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Semi-analytic solutions of nonlinear multidimensional fractional differential equations


  • In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a class of nonlinear multidimensional fractional differential equations with Caputo-Fabrizio fractional derivative. The main advantage of the Caputo-Fabrizio fractional derivative appears in its non-singular kernel of a convolution type. The sufficient condition that guarantees a unique solution is obtained, the convergence of the series solution is discussed, and the maximum absolute error is estimated. Several numerical problems with an unknown exact solution are solved using the two techniques. A comparative study between the two solutions is presented. A comparative study shows that the time consumed by ADM is much smaller compared with the Picard technique.

    Citation: M. Botros, E. A. A. Ziada, I. L. EL-Kalla. Semi-analytic solutions of nonlinear multidimensional fractional differential equations[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13306-13320. doi: 10.3934/mbe.2022623

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  • In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a class of nonlinear multidimensional fractional differential equations with Caputo-Fabrizio fractional derivative. The main advantage of the Caputo-Fabrizio fractional derivative appears in its non-singular kernel of a convolution type. The sufficient condition that guarantees a unique solution is obtained, the convergence of the series solution is discussed, and the maximum absolute error is estimated. Several numerical problems with an unknown exact solution are solved using the two techniques. A comparative study between the two solutions is presented. A comparative study shows that the time consumed by ADM is much smaller compared with the Picard technique.



    Fractional calculus is a main branch of mathematics that can be considered as the generalisation of integration and differentiation to arbitrary orders. This hypothesis begins with the assumptions of L. Euler (1730) and G. W. Leibniz (1695). Fractional differential equations (FDEs) have lately gained attention and publicity due to their realistic and accurate computations [1,2,3,4,5,6,7]. There are various types of fractional derivatives, including Riemann–Liouville, Caputo, Grü nwald–Letnikov, Weyl, Marchaud, and Atangana. This topic's history can be found in [8,9,10,11]. Undoubtedly, fractional calculus applies to mathematical models of different phenomena, sometimes more effectively than ordinary calculus [12,13]. As a result, it can illustrate a wide range of dynamical and engineering models with greater precision. Applications have been developed and investigated in a variety of scientific and engineering fields over the last few decades, including bioengineering [14], mechanics [15], optics [16], physics [17], mathematical biology, electrical power systems [18,19,20] and signal processing [21,22,23].

    One of the definitions of fractional derivatives is Caputo-Fabrizo, which adds a new dimension in the study of FDEs. The new derivative's feature is that it has a nonsingular kernel, which is made from a combination of an ordinary derivative with an exponential function, but it has the same supplementary motivating properties with various scales as in the Riemann-Liouville fractional derivatives and Caputo. The Caputo-Fabrizio fractional derivative has been used to solve real-world problems in numerous areas of mathematical modelling for example, numerical solutions for groundwater pollution, the movement of waves on the surface of shallow water modelling [24], RLC circuit modelling [25], and heat transfer modelling [26,27] were discussed.

    Rach (1987), Bellomo and Sarafyan (1987) first compared the Adomian Decomposition method (ADM) [28,29,30,31,32] to the Picard method on a variety of examples. These methods have many benefits: they effectively work with various types of linear and nonlinear equations and also provide an analytic solution for all of these equations with no linearization or discretization. These methods are more realistic compared with other numerical methods as each technique is used to solve a specific type of equations, on the other hand ADM and Picard are useful for many types of equations. In the numerical examples provided, we compare ADM and Picard solutions of multidimentional fractional order equations with Caputo-Fabrizio.

    The fractional derivative of Caputo-Fabrizio for the function x(t) is defined as [33]

    CFDα0x(t)=B(α)1αt0dds(x(s)) eα1α(ts)ds, (1.1)

    and its corresponding fractional integral is

    CFIαx(t)=1αB(α)x(t)+αB(α)t0x (s)ds,    0<α<1, (1.2)

    where x(t) be continuous and differentiable on [0, T]. Also, in the above definition, the function B(α)>0 is a normalized function which satisfy the condition B(0)=B(1)=0. The relation between the Caputo–Fabrizio fractional derivate and its corresponding integral is given by

    (CFIα0)(CFDα0f(t))=f(t)f(a). (1.3)

    In this section, we will introduce a multidimentional FDE subject to the initial condition. Let α(0,1], 0<α1<α2<...,αm<1, and m is integer real number,

    CFDx=f(t,x,CFDα1x,CFDα2x,...,CFDαmx,) ,x(0)=c0, (2.1)

    where x=x(t),tJ=[0,T],TR+,xC(J).

    To facilitate the equation and make it easy for the calculation, we let x(t)=c0+X(t) so Eq (2.1) can be witten as

    CFDαX=f(t,c0+X,CFDα1X,CFDα2X,...,CFDαmX), X(0)=0. (2.2)

    the algorithm depends on converting the initial condition from a constant c0 to 0.

    Let CFDαX=y(t) then X=CFIαy, so we have

    CFDαiX= CFIααi CFDαX= CFIααiy,  i=1,2,...,m. (2.3)

    Substituting in Eq (2.2) we obtain

    y=f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy). (2.4)

    Assume f satisfies Lipschtiz condition with Lipschtiz constant L given by,

    |f(t,y0,y1,...,ym)||f(t,z0,z1,...,zm)|Lmi=0|yizi|, (2.5)

    which implies

    |f(t,c0+CFIαy,CFIαα1y,..,CFIααmy)f(t,c0+CFIαz,CFIαα1z,..,CFIααmz)|Lmi=0| CFIααiy CFIααiz|. (2.6)

    The solution algorithm of Eq (2.4) using ADM is,

    y0(t)=a(t)yn+1(t)=An(t), j0. (2.7)

    where a(t) pocesses all free terms in Eq (2.4) and An are the Adomian polynomials of the nonlinear term which takes the form [34]

    An=f(Sn)n1i=0Ai, (2.8)

    where f(Sn)=ni=0Ai. Later, this accelerated formula of Adomian polynomial will be used in convergence analysis and error estimation. The solution of Eq (2.4) can be written in the form,

    y(t)=i=0yi(t). (2.9)

    lastly, the solution of the Eq (2.4) takes the form

    x(t)=c0+X(t)=c0+ CFIαy(t). (2.10)

    At which we convert the parameter to the initial form y to x in Eq (2.10), so we have the solution of the original Eq (2.1).

    Define a mapping F:EE where E=(C[J],) is a Banach space of all continuous functions on J with the norm x= maxtϵJx(t).

    Theorem 3.1. Equation (2.4) has a unique solution whenever 0<ϕ<1 where ϕ=L(mi=0[(ααi)(T1)]+1B(ααi)).

    Proof. First, we define the mapping F:EE as

    Fy=f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy).

    Let y and zE are two different solutions of Eq (2.4). Then

    FyFz=f(t,c0+CFIαy,CFIαα1y,..,CFIααmy)f(t,c0+CFIαz,CFIαα1z,...,CFIααmz)

    which implies that

    |FyFz|=|f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy)f(t,c0+ CFIαz, CFIαα1z,..., CFIααmz)|Lmi=0| CFIααiy CFIααiz|Lmi=0|1(ααi)B(ααi)(yz)+ααiB(ααi)t0(yz)ds|FyFzLmi=01(ααi)B(ααi)maxtϵJ|yz|+ααiB(ααi)maxtϵJ|yz|t0dsLmi=01(ααi)B(ααi)yz+ααiB(ααi)yzTLyz(mi=01(ααi)B(ααi)+ααiB(ααi)T)Lyz(mi=0[(ααi)(T1)]+1B(ααi))ϕyz.

    under the condition 0<ϕ<1, the mapping F is contraction and hence there exists a unique solution yC[J] for the problem Eq (2.4) and this completes the proof.

    Theorem 3.2. The series solution of the problem Eq (2.4)converges if |y1(t)|<c and c isfinite.

    Proof. Define a sequence {Sp} such that Sp=pi=0yi(t) is the sequence of partial sums from the series solution i=0yi(t), we have

    f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy)=i=0Ai,

    So

    f(t,c0+ CFIαSp, CFIαα1Sp,..., CFIααmSp)=pi=0Ai,

    From Eq (2.7) we have

    i=0yi(t)=a(t)+i=0Ai1

    let Sp,Sq be two arbitrary sums with pq. Now, we are going to prove that {Sp} is a Caushy sequence in this Banach space. We have

    Sp=pi=0yi(t)=a(t)+pi=0Ai1,Sq=qi=0yi(t)=a(t)+qi=0Ai1.
    SpSq=pi=0Ai1qi=0Ai1=pi=q+1Ai1=p1i=qAi1=f(t,c0+ CFIαSp1, CFIαα1Sp1,..., CFIααmSp1)f(t,c0+ CFIαSq1, CFIαα1Sq1,..., CFIααmSq1)
    |SpSq|=|f(t,c0+ CFIαSp1, CFIαα1Sp1,..., CFIααmSp1)f(t,c0+ CFIαSq1, CFIαα1Sq1,..., CFIααmSq1)|Lmi=0| CFIααiSp1 CFIααiSq1|Lmi=0|1(ααi)B(ααi)(Sp1Sq1)+ααiB(ααi)t0(Sp1Sq1)ds|Lmi=01(ααi)B(ααi)|Sp1Sq1|+ααiB(ααi)t0|Sp1Sq1|ds
    SpSqLmi=01(ααi)B(ααi)maxtϵJ|Sp1Sq1|+ααiB(ααi)maxtϵJ|Sp1Sq1|t0dsLSpSqmi=0(1(ααi)B(ααi)+ααiB(ααi)T)LSpSq(mi=0[(ααi)(T1)]+1B(ααi))ϕSpSq

    let p=q+1 then,

    Sq+1SqϕSqSq1ϕ2Sq1Sq2...ϕqS1S0

    From the triangle inequality we have

    SpSqSq+1Sq+Sq+2Sq+1+...SpSp1[ϕq+ϕq+1+...+ϕp1]S1S0ϕq[1+ϕ+...+ϕpq+1]S1S0ϕq[1ϕpq1ϕ]y1(t)

    Since 0<ϕ<1,pq then (1ϕpq)1. Consequently

    SpSqϕq1ϕy1(t)ϕq1ϕmaxtϵJ|y1(t)| (3.1)

    but |y1(t)|< and as q then, SpSq0 and hence, {Sp} is a Caushy sequence in this Banach space then the proof is complete.

    Theorem 3.3. The maximum absolute truncated error Eq (2.4)is estimated to be maxtϵJ|y(t)qi=0yi(t)|ϕq1ϕmaxtϵJ|y1(t)|

    Proof. From the convergence theorm inequality (Eq 3.1) we have

    SpSqϕq1ϕmaxtϵJ|y1(t)|

    but, Sp=pi=0yi(t) as p then, Spy(t) so,

    y(t)Sqϕq1ϕmaxtϵJ|y1(t)|

    so, the maximum absolute truncated error in the interval J is,

    maxtϵJ|y(t)qi=0yi(t)|ϕq1ϕmaxtϵJ|y1(t)| (3.2)

    and this completes the proof.

    In this part, we introduce several numerical examples with unkown exact solution and we will use inequality (Eq 3.2) to estimate the maximum absolute truncated error.

    Example 4.1. Application of linear FDE

    CFDx(t)+2aCFD1/2x(t)+bx(t)=0,       x(0)=1. (4.1)

    A Basset problem in fluid dynamics is a classical problem which is used to study the unsteady movement of an accelerating particle in a viscous fluid under the action of the gravity [36]

    Set

    X(t)=x(t)1

    Equation (4.1) will be

    CFDX(t)+2aCFD1/2X(t)+bX(t)=0,       X(0)=0. (4.2)

    Appling Eq (2.3) to Eq (4.2), and using initial condition, also we take a = 1, b = 1/2,

    y=122I1/2y12I y (4.3)

    Appling ADM to Eq (4.3), we find the solution algorithm become

    y0(t)=12,yi(t)=2 CFI1/2yi112 CFI yi1,     i1. (4.4)

    Appling Picard solution to Eq (4.2), we find the solution algorithm become

    y0(t)=12,yi(t)=122 CFI1/2yi112 CFI yi1,     i1. (4.5)

    From Eq (4.4), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.5), the solution using Picard technique is given by y(t)=Limiyi(t). Lately, the solution of the original problem Eq (4.2), is

    x(t)=1+ CFI y(t).

    One the same processor (q = 20), the time consumed using ADM is 0.037 seconds, while the time consumed using Picard is 7.955 seconds.

    Figure 1 gives a comparison between ADM and Picard solution of Ex. 4.1.

    Figure 1.  ADM and Picard solution of Ex. 4.1.

    Example 4.2. Consider the following nonlinear FDE [35]

    CFD1/2x=8t3/23πt7/44Γ(114)t44+18 CFD1/4x+14x2, x(0)=0. (4.6)

    Appling Eq (2.3) to Eq (4.6), and using initial condition,

    y=8t3/23πt7/44Γ(114)t44+18 CFI1/4y+14(CFI1/2y)2. (4.7)

    Appling ADM to Eq (4.7), we find the solution algorithm will be become

    y0(t)=8t3/23πt7/44Γ(114)t44,yi(t)=18 CFI1/4yi1+14(Ai1),     i1. (4.8)

    at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.

    Appling Picard solution to Eq (4.7), we find the the solution algorithm become

    y0(t)=8t3/23πt7/44Γ(114)t44,yi(t)=y0(t)+18 CFI1/4yi1+14(CFI1/2yi1)2,     i1. (4.9)

    From Eq (4.8), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.9), the solution using Picard technique is given by y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.7), is.

    x(t)= CFI1/2y.

    One the same processor (q = 2), the time consumed using ADM is 65.13 seconds, while the time consumed using Picard is 544.787 seconds.

    Table 1 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 2):

    Table 1.  Max. absolute error.
    q max. absolute error
    2 0.114548
    5 0.099186
    10 0.004363

     | Show Table
    DownLoad: CSV

    Figure 2 gives a comparison between ADM and Picard solution of Ex. 4.2.

    Figure 2.  ADM and Picard solution of Ex. 4.2.

    Example 4.3. Consider the following nonlinear FDE [35]

    CFDαx=3t2128125πt5+110(CFD1/2x)2,x(0)=0. (4.10)

    Appling Eq (2.3) to Eq (4.10), and using initial condition,

    y=3t2128125πt5+110(CFI1/2y)2 (4.11)

    Appling ADM to Eq (4.11), we find the solution algorithm become

    y0(t)=3t2128125πt5,yi(t)=110(Ai1),     i1 (4.12)

    at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.

    Then appling Picard solution to Eq (4.11), we find the solution algorithm become

    y0(t)=3t2128125πt5,yi(t)=y0(t)+110(CFI1/2yi1)2,     i1. (4.13)

    From Eq (4.12), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.13), the solution is y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.11), is

    x(t)=CFIy(t).

    One the same processor (q = 4), the time consumed using ADM is 2.09 seconds, while the time consumed using Picard is 44.725 seconds.

    Table 2 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 4):

    Table 2.  Max. absolute error.
    q max. absolute error
    2 0.00222433
    5 0.0000326908
    10 2.88273*108

     | Show Table
    DownLoad: CSV

    Figure 3 gives a comparison between ADM and Picard solution of Ex. 4.3 with α=1.

    Figure 3.  ADM and Picard solution where of Ex. 4.3.

    Example 4.4. Consider the following nonlinear FDE [35]

    CFDαx=t2+12 CFDα1x+14 CFDα2x+16 CFDα3x+18x4,x(0)=0. (4.14)

    Appling Eq (2.3) to Eq (4.10), and using initial condition,

    y=t2+12(CFIαα1y)+14(CFIαα2y)+16(CFIαα3y)+18(CFIαy)4, (4.15)

    Appling ADM to Eq (4.15), we find the solution algorithm become

    y0(t)=t2,yi(t)=12(CFIαα1y)+14(CFIαα2y)+16(CFIαα3y)+18Ai1,  i1 (4.16)

    where Ai are Adomian polynomial of the nonliner term (CFIαy)4.

    Then appling Picard solution to Eq (4.15), we find the solution algorithm become

    y0(t)=t2,yi(t)=t2+12(CFIαα1yi1)+14(CFIαα2yi1)+16(CFIαα3yi1)+18(CFIαyi1)4     i1. (4.17)

    From Eq (4.16), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.17), the solution using Picard technique is y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.14), is

    x(t)=CFIαy(t).

    One the same processor (q = 3), the time consumed using ADM is 0.437 seconds, while the time consumed using Picard is (16.816) seconds. Figure 4 shows a comparison between ADM and Picard solution of Ex. 4.4 atα=0.7,α1=0.1,α2=0.3,α3=0.5.

    Figure 4.  ADM and Picard solution where of Ex. 4.4.

    The Caputo-Fabrizo fractional deivative has a nonsingular kernel, and consequently, this definition is appropriate in solving nonlinear multidimensional FDE [37,38]. Since the selected numerical problems have an unkown exact solution, the formula (3.2) can be used to estimate the maximum absolute truncated error. By comparing the time taken on the same processor (i7-2670QM), it was found that the time consumed by ADM is much smaller compared with the Picard technique. Furthermore Picard gives a more accurate solution than ADM at the same interval with the same number of terms.

    The authors declare there is no conflict of interest.



    [1] S. Narayanamoorthy, D. Baleanu, K. Thangapandi, S. S. N. Perera, Analysis for fractional-order predator-prey model with uncertainty, IET Syst. Biol., 13 (2019), 277–289. https://doi.org/10.1049/iet-syb.2019.0055 doi: 10.1049/iet-syb.2019.0055
    [2] D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional Dynamics and Control, Springer New York, NY, 2012. https://doi.org/10.1007/978-1-4614-0457-6
    [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 204 (2006), 1–523.
    [4] A. M. A. El-Sayed, I. L. El-Kalla, E. A. A.Ziada, Analytical and numerical solutions of nonlinear fractional differentail equations, Appl. Numer. Math., 60 (2010), 788–797. https://doi.org/10.1016/j.apnum.2010.02.007 doi: 10.1016/j.apnum.2010.02.007
    [5] I. L. El-Kalla, Error Estimate of the series solution to a class of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1408–1413. https://doi.org/10.1016/j.cnsns.2010.05.030 doi: 10.1016/j.cnsns.2010.05.030
    [6] H. Ye, R. Huang, On the nonlinear fractional differential equations with Caputo sequential fractional derivative, Adv. Math. Phys., 2015 (2015). https://doi.org/10.1155/2015/174156
    [7] C. Cesarano, Generalized special functions in the description of fractional diffusive equations, Commun. Appl. Ind. Math., 10 (2019), 31–40. https://doi.org/10.1515/caim-2019-0010 doi: 10.1515/caim-2019-0010
    [8] K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, Amsterdam, The Netherlands, 111 (1974), 1–234.
    [9] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, Hoboken, NJ, USA, 1993.
    [10] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Taylor & Francis: Oxfordshire, UK, 1993.
    [11] J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
    [12] M. Hosseininia, M. H. Heydari, Z. Avazzadeh, F. M. Ghaini, A hybrid method based on the orthogonal Bernoulli polynomials and radial basis functions for variable order fractional reaction-advection-diffusion equation, Eng. Anal. Boundary Elem., 127 (2021), 18–28. https://doi.org/10.1016/j.enganabound.2021.03.006 doi: 10.1016/j.enganabound.2021.03.006
    [13] M. Inc, M. Partohaghighi, M. A. Akinlar, P. Agarwal, Y. M. Chu, New solutions of fractional-order Burger-Huxley equation, Results Phys., 18 (2020), 103290. https://doi.org/10.1016/j.rinp.2020.103290 doi: 10.1016/j.rinp.2020.103290
    [14] R. L. Magin, Fractional calculus in bioengineering, part 1, Crit. Rev. Biomed. Eng., 32 (2004). https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10
    [15] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Vienna, 378 (2014). https://doi.org/10.1007/978-3-7091-2664-6
    [16] H. Bulut, T. A. Sulaiman, H. M. Baskonus, H. Rezazadeh, M. Eslami, M. Mirzazadeh, Optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation, Optik, 172 (2018), 20–27. https://doi.org/10.1016/j.ijleo.2018.06.108 doi: 10.1016/j.ijleo.2018.06.108
    [17] H. Rudolf, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. https://doi.org/10.1142/3779
    [18] I. Dassios, T. Kerci, D. Baleanu, F. Milano, Fractional-order dynamical model for electricity markets, Math. Methods Appl. Sci., 2021 (2021). https://doi.org/10.1002/mma.7892
    [19] I. Dassios, F. Milano, Singular dual systems of fractional-order differential equations, Math. Methods Appl. Sci., 2021 (2021). https://doi.org/10.1002/mma.7584
    [20] I. Dassios, G. Tzounas, F. Milano, Generalized fractional controller for singular systems of differential equations, J. Comput. Appl. Math., 378 (2020), 112919. https://doi.org/10.1016/j.cam.2020.112919 doi: 10.1016/j.cam.2020.112919
    [21] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
    [22] A. Akgül, I. Siddique, Analysis of MHD couette flow by fractal-fractional differential operators, Chaos Solitons Fractals, 146 (2021). https://doi.org/10.1016/j.chaos.2021.110893
    [23] S. Ahmad, A. Ullah, A. Akgül, Investigating the complex behaviour of multiscroll chaotic system with Caputo fractal-fractional operator, Chaos Solitons Fractals, 146 (2021), 110900. https://doi.org/10.1016/j.chaos.2021.110900 doi: 10.1016/j.chaos.2021.110900
    [24] B. S. T. Alkahtani, A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 539–546. https://doi.org/10.1016/j.chaos.2016.03.012 doi: 10.1016/j.chaos.2016.03.012
    [25] A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1–7. https://doi.org/10.1177/1687814015613758 doi: 10.1177/1687814015613758
    [26] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
    [27] J. Hristov, Transient heal diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Therm. Sci., 20 (2016), 757–762. https://doi.org/10.2298/TSCI160112019H doi: 10.2298/TSCI160112019H
    [28] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Springer Dordrecht, 1995. https://doi.org/10.1007/978-94-015-8289-6
    [29] E. A. Az-Zobi, K. Al-Khaled, A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws, Appl. Math. Comput., 217 (2010), 4248–4256. https://doi.org/10.1016/j.amc.2010.10.040 doi: 10.1016/j.amc.2010.10.040
    [30] G. Adomian, R. Rach, Modified adomian polynomials, Math. Comput. Modell., 24 (1996), 39–46. https://doi.org/10.1016/S0895-7177(96)00171-9
    [31] X. G. Luo, A two-step Adomian decomposition method, Appl. Math. Comput., 170 (2005), 570–583. https://doi.org/10.1016/j.amc.2004.12.010 doi: 10.1016/j.amc.2004.12.010
    [32] I. L. El-Kalla, New results on the analytic summation of Adomian series for some classes of differential and integral equations, Appl. Math. Comput., 217 (2010), 3756–3763. https://doi.org/10.1016/j.amc.2010.09.034 doi: 10.1016/j.amc.2010.09.034
    [33] M. Al-Refai, K. Pal, New aspects of Caputo–Fabrizio fractional derivative, Prog. Fract. Differ. Appl., 5 (2019), 157–66. https://doi.org/10.18576/pfda/050206 doi: 10.18576/pfda/050206
    [34] I. L. El-Kalla, Error estimate for series solutions to a class of nonlinear integral equations of mixed type, J. Appl. Math. Comput., 38 (2012), 341–351. https://doi.org/10.1007/s12190-011-0482-3 doi: 10.1007/s12190-011-0482-3
    [35] E. E. Ziada, Solution of some fractional order differential and integral equations, Lambert Academic Publishing GmbH & Co. KG, 2012.
    [36] A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer Verlag, 378 (1997), 223–276. https://doi.org/10.1007/978-3-7091-2664-6
    [37] D. Assante, C. Cesarano, C. Fornaro, L. Vázquez, Higher order and fractional diffusive equations, J. Eng. Sci. Technol. Rev., 8 (2015), 202–204. https://doi.org/10.25103/JESTR.085.25 doi: 10.25103/JESTR.085.25
    [38] W. W. Mohammed, M. Alshammari, C. Cesarano, S. Albadrani, M. El-Morshedy, Brownian motion effects on the stabilization of stochastic solutions to fractional diffusion equations with polynomials, Mathematics, 10 (2022). https://doi.org/10.3390/math10091458
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