Feklistova, Computationally efficient delamination detection in composite beams using haar wavelets, Mech. Syst. Signal Pr., 25 (2011), 2257-2270. doi: 10.1016/j.ymssp.2011.02.003
![]() |
[40] | Z. Gao, X. Liao, Discretization algorithm for fractional order integral by haar wavelet approximation, Appl. Math. Comput., 218 (2011), 1917-1926. |
[41] |
C. Chen, C. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proceedings-Control Theory and Applications, 144 (1997), 87-94. doi: 10.1049/ip-cta:19970702
![]() |
[42] |
Y. Chen, M. Yi, C. Yu, Error analysis for numerical solution of fractional differential equation by haar wavelets method, J. Comput. Sci., 3 (2012), 367-373. doi: 10.1016/j.jocs.2012.04.008
![]() |
[43] |
X. Xie, G. Jin, Y. Yan, et al. Free vibration analysis of composite laminated cylindrical shells using the haar wavelet method, Composite Structures, 109 (2014), 169-177. doi: 10.1016/j.compstruct.2013.10.058
![]() |
[44] | U. Saeed, M. ur Rehman, Haar wavelet-quasilinearization technique for fractional nonlinear differential equations, Appl. Math. Comput., 220 (2013), 630-648. |
[45] |
İ. Çelik, Haar wavelet approximation for magnetohydrodynamic flow equations, Appl. Math. Model., 37 (2013), 3894-3902. doi: 10.1016/j.apm.2012.07.048
![]() |
[46] |
İ. Çelik, Haar wavelet method for solving generalized burgers-huxley equation, Arab Journal of Mathematical Sciences, 18 (2012), 25-37. doi: 10.1016/j.ajmsc.2011.08.003
![]() |
[47] | G. Hariharan, K. Kannan, K. Sharma, Haar wavelet method for solving fisher's equation, Appl. Math. Comput., 211 (2009), 284-292. |
[48] | L. Wang, Y. Ma, Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Appl. Math. Comput., 227 (2014), 66-76. |
[49] |
H. Kaur, R. Mittal, V. Mishra, Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Model., 38 (2014), 4958-4971. doi: 10.1016/j.apm.2014.03.019
![]() |
[50] |
Z. Shi, Y. Y. Cao, Q. j. Chen, Solving 2d and 3d poisson equations and biharmonic equations by the haar wavelet method, Appl. Math. Model., 36 (2012), 5143-5161. doi: 10.1016/j.apm.2011.11.078
![]() |
[51] |
G. Jin, X. Xie, Z. Liu, The haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory, Composite Structures, 108 (2014), 435-448. doi: 10.1016/j.compstruct.2013.09.044
![]() |
[52] |
S. Kumar, A. Kumar, S. Abbas, et al. A modified analytical approach with existence and uniqueness for fractional cauchy reaction-diffusion equations, Advances in Difference Equations, 2020 (2020), 1-18. doi: 10.1186/s13662-019-2438-0
![]() |
[53] | S. Kumar, A. Kumar, S. Momani, et al. Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems, Advances in Difference Equations, 2019 (2019), 413. |
[54] | M. Jleli, S. Kumar, R. Kumar, et al. Analytical approach for time fractional wave equations in the sense of yang-abdel-aty-cattani via the homotopy perturbation transform method, Alexan. Eng. J., 2019. |
[55] | S. Kumar, K. S. Nisar, R. Kumar, et al. A new rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force, Mathematical Methods in Applied Science, 2020. |
[56] |
B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos, Soliton and Fractals, 133 (2020), 109619. doi: 10.1016/j.chaos.2020.109619
![]() |
[57] |
A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, et al. Solitary solutions for time-fractional nonlinear dispersive pdes in the sense of conformable fractional derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 093102. doi: 10.1063/1.5100234
![]() |
[58] | E. F. D. Goufo, S. Kumar, S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Soliton and Fractals, 130 (2020), 109467. |
[59] |
S. Bhatter, A. Mathur, D. Kumar, et al. A new analysis of fractional drinfeld-sokolov-wilson model with exponential memory, Physica A: Statistical Mechanics and its Applications, 537 (2020), 122578. doi: 10.1016/j.physa.2019.122578
![]() |
[60] | P. Veeresha, D. G. Prakasha, D. Kumar, An efficient technique for nonlinear time-fractional klein- fock-gordon equation, Appl. Math. Comput., 364 (2020), 124637. |
[61] | D. Kumar, J. Singh, M. A. Al-Qurashi, et al. A new fractional sirs-si malaria disease model with application of vaccines, antimalarial drugs, and spraying, Advances in Difference Equations, 2019 (2019), 278. |
[62] |
T. Hull, W. Enright, B. Fellen, et al. Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal., 9 (1972), 603-637. doi: 10.1137/0709052
![]() |
[63] | O. S. Board, Ocean acidification: a national strategy to meet the challenges of a changing ocean, National Academies Press, 2010. |
[64] |
S. Muthukaruppan, I. Krishnaperumal, R. Lakshmanan, Theoretical analysis of mass transfer with chemical reaction using absorption of carbon dioxide into phenyl glycidyl ether solution, Appl. Math. Ser. B, 3 (2012), 1179-1186. doi: 10.4236/am.2012.310172
![]() |
[65] |
S. Kumar, M. M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun., 185 (2014), 1947-1954. doi: 10.1016/j.cpc.2014.03.025
![]() |
[66] | Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276-2285. |
[67] | M. Srivastava, S. K. Agrawal, S. Das, Synchronization of chaotic fractional order lotka-volterra system, Int. J. Nonlinear Sci., 13 (2012), 482-494. |
[68] | K. Diethelm, N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621-640. |
[69] |
H. Aminikhah, An analytical approximation to the solution of chemical kinetics system, J. King Saud Univ. Sci., 23 (2011), 167-170. doi: 10.1016/j.jksus.2010.07.003
![]() |
[70] |
J. S. Duan, R. Rach, A. M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the adomian decomposition method, J. Math. Chem., 53 (2015), 1054-1067. doi: 10.1007/s10910-014-0469-z
![]() |
[71] | M. A. AL-Jawary, G. H. Radhi, The variational iteration method for calculating carbon dioxide absorbed into phenyl glycidyl ether, Iosr Journal of Mathematics, 11 (2015), 99-105. |
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t | Θ1(HWM) | Θ1(ABM) | Θ2(HWM) | Θ2(ABM) |
0.1 | 0.9901 | 0.9893 | 0.0100 | 0.0107 |
0.2 | 0.9802 | 0.9794 | 0.0198 | 0.0206 |
0.3 | 0.9704 | 0.9697 | 0.0296 | 0.0303 |
0.4 | 0.9608 | 0.9600 | 0.0393 | 0.0400 |
0.5 | 0.9512 | 0.9505 | 0.0489 | 0.0495 |
0.6 | 0.9418 | 0.9410 | 0.0585 | 0.0590 |
0.7 | 0.9324 | 0.9317 | 0.0680 | 0.0683 |
0.8 | 0.9231 | 0.9224 | 0.0775 | 0.0776 |
0.9 | 0.9139 | 0.9132 | 0.0869 | 0.0868 |
1.0 | 0.9048 | 0.9041 | 0.0963 | 0.0962 |
t | Θ3(HWM) | Θ3(ABM) |
0.1 | 3.0×10−8 | 4.0×10−8 |
0.2 | 2.4×10−7 | 2.7×10−7 |
0.3 | 7.9×10−7 | 8.6×10−7 |
0.4 | 1.8×10−6 | 1.9×10−6 |
0.5 | 3.6×10−6 | 3.8×10−6 |
0.6 | 6.2×10−6 | 6.4×10−6 |
0.7 | 9.8×10−6 | 1.0×10−5 |
0.8 | 1.5×10−5 | 1.5×10−5 |
0.9 | 2.0×10−5 | 2.0×10−5 |
1.0 | 2.8×10−5 | 2.8×10−5 |
t | Θ1(HWM) | Θ1(ABM) | Θ2(HWM) | Θ2(ABM) |
0.1 | 0.9901 | 0.9893 | 0.0100 | 0.0107 |
0.2 | 0.9802 | 0.9794 | 0.0198 | 0.0206 |
0.3 | 0.9704 | 0.9697 | 0.0296 | 0.0303 |
0.4 | 0.9608 | 0.9600 | 0.0393 | 0.0400 |
0.5 | 0.9512 | 0.9505 | 0.0489 | 0.0495 |
0.6 | 0.9418 | 0.9410 | 0.0585 | 0.0590 |
0.7 | 0.9324 | 0.9317 | 0.0680 | 0.0683 |
0.8 | 0.9231 | 0.9224 | 0.0775 | 0.0776 |
0.9 | 0.9139 | 0.9132 | 0.0869 | 0.0868 |
1.0 | 0.9048 | 0.9041 | 0.0963 | 0.0962 |
t | Θ3(HWM) | Θ3(ABM) |
0.1 | 3.0×10−8 | 4.0×10−8 |
0.2 | 2.4×10−7 | 2.7×10−7 |
0.3 | 7.9×10−7 | 8.6×10−7 |
0.4 | 1.8×10−6 | 1.9×10−6 |
0.5 | 3.6×10−6 | 3.8×10−6 |
0.6 | 6.2×10−6 | 6.4×10−6 |
0.7 | 9.8×10−6 | 1.0×10−5 |
0.8 | 1.5×10−5 | 1.5×10−5 |
0.9 | 2.0×10−5 | 2.0×10−5 |
1.0 | 2.8×10−5 | 2.8×10−5 |