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Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative


  • This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.

    Citation: Muhammad Nadeem, Ji-Huan He, Hamid. M. Sedighi. Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8190-8207. doi: 10.3934/mbe.2023356

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  • This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.



    The study of fractional calculus (FC) yields the development of ordinary calculus with the history of more than 300 years earlier. In real-world, fractional-order derivatives are nonlocal, whereas integer-order derivatives are local. Many physical phenomena are designed by fractional partial differential equations arising in biology, sociology, medicine, hydrodynamics, computational modeling, chemical kinetics and among others [1,2,3]. One of the most exciting and challenging study to investigate the exact solution of some differential problem in physical science. Dong and Gao [4] derived an integral formulation of the nonlocal operator Ginzburg-Landau equation with the half Laplacian. To overcome this situation, numerous mathematical strategies have been put forth to configure the approximate solutions of these problems, such that Laplace iterative transform method [5], q-homotopy analysis Sumudu transform method [6], ρ-Laplace transform method [7,8], Haar wavelet method [9], Chebyshev spectral collocation method [10], extended modified auxiliary [11] and many others. Recently, various type of concepts and formulas of fractional operators are studied such as Riemann and Liouville [12], Caputo and Fabrizio [13], Atangana and Baleanu [14], and Liouville and Caputo [15]. Later, Abro and Atangana [16] showed that Liouville-Caputo and Atangana-Baleanu operators have excellent fractional retrieves. Caputo and Fabrizio [17] proposed a new concept of fractional derivative with a stabilize kernel to represent the temporal and spatial variables in two different ways. Toufik and Atangana [18] established a novel notion of fractional differentiation with a non-local and non-singular kernel to expand the limitations of the traditional Riemann-Liouville and Caputo fractional derivatives to solve linear and non-linear fractional differential equations. Gao et al. [19,20] presented a new method to achieve a smooth decay rates for the damped wave problems with nonlinear acoustic boundary conditions.

    The diffusion equation with time fractional derivative presents the density dynamics in a material undergoing diffusion. Jaradat et al. [21] provided the extended fractional power series approach for the analytical solution of 2D diffusion, wave-like, telegraph, and Burgers models. They obtained the results and claimed that both schemes are in excellent agreement. Dehghan and Shakeri [22] provided variational iteration method for solving the Cauchy reaction–diffusion problem. Singh and Srivastava [23] obtained the approximate series solution of multi-dimensional with time-fractional derivative using reduced differential transform method. Shah et al. [24] used natural transform method for the analytical solution of fractional order diffusion equations. Kumar et al. [25] used Laplace transform for the analytical solution of fractional multi-dimensional diffusion equations.

    He [26,27] studied an idea of the HPS for the analytical results of ordinary and partial differential problems. HPS provided the excellent findings and show the rate of convergence toward the precise solution than other analytical approaches in literature. Odibat and Momani [28] have demonstrated the significance of HPS in large number of fields and showed that HPS has an excellent treatment in providing the exact solution of these problems. Tarig M. Elzaki [29] established a new approach named as Elzaki transform (ET) to evaluate the approximate solutions in a wide range of areas. The ET is a remarkable tool in order to show the physical nature of the differential problems compared to other schemes. Recently, many authors studied the Elzaki transform involving Atangana-Baleanu fractional derivative operator for various fields such as alcohol drinking model [30], Hirota-Satsuma KdV equations [31], nonlinear regularized long-wave models, but all these approaches have some limitations and restrictions.

    In this paper, we eliminate these draw backs and study the Elzaki transform combined with the HPS involving Atangana-Baleanu fractional derivative operator in Caputo sense for the approximate solution multi-dimensional diffusion problems. The reason for using Atangana-Baleanu fractional derivatives is its nonlocal properties and its capability to deal the complex behavior more efficiently than other operators. The obtained series show the significant results and we see that the computational series approaches the precise results with few repetitions. This paper is designed as: In Section 2, we define a few basic definitions of Atangana-Baleanu fractional derivative operator in Caputo sense and Elazki transform. We formulate the strategy of EHPTS to achieve the numerical solution of the differential problems in Section 3. We provide a three-example approach for assessing the validity and dependability of EHPTS in Section 4 and we depict the conclusion in last Section 5.

    Definition 2.1. The Caputo fractional derivative (CFD) is given as [32]

    Dαηϑ(η)=1(mα)η0ϑm(v)(ηv)α+1+mdv,   m1<αm. (2.1)

    Definition 2.2. The Atangana-Baleanu Caputo (ABC) operator is defined as [33]

    Dαηϑ(η)=N(α)1αηmϑ(v)Eα[α(ηv)α1α]dv, (2.2)

    where ϑH1(α,β), β>α,α[0,1], Eα is Mittag Leffler function, N(α) is normalisation function and N(0)=N(1)=1.

    Definition 2.3. The fractional integral operator in ABC sense is given as [33]

    Iαη(ϑ(η))=1αN(α)ϑ(η)+αΓ(α)N(α)ηmϑ(v)(ηv)α1dv. (2.3)

    Definition 2.4. The Elzaki transform is given as [34]

    E[ϑ(η)]=R(f)=f0eηfϑ(η)dη,   k1fk2. (2.4)

    Propositions: The differential properties of ET are defined as [35]

    E[ηn]=n!fn+2,E[ϑ(η)]=E[ϑ(η)]ffϑ(0),E[ϑ(η)]=E[ϑ(η)]f2ϑ(0)fϑ(0), (2.5)

    Definition 2.5. The Elzaki transform of Dαηϑ(η) CFD operator is as

    E[Dαηϑ(η)]=fαR(f)m1k=0f2α+kϑk(0),m1<α<m. (2.6)

    Definition 2.6. The Elzaki transform of ABCDαηϑ(η) under ABC operator is as

    E[ABCDαηϑ(η)]=N(α)αfα+1α(R(f)ffϑ(0)), (2.7)

    where f is the transfer parameter of η such that E[ϑ(η)]=R(f).

    Consider a fractional partial differential equation in the following form,

    ABCDαηϑ(θ1,η)+Lϑ(θ1,η)+Mϑ(θ1,η)=g(θ1,η), (3.1)

    with the following initial condition

    ϑ(θ1,0)=a, (3.2)

    here ABCDαηϑ represents ABC fractional derivative operator, a is constants. where L and M are linear and nonlinear operators, g(θ1,η) in known term.

    Employing ET on Eq (3.1), we obtain

    E[ABCDαηϑ(θ1,η)+Lϑ(θ1,η)+Mϑ(θ1,η)]=E[g(θ1,η)]. (3.3)

    By the property of the ET differentiation, we have

    N(α)αfα+1α[E[ϑ(θ1,η)]f2ϑ(θ1,0)]=E[g(θ1,η)]E[Lϑ(θ1,η)+Mu(θ1,η)],

    which can be written as

    E[ϑ(θ1,η)]=f2ϑ(θ1,0)+αfα+1αN(α)E[g(θ1,η)]αfα+1αN(α)E[Lϑ(θ1,η)+Mϑ(θ1,η)].

    Employing the inverse ET, we get

    ϑ(θ1,η)=E1[f2ϑ(θ1,0)+αfα+1αN(α)E[g(θ1,η)]][αfα+1αN(α)E{Lϑ(θ1,η)+Mϑ(θ1,η)}].

    In other words, we may also write it as

    ϑ(θ1,η)=G(θ1,η)[αfα+1αN(α)E{Lϑ(θ1,η)+Mϑ(θ1,η)}]. (3.4)

    where

    ϑ(θ1,η)=G(θ1,η)E1[f2ϑ(θ1,0)+αfα+1αN(α)E[g(θ1,η)]].

    Now, we apply HPS on Eq (3.4). Let

    ϑ(η)=i=0piϑi(n)=ϑ0+p1ϑ1+p2ϑ2+, (3.5)

    where p is homotopy parameter and Mϑ(θ1,η) can be calculated by using formula,

    Mϑ(θ1,η)=i=0piHi(ϑ)=H0+p1H1+p2H2+, (3.6)

    where He's polynomial are calculated as

    Hn(ϑ0+ϑ1++ϑn)=1n!npn(M(i=0piϑi))p=0,    n=0,1,2, (3.7)

    Put Eqs (3.5)–(3.7) in Eq (3.4), we get

    i=0piϑ(θ1,η)=G(θ1,η)[αfα+1αN(α)E{Li=0piϑi(θ1,η)+i=0piHi}]. (3.8)

    and similar power of p produces the following iterations, we get

    p0:ϑ0(θ1,η)=G(θ1,η),p1:ϑ1(θ1,η)=E1[αfα+1αN(α)E{ϑ0(θ1,η)+H0(ϑ)}],p2:ϑ2(θ1,η)=E1[αfα+1αN(α)E{ϑ1(θ1,η)+H1(ϑ)}],p3:ϑ3(θ1,η)=E1[αfα+1αN(α)E{ϑ2(θ1,η)+H2(ϑ)}], (3.9)

    on continuing, these iterations can be written in the following series

    ϑ(θ1,η)=ϑ0(θ1,η)+ϑ1(θ1,η)+ϑ2(θ1,η)+ϑ3(θ1,η)+=i=0ϑi. (3.10)

    which represents the approximate solution of the differential problem (3.1).

    Some numerical applications are provided to confirm the significance of EHPTS and the physical behavior through the graphical representation. It is noticed that only few iterations are enough to demonstrate the accuracy of EHPTS.

    Consider a one-dimensional fractional diffusion problem

    αϑηα=2ϑθ21+sinθ1, (4.1)

    with the initial condition

    ϑ(θ1,0)=cosθ1, (4.2)

    and boundary condition

    ϑ(0,η)=Eη,ϑ(π,η)=Eη. (4.3)

    Taking ET on Eq (4.1), we get

    E[αϑηα]=E[2ϑθ21+sinθ1].

    Employing the differential properties of ET under ABC operator, we get

    N(α)αfα+1α[E[ϑ(θ1,η)]f2ϑ(θ1,0)]=E[2ϑθ21]+f2sinθ1,

    it may also be written as

    E[ϑ(θ1,η)]=f2ϑ(θ1,0))+αfα+1αN(α)f2sinθ1+αfα+1αN(α)E[2ϑθ21]. (4.4)

    Taking inverse ET on Eq (4.4), we get, we get

    ϑ(θ1,η)=cosθ1+sinθ1[αηαΓ(α+1)+(1α)]+E1[αfα+1αN(α)E{2ϑθ21}]. (4.5)

    Applying HPS on Eq (4.5), we get

    i=0piϑ(θ1,η)=cosθ1+sinθ1[αηαΓ(α+1)+(1α)]+E1[αfα+1αN(α)E{i=0pi2ϑθ21}]. (4.6)

    Equating p on both sides, we have

    p0:ϑ0(θ1,η)=cosθ1+sinθ1[αηαΓ(α+1)+(1α)],p1:ϑ1(θ1,η)=E1[αfα+1αN(α)E{2ϑ0θ21}],p2:ϑ2(θ1,η)=E1[αfα+1αN(α)E{2ϑ1θ21}],.
    ϑ0(θ1,η)=cosθ1+sinθ1[αηαΓ(α+1)+(1α)],ϑ1(θ1,η)=αηαΓ(α+1)cosθ1+(1α)cosθ1sinθ1[2η2αΓ(2α+1)+(1α2)ηαΓ(α+1)+(1α)2],ϑ2(θ1,η)=αη2αΓ(2α+1)cosθ1+(1α)ηαΓ(α+1)cosθ1+α(1α)ηαΓ(α+1)+(1α2)cosθ1+sinθ1[αη3αΓ(3α+1)+α(1α)η2αΓ(2α+1)+α(1α2)η2αΓ(2α+1)+(1α)(1α2)ηαΓ(α+1)+α(1α)2ηαΓ(α+1)+(1α)3],

    similarly proceeding this process, we can obtain this iteration series such as

    ϑ(θ1,η)=cosθ1+sinθ1[αηαΓ(α+1)+(1α)]αηαΓ(α+1)cosθ1+(1α)cosθ1sinθ1[αη2αΓ(2α+1)+(1α2)ηαΓ(α+1)+(1α)2]+αη2αΓ(2α+1)cosθ1+(1α)ηαΓ(α+1)cosθ1+α(1α)ηαΓ(α+1)+(1α2)cosθ1+sinθ1[αη3αΓ(3α+1)+α(1α)η2αΓ(2α+1)+α(1α2)η2αΓ(2α+1)+(1α)(1α2)ηαΓ(α+1)+α(1α)2ηαΓ(α+1)+(1α)3], (4.7)

    which provides the close contact at α=1 such that

    ϑ(θ1,η)=(1eη)sinθ1+eηcosθ1. (4.8)
    Table 1.  The EHPTS, exact and absolute error of ϑ(θ1,η) for Problem 1 at various value of θ1 with α=1 and η=0.01.
    θ1 EHPTS values Exact values Absolute error
    0.1 0.986099 0.986097 2×107
    0.2 0.972295 0.972292 3×106
    0.3 0.948776 0.948771 5×106
    0.4 0.915778 0.915771 7×106
    0.5 0.873629 0.873621 8×106
    0.6 0.822751 0.822742 9×106
    0.7 0.763653 0.763642 1.1×106
    0.8 0.696924 0.696912 1.2×105
    0.9 0.623232 0.623219 1.3×105
    1.0 0.543313 0.543299 8.6×104

     | Show Table
    DownLoad: CSV

    In Figure 1, we plot (a) surface solution for approximate results (b) surface solution for exact results. We indicate the performance of the EHPTS at α=1 with 1θ11 and 0η1 respectively. Figure 2 represents the graphical error between the approximate solution obtained by the EHPTS for (4.7) under ABC fractional derivative operators and the exact solutions for (4.8) at 0θ15 and η=0.5. We observe that both solutions are in close contact and present that EHPTS is extremely reliable and achieves the convenient findings.

    Figure 1.  The approximate and exact solution surface for one-dimensional equation.
    Figure 2.  Graphical error between the EHPTS and the exact results.

    Next, consider a two-dimensional fractional diffusion problem

    αϑηα=2ϑθ21+2ϑθ22ϑ, (4.9)

    with the initial condition

    ϑ(θ1,θ2,0)=sinθ1cosθ2, (4.10)

    and boundary condition

    ϑ(θ1,0,η)=ϑ(θ1,π,η)=e3ηsinθ1,ϑ(0,θ2,η)=ϑ(π,θ2,η)=0, (4.11)

    Taking ET on Eq (4.9), we get

    E[αϑηα]=E[2ϑθ21+2ϑθ22ϑ],

    Employing the differential properties of ET under ABC operator, we get

    N(α)αfα+1α[E[ϑ(θ1,η)]f2ϑ(θ1,0)]=E[2ϑθ21+2ϑθ22ϑ],

    it may also be written as

    E[ϑ(θ1,η)]=f2ϑ(θ1,0))+αfα+1αN(α)E[2ϑθ21+2ϑθ22ϑ], (4.12)

    Taking inverse ET on Eq (4.12), we get

    ϑ(θ1,η)=ϑ(θ1,0)+E1[αfα+1αN(α)E{2ϑθ21+2ϑθ22ϑ}]. (4.13)

    Applying HPS on Eq (4.13), we get

    i=0piϑ(θ1,η)=sinθ1cosθ2+E1[αfα+1αN(α)E{i=0pi2ϑθ21+i=0pi2ϑθ22i=0piϑ}].

    Equating p on both sides, we have

    p0:ϑ0(θ1,θ2,η)=ϑ(θ1,0,η),p1:ϑ1(θ1,θ2,η)=E1[αfα+1αN(α)E{2ϑ0θ21+2ϑ0θ22ϑ0}],p2:ϑ2(θ1,θ2,η)=E1[αfα+1αN(α)E{2ϑ1θ21+2ϑ1θ22ϑ1}],p3:ϑ3(θ1,θ2,η)=E1[αfα+1αN(α)E{2ϑ2θ21+2ϑ2θ22ϑ2}],p4:ϑ4(θ1,θ2,η)=E1[αfα+1αN(α)E{2ϑ3θ21+2ϑ3θ22ϑ3}],.
    ϑ0(θ1,θ2,η)=sinθ1cosθ2,ϑ1(θ1,θ2,η)=3sinθ1cosθ2[αηαΓ(α+1)+(1α)],ϑ2(θ1,θ2,η)=9sinθ1cosθ2[α2η2αΓ(2α+1)+2α(1α)ηαΓ(α+1)+(1α)2],ϑ3(θ1,θ2,η)=27sinθ1cosθ2[α3η3αΓ(3α+1)+3α2(1α)η2αΓ(2α+1)+3α(1α)2ηαΓ(α+1)+(1α)3],ϑ4(θ1,θ2,η)=81sinθ1cosθ2[α4η3αΓ(4α+1)+4α3(1α)η3αΓ(3α+1)+6α2(1α)2η2αΓ(2α+1)+2α(1α)3ηαΓ(α+1)+(1α)4],

    similarly proceeding this process, we can obtain this iteration series such as

    ϑ(θ1,θ2,η)=sin(θ1)cos(θ2)3sinθ1cosθ2[αηαΓ(α+1)+(1α)]+9sinθ1cosθ2[α2η2αΓ(2α+1)+2α(1α)ηαΓ(α+1)+(1α)2]27sinθ1cosθ2[α3η3αΓ(3α+1)+3α2(1α)η2αΓ(2α+1)+3α(1α)2ηαΓ(α+1)+(1α)3]+81sinθ1cosθ2[α4η3αΓ(4α+1)+4α3(1α)η3αΓ(3α+1)+6α2(1α)2η2αΓ(2α+1)+2α(1α)3ηαΓ(α+1)+(1α)4], (4.14)
    Table 2.  The EHPTS, exact and absolute error of ϑ(θ1,θ2,η) for Problem 2 at various value of θ1 with α=1 and θ2=0.1, η=0.1.
    θ1 EHPTS values Exact values Absolute error
    0.1 0.0735908 0.0735889 1.9×106
    0.2 0.146446 0.1046443 3×106
    0.3 0.217839 0.217833 6×106
    0.4 0.287054 0.287047 7×106
    0.5 0.353402 0.353393 9×106
    0.6 0.416219 0.416208 1.1×105
    0.7 0.474876 0.474864 1.2×105
    0.8 0.528789 0.528775 1.4×105
    0.9 0.577419 0.577404 1.5×105
    1.0 0.620279 0.620263 1.6×105

     | Show Table
    DownLoad: CSV

    which provides the close contact at α=1 such that

    ϑ(θ1,θ2,η)=e3ηsinθ1cosθ2. (4.15)

    In Figure 3, we plot (a) surface solution for approximate results (b) surface solution for exact results. We indicate the performance of the EHPTS at α=1 with 1θ11, θ2=0.1 and 0η0.5 respectively. Figure 4 represents the graphical error between the approximate solution obtained by the EHPTS for (4.14) under ABC fractional derivative operators and the exact solutions for (4.23) at 0θ15, θ2=0.5 and η=0.25,0.50,0.75 and 1. We observe that both solutions are in close contact and present that EHPTS is extremely reliable and achieves the convenient findings.

    Figure 3.  The approximate and exact solution surface for two-dimensional equation.
    Figure 4.  Graphical error between the EHPTS and the exact results.

    Finally, consider a three-dimensional fractional diffusion problem

    αϑηα=2ϑθ21+2ϑθ22+2ϑθ232ϑ, (4.16)

    with the initial condition

    ϑ(θ1,θ2,θ3,0)=sinθ1sinθ2sinθ3, (4.17)

    and boundary condition

    ϑ(0,θ2,θ3,η)=ϑ(π,θ2,θ3,η)=0,ϑ(θ1,0,θ3,η)=ϑ(θ1,π,θ3,η)=0,ϑ(θ1,θ2,0,η)=ϑ(θ1,θ2,π,η)=0, (4.18)

    Taking ET on Eq (4.16), we get

    E[αϑηα]=E[2ϑθ21+2ϑθ22+2ϑθ232ϑ],

    Employing the differential properties of ET under ABC operator, we get

    N(α)αfα+1α[E[ϑ(θ1,η)]f2ϑ(θ1,0)]=E[2ϑθ21+2ϑθ22+2ϑθ232ϑ], (4.19)

    it may also be written as

    E[ϑ(θ1,η)]=f2ϑ(θ1,0))+αfα+1αN(α)E[2ϑθ21+2ϑθ22+2ϑθ232ϑ], (4.20)

    Taking inverse ET on Eq (4.20), we get, we get

    ϑ(θ1,η)=ϑ(θ1,0)+E1[αfα+1αN(α)E{2ϑθ21+2ϑθ22+2ϑθ232ϑ}]. (4.21)

    Applying HPS on Eq (4.21), we get, we get

    i=0piϑ(θ1,η)=sinθ1sinθ2sinθ3+E1[αfα+1αN(α)E{i=0pi2ϑθ21+i=0pi2ϑθ22+i=0pi2ϑθ232i=0piϑ}].

    Equating p on both sides, we have

    p0:ϑ0(θ1,θ2,θ3,η)=ϑ(θ1,θ2,θ3,0),p1:ϑ1(θ1,θ2,θ3,η)=E1[αfα+1αN(α)E{2ϑ0θ21+2ϑ0θ22+2ϑ0θ232ϑ0}],p2:ϑ2(θ1,θ2,θ3,η)=E1[αfα+1αN(α)E{2ϑ1θ21+2ϑ1θ22+2ϑ1θ232ϑ1}],p3:ϑ3(θ1,θ2,θ3,η)=E1[αfα+1αN(α)E{2ϑ2θ21+2ϑ2θ22+2ϑ2θ232ϑ2}],p4:ϑ4(θ1,θ2,θ3,η)=E1[αfα+1αN(α)E{2ϑ3θ21+2ϑ3θ22+2ϑ3θ232ϑ3}],.
    ϑ0(θ1,θ2,θ3,η)=sinθ1sinθ2sinθ3,ϑ1(θ1,θ2,θ3,η)=5sinθ1sinθ2sinθ3[αηαΓ(α+1)+(1α)],ϑ2(θ1,θ2,θ3,η)=25sinθ1sinθ2sinθ3[α2η2αΓ(2α+1)+2α(1α)ηαΓ(α+1)+(1α)2],ϑ3(θ1,θ2,θ3,η)=125sinθ1sinθ2sinθ3[α3η3αΓ(3α+1)+3α2(1α)η2αΓ(2α+1)+3α(1α)2ηαΓ(α+1)+(1α)3],ϑ4(θ1,θ2,θ3,η)=625sinθ1sinθ2sinθ3[α4η3αΓ(4α+1)+4α3(1α)η3αΓ(3α+1)+6α2(1α)2η2αΓ(2α+1)+2α(1α)3ηαΓ(α+1)+(1α)4],

    similarly proceeding this process, we can obtain this iteration series such as

    ϑ(θ1,θ2,θ3,η)=sinθ1sinθ2sinθ35sinθ1sinθ2sinθ3[αηαΓ(α+1)+(1α)]+25sinθ1sinθ2sinθ3[α2η2αΓ(2α+1)+2α(1α)ηαΓ(α+1)+(1α)2]125sinθ1sinθ2sinθ3[α3η3αΓ(3α+1)+3α2(1α)η2αΓ(2α+1)+3α(1α)2ηαΓ(α+1)+(1α)3]+625sinθ1sinθ2sinθ3[α4η3αΓ(4α+1)+4α3(1α)η3αΓ(3α+1)+6α2(1α)2η2αΓ(2α+1)+2α(1α)3ηαΓ(α+1)+(1α)4], (4.22)
    Table 3.  The EHPTS, exact and absolute error of ϑ(θ1,θ2,θ3,η) for Problem 3 at various value of θ1 with α=1 and θ2=0.1, θ3=0.1 and η=0.0.5.
    x Approximate values Exact values Absolute error
    0.1 0.00077427 0.000774393 2.7×107
    0.2 0.0015408 0.00154105 2.5×107
    0.3 0.00229194 0.00229231 3.7×107
    0.4 0.00302018 0.00302066 4.8×107
    0.5 0.00371824 0.00371883 5.9×107
    0.6 0.00437915 0.00437985 7×107
    0.7 0.00499631 0.0049971 7.9×107
    0.8 0.00556354 0.00556443 8.9×107
    0.9 0.00607518 0.00607615 9.7×107
    1.0 0.00652613 0.00652717 1.04×106

     | Show Table
    DownLoad: CSV

    which provides the close contact at α=1 such that

    ϑ(θ1,θ2,θ3,η)=e5ηsinθ1sinθ2sinθ3. (4.23)

    In Figure 5, we plot (a) surface solution for approximate results at α=0.50,0.75,1 (b) surface solution for exact results. We indicate the performance of the EHPTS at α=1 with 0θ110, θ2=0.1, θ3=0.1 and 0η0.1 respectively. Figure 6 represents the graphical error between the approximate solution obtained by the EHPTS for (4.22) under ABC fractional derivative operators and the exact solutions for (4.23) at 0θ110, θ2=0.1, θ3=0.1 and η=0.25,0.50,0.75 and 1. We observe that both solutions are in close contact and present that EHPTS is extremely reliable and achieves the convenient findings.

    Figure 5.  The approximate and exact solution surface for three-dimensional equation.
    Figure 6.  Graphical error between the EHPTS and the exact results.

    This paper presents the study of EHPTS for obtaining the approximate solution of multi-dimensional diffusion problems under ABC fractional order derivative. In addition, HPS produces successive iterations and shows the results in the form of a series. This strategy does not involve rectified constants, steady constraints, or massive integrals due to the noise-free results. Some examples are carried out to provide the efficiency of EHPTS and showed the results in better obligations towards the precise results. We compute the values of iterations and graphical results using the Mathematica software 11. The physical solutions behavior of the graphical representation and plot distribution yield that EHPTS is a very powerful and efficient method to produce the approximate solution of partial differential equations that arise in science and engineering. This method evaluates and controls the series of solutions that quickly arrive at the precise solution in a condensed acceptable domain. In future, we consider the strategy of EHPTS for other fractional differential problems and compete with other exceedingly fractional order systems of equations.

    Hamid.M. Sedighi is grateful to the Research Council of Shahid Charman University of Ahvaz, Iran. This paper is supported by (Grant No. SCU.EM1401.98) and Natural Science Foundation of Shaanxi Provincial Department of Education in 2022 (22JK0437).

    The authors declare there is no conflict of interest.



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