Research article

Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations

  • These authors contributed equally to this work and are co-first authors
  • Received: 21 September 2022 Revised: 05 November 2022 Accepted: 17 November 2022 Published: 14 December 2022
  • MSC : 33B15, 34A34, 35A20, 35A22, 44A10

  • This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.

    Citation: M. Mossa Al-Sawalha, Osama Y. Ababneh, Rasool Shah, Nehad Ali Shah, Kamsing Nonlaopon. Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations[J]. AIMS Mathematics, 2023, 8(3): 5266-5280. doi: 10.3934/math.2023264

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  • This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.



    Due to its well-established applications in various scientific and technical fields, fractional calculus has gained prominence during the last three decades. Many pioneers have shown that when adjusted by integer-order models, fractional-order models may accurately represent complex events [1,2]. The Caputo fractional derivatives are nonlocal in contrast to the integer-order derivatives, which are local in nature [1]. In other words, the integer-order derivative may be used to analyze changes in the area around a point, but the Caputo fractional derivative can be used to analyze changes in the whole interval. Senior mathematicians including Riemann [4], Caputo [5], Podlubny [6], Ross [7], Liouville [8], Miller and others, collaborated to create the fundamental foundation for fractional order integrals and derivatives. The theory of fractional-order calculus has been related to real-world projects, and it has been applied to chaos theory [9], signal processing [10], electrodynamics [11], human diseases [12,13], and other areas [14,15,16].

    Due to the numerous applications of fractional differential equations in engineering and science such as electrodynamics [17], chaos ideas [18], accounting [19], continuum and fluid mechanics [20], digital signal [21] and biological population designs [22] fractional differential equations are now more widely known. For such issues to be resolved, efficient tools are needed [23,24,25]. Because of this, we will attempt to apply an efficient analytical technique to solve nonlinear arbitrary order differential equations in this article. Many strategies in collaboration fields may be delightfully and even more accurately analyzed using fractional differential equations. Various strategies have been developed in this regard, some of them are as follows, such as the fractional Reduced differential transformation technique [26], Adomian decomposition technique [27], the fractional Variational iteration technique [28], Elzaki decomposition technique [29,30], iterative transformation technique [31], the fractional natural decomposition method (FNDM) [32], and the fractional homotopy perturbation method [33].

    The power series solution is used to solve some classes of the differential and integral equations of fractional or non-fractional order, and it is based on assuming that the solution of the equation can be expanded as a power series. RPS is an easy and fast technique for determining the coefficients of the power series solution. The Jordanian mathematician Omar Abu Arqub created the residual power series method in 2013, as a technique for quickly calculating the coefficients of the power series solutions for 1st and 2nd-order fuzzy differential equations [34]. Without perturbation, linearization, or discretization, the residual power series method provides a powerful and straightforward power series solution for highly linear and nonlinear equations [35,36,37,38]. The residual power series method has been used to solve an increasing variety of nonlinear ordinary and partial differential equations of various sorts, orders, and classes during the past several years. It has been used to make non-linear fractional dispersive partial differential equation have solitary pattern results and to predict them [39], to solve the highly nonlinear singular differential equation known as the generalized Lane-Emden equation [40], to solve higher-order ordinary differential equations numerically [41], to approximate solve the fractional nonlinear KdV-Burger equations, to predict and represent the RPSM differs from several other analytical and numerical approaches in some crucial ways [42]. First, there is no requirement for a recursion connection or for the RPSM to compare the coefficients of the related terms. Second, by reducing the associated residual error, the RPSM offers a straightforward method to guarantee the convergence of the series solution. Thirdly, the RPSM doesn't suffer from computational rounding mistakes and doesn't use a lot of time or memory. Fourth, the approach may be used immediately to the provided issue by selecting an acceptable starting guess approximation since the residual power series method does not need any converting when transitionary from low-order to higher-order and from simple linearity to complicated nonlinearity [43,44,45]. The process of solving linear differential equations using the LT method consists of three steps. The first step depends on transforming the original differential equation into a new space, called the Laplace space. In the second step, the new equation is solved algebraically in the Laplace space. In the last step, the solution in the second step is transformed back into the original space, resulting in the solution of the given problem.

    In this article, we apply the Laplace residual power series method to achieve the definitive solution of the fractional-order nonlinear partial differential equations. The Laplace transformation efficiently integrates the residual power series method for the renewability algorithmic technique. This proposed technique produces interpretive findings in the sense of a convergent series. The Caputo fractional derivative operator explains quantitative categorizations of the partial differential equations. The offered methodology is well demonstrated in modelling and enumeration investigations. The exact-analytical findings are a valuable way to analyze the problematic dynamics of systems, notably for computational fractional partial differential equations.

    Definition 2.1. The fractional Caputo derivative of a function u(ζ,t) of order α is given as [46]

    CDαtu(ζ,t)=Jmαtum(ζ,t),m1<αm,t>0, (2.1)

    where mN and Jαt is the fractional integral Riemann-Liouville (RL) of u(ζ,t) of order α is given as

    Jσtu(ζ,t)=1Γ(α)t0(tτ)α1u(φ,τ)dτ (2.2)

    Definition 2.2. The Laplace transformation (LT) of u(ζ,t) is given as [46]

    u(ζ,s)=Lt[u(ζ,t)]=0estu(ζ,t)dt,s>α, (2.3)

    where the Laplace transform inverse is defined as

    u(ζ,t)=L1t[u(ζ,s)]=l+iliestu(ζ,s)ds,l=Re(s)>l0. (2.4)

    Lemma 2.1. Suppose that u(ζ,t) is piecewise continue term and U(ζ,s)=Lt[u(ζ,t)], we get

    (1) Lt[Jαtu(ζ,t)]=U(ζ,s)sα,α>0.

    (2) Lt[Dαtu(ζ,t)]=sσU(ζ,s)m1k=0sαk1uk(ζ,0),m1<αm.

    (3) Lt[Dnαtu(ζ,t)]=snαU(ζ,s)n1k=0s(nk)α1Dkαtu(ζ,0),0<α1.

    Proof. For proof see Refs. [46].

    Theorem 2.1. Let u(ζ,t) be a piecewise continuous function on I×[0,) with exponential order ζ. Assume that the fractional expansion of the function U(ζ,s)=Lt[u(ζ,t)] is as follows:

    U(ζ,s)=n=0fn(ζ)s1+nα,0<α1,ζI,s>ζ. (2.5)

    Then, fn(ζ)=Dnσtu(ζ,0).

    Proof. For proof see Refs. [46].

    Remark 2.1. The inverse Laplace transform of the Eq (2.5) is represented as [46]

    u(ζ,t)=i=0Dαtu(ζ,0)Γ(1+iα)ti(ζ),0<ζ1,t0. (2.6)

    Consider the fractional order partial differential equation,

    DαtU(ζ,t)+3U(ζ,t)tζ24U(ζ,t)t2ζ2+4U(ζ,t)ζ4+a(2U(ζ,t)ζ2)2b(2U(ζ,t)t2)3+cU(ζ,t)=0. (3.1)

    Applying LT of Eq (3.1), we get

    U(ζ,s)+f0(ζ,s)s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2U(ζ,s)ζ2))2bL(L1t(2U(ζ,s)t2))3+cU(ζ,s)]=0. (3.2)

    Suppose that the result of Eq (3.2), we get

    U(ζ,s)=n=0fn(ζ,s)snα+1. (3.3)

    The kth-truncated term series are

    U(ζ,s)=f0(ζ,s)s+kn=1fn(ζ,s)snα+1,k=1,2,3,4. (3.4)

    Residual Laplace function (RLF) is given as

    LtResu(ζ,s)=U(ζ,s)+f0(ζ,s)s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2U(ζ,s)ζ2))2bL(L1t(2U(ζ,s)t2))3+cU(ζ,s)]. (3.5)

    And the kth-LRFs as

    LtResk(ζ,s)=Uk(ζ,s)+f0(ζ,s)s+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2Uk(ζ,s)ζ2))2bL(L1t(2Uk(ζ,s)t2))3+cUk(ζ,s)]. (3.6)

    To illustrate a few facts, the following LRPSM features are provided:

    (1) LtRes(ζ,s)=0 and limjLtResk(ζ,s)=LtResu(ζ,s) for each s>0.

    (2) limssLtResu(ζ,s)=0limssLtResu,k(ζ,s)=0.

    (3) limsskα+1LtResu,k(ζ,s)=limsskα+1LtResu,k(ζ,s)=0,0<α1,k=1,2,3,.

    To calculate the coefficients using fn(ζ,s), gn(ζ,s), hn(ζ,s) and ln(ζ,s), the following system is recursively solved:

    limsskα+1LtResu,k(α,s)=0,k=1,2,. (3.7)

    In finally inverse Laplace transform to Eq (3.4), to get the kth analytical result of uk(ζ,t).

    Example 4.1. Consider the fractional partial differential equations [47],

    Dαtu(ζ,t)3u(ζ,t)tζ24u(ζ,t)t2ζ2+4u(ζ,t)ζ4+19(2u(ζ,t)ζ2)21216(2u(ζ,t)t2)3+16u(ζ,t)=0, where 2<α3, (4.1)

    with the following IC's:

    u(x,0)=ζ4,tu(ζ,0)=0,2t2u(ζ,0)=0. (4.2)

    Using Laplace transform to Eq (4.1), we get

    U(ζ,s)+ζ4s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2U(ζ,s)ζ2))21216L(L1t(2U(ζ,s)t2))3+16U(ζ,s)]=0, (4.3)

    and so the kth-truncated term series are

    ζu(ζ,s)=ζ4s+kn=1fn(ζ,s)snα+1,k=1,2,3,4. (4.4)

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)+ζ4s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2U(ζ,s)ζ2))21216L(L1t(2U(ζ,s)t2))3+16U(ζ,s)], (4.5)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)+ζ4s+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2Uk(ζ,s)ζ2))21216L(L1t(2Uk(ζ,s)t2))3+16Uk(ζ,s)]. (4.6)
    Table 1.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.25 for Example 4.1.
    ζ α=2.5 α=2.7 α=2.9 α=3 HPM [47] Exact
    0 0.0222397 0.0111683 0.00553658 0.00388069 0.00388069 0.0038812
    0.2 0.0206397 0.00956834 0.00393658 0.00228069 0.00228069 0.0022812
    0.4 -0.00336028 -0.0144317 -0.0200634 -0.0217193 -0.0217193 -0.0217188
    0.6 -0.10736 -0.118432 -0.124063 -0.125719 -0.125719 -0.125719
    0.8 -0.38736 -0.398432 -0.404063 -0.405719 -0.405719 -0.405719
    1.0 -0.97776 -0.988832 -0.994463 -0.996119 -0.996119 -0.996119

     | Show Table
    DownLoad: CSV

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.4) into the kth residual Laplace term Eq (4.6), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=24,f2(ζ,s)=384,f3(ζ,s)=6144. (4.7)

    Putting the value of fk(ζ,s), k=1,2,3,, in Eq (4.4), we get

    U(ζ,s)=ζ4s+24sα+1384s2α+1+6144s3α+1+. (4.8)

    Using inverse LT, we get

    u(ζ,t)=ζ4+24tαΓ(α+1)384t2αΓ(2α+1)+6144t3αΓ(3α+1)+, (4.9)

    and the exact solution are

    u=ζ4+4t3. (4.10)

    In Figure 1, the exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3 of Example 4.1. In Figure 2, analytical solution for u(ζ,t) at different value of α=2.8 and 2.6 at ζ and t=0.3. In Figure 3, analytical solution for u(ζ,t) at various value of α at t=0.3 of Example 4.1.

    Figure 1.  The actual and LRPSM results for u(ζ,t) at α=3 at ζ and t=0.3.
    Figure 2.  Analytical solution for u(ζ,t) at different value of α=2.8 and 2.6 at ζ and t=0.3.
    Figure 3.  Analytical solution for u(ζ,t) at various value of α at t=0.3.

    Example 4.2. Consider the fractional partial differential equations [47]:

    DαtU(ζ,t)3U(ζ,t)tζ24U(ζ,t)t2ζ2+4U(ζ,t)ζ4+(2U(ζ,t)ζ2)2(2U(ζ,t)t2)2+2U2(ζ,t)=0, where 2<α3, (4.11)

    with the following IC's:

    U(ζ,0)=eζ,tU(ζ,0)=eζ,2t2U(ζ,0)=eζ. (4.12)

    Using Laplace transform to Eq (4.11), we get

    U(ζ,s)eζseζs2eζs3+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+Lt(L1t(2U(ζ,s)ζ2))2Lt(L1t(2U(ζ,s)t2))2+2Lt(L1t(U(ζ,s)))2]=0. (4.13)
    Table 2.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.099 for Example 4.2.
    ζ α=2.5 α=2.7 α=2.9 α=3 Exact
    0 1.08911 1.09052 1.09122 1.09142 1.09199
    0.2 1.32968 1.33169 1.33268 1.33297 1.33376
    0.4 1.62325 1.62612 1.62754 1.62795 1.62905
    0.6 1.98139 1.98553 1.98759 1.98818 1.98973
    0.8 2.41823 2.42422 2.42721 2.42807 2.43026
    1 2.95086 2.95959 2.96394 2.96519 2.96833

     | Show Table
    DownLoad: CSV

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)eζseζs2eζs3+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+Lt(L1t(2U(ζ,s)ζ2))2Lt(L1t(2U(ζ,s)t2))2+2Lt(L1t(U(ζ,s)))2], (4.14)

    and so the kth-truncated term series are

    u(ζ,s)=eζs+eζs2+eζs3+kn=1fn(ζ,s)snα+1,k=1,2,3,4, (4.15)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)eζseζs2eζs3+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4Uk(ζ,s)ζ4+Lt(L1t(2Uk(ζ,s)ζ2))2Lt(L1t(2Uk(ζ,s)t2))2+2Lt(L1t(Uk(ζ,s)))2]. (4.16)

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.15) into the kth residual Laplace term Eq (4.16), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=(eζ+3e2ζ),f2(ζ,s)=eζ+54e2ζ+36e3ζ,f3(ζ,s)=(eζ+870e2ζ+3564e3ζ+792e4ζ).   (4.17)

    Putting the value of fk(ζ,s), k=1,2,3,, in Eq (4.15), we get

    U(ζ,s)=eζs+eζs2+eζs3eζ+3e2ζsα+1eζ+54e2ζ+36e3ζs2α+1eζ+870e2ζ+3564e3ζ+792e4ζs3α+1+. (4.18)

    Using inverse LT, we get

    u(ζ,t)=eζ+eζt+eζt2(eζ+3e2ζ)tαΓ(α+1)+(eζ+54e2ζ+36e3ζ)t2αΓ(2α+1)+(eζ+870e2ζ+3564e3ζ+792e4ζ)t3αΓ(3α+1)+, (4.19)

    and the exact solution are

    u=eζ+t. (4.20)

    In Figure 4, the exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3 of Example 4.2. In Figure 5, LRPSM solutions for u(ζ,t) at α=2.5 and α=2.8 and t=0.3 of Example 4.2.

    Figure 4.  Exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3.
    Figure 5.  LRPSM solutions for u(ζ,t) at α=2.5 and α=2.8 and t=0.3.

    Example 4.3. Consider the fractional partial differential equations [47]:

    Dαtu(ζ,t)3u(ζ,t)tζ24u(ζ,t)t2x2+4u(ζ,t)ζ4(2u(ζ,t)t2)(u(ζ,t)ζ)u(ζ,t)(u(ζ,t)t)=0, where 2<α3, (4.21)

    with the following IC's:

    U(ζ,0)=cosζ,tU(ζ,0)=sinζ,2t2U(ζ,0)=cosζ. (4.22)
    Table 3.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.22 for Example 4.3.
    ζ α=2.5 α=2.7 α=2.9 α=3 Exact
    0 0. 0.97178 0.973463 0.974025 0.975897
    0.2 -0.04370738 0.908702 0.910351 0.910903 0.913089
    0.4 -0.085672 0.809397 0.810946 0.811465 0.813878
    0.6 -0.124221 0.677823 0.679212 0.679677 0.682221
    0.8 -0.157818 0.519227 0.5204 0.520792 0.523366
    1 -0.185124 0.339931 0.34084 0.341145 0.343646

     | Show Table
    DownLoad: CSV

    Using Laplace transform to Eq (4.21), we get

    U(ζ,s)cosζs+sinζs2+cosζs3+1sσ[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+L(L1t(2u(ζ,s)t2)L1t(u(ζ,s)ζ))L(L1t(U(ζ,s))L1t(U(ζ,s)t))]=0. (4.23)

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)cosζs+sinζs2+cosζs3+1sσ[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+L(L1t(2u(ζ,s)t2)L1t(u(ζ,s)ζ))L(L1t(U(ζ,s))L1t(U(ζ,s)t))], (4.24)

    and so the kth-truncated term series are

    u(ζ,s)=cosζs+sinζs2+cosζs3+kn=1fn(ζ,s)snα+1,k=1,2,3,4, (4.25)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)cosζs+sinζs2+cosζs3+1sα[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4Uk(ζ,s)ζ4+Lt(L1t(2Uk(ζ,s)t2)L1t(Uk(ζ,s)ζ))Lt(L1t(Uk(ζ,s))L1t(Uk(ζ,s)t))]. (4.26)

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.25) into the kth residual Laplace term Eq (4.26), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=cosζ,f2(ζ,s)=cosζ,f3(ζ,s)=cosζ. (4.27)

    Putting the value of fk(x,s), k=1,2,3,, in Eq (4.25), we get

    U(ζ,s)=cosζssinζs2cosζs3cosζsα+1+cosζs2α+1cosζs3α+1+. (4.28)

    Using inverse LT, we get

    u(ζ,t)=cosζtsinζt2cosζ2tαcosζΓ(α+1)+t2αcosζΓ(2α+1)t3αcosζΓ(3α+1)+, (4.29)

    and the exact solution are

    u=cos(ζ+t). (4.30)

    In Figure 6, exact and LRPSM solutions for u(ζ,t) at α=3 and t=0.3 of Example 4.3. Figure 7, LRPSM solutions for u(ζ,t) at α=2.5, α=2.8, and t=0.3.

    Figure 6.  Exact and LRPSM solutions for u(ζ,t) at α=3 at and t=0.3.
    Figure 7.  LRPSM solutions for u(ζ,t) at α=2.5 α=2.8, and t=0.3.

    In this article, the fractional partial differential equation has been solved analytically by employing the Laplace residual power series method in conjunction with the Caputo operator. To demonstrate the validity of the recommended method, we analyzed three distinct partial differential equation problems. The simulation results demonstrate that the outcomes of our method are in close accordance with the exact answer. The new method is highly straightforward, efficient, and suitable for getting numerical solutions to partial differential equations. The primary advantage of the proposed approach is the series form solution, which rapidly converges to the exact answer. We can therefore conclude that the suggested approach is quite methodical and efficient for a more thorough investigation of fractional-order mathematical models.

    The authors declare no conflicts of interest.



    [1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, In: North-Holland mathematics studies, Elsevier, 204 (2006), 1–523.
    [2] D. Baleanu, Z. B. Guvenc, J. A. Tenreiro Machado, New trends in nanotechnology and fractional calculus applications, New York: Springer, 2010.
    [3] M. Alqhtani, K. M. Saad, R. shah, W. Weera, W. M. Hamanah, Analysis of the fractional-order local poisson equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
    [4] B. Riemann, Versuch einer allgemeinen auffassung der integration und differentiation, Cambridge: Cambridge University Press, 2014.
    [5] A. S. Alshehry, R. Shah, N. A. Shah, I. Dassios, A reliable technique for solving fractional partial differential equation, Axioms, 11 (2022), 574. https://doi.org/10.3390/axioms11100574 doi: 10.3390/axioms11100574
    [6] I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
    [7] K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, Wiley-Interscience, 1993.
    [8] J. Liouville, Memoire sur quelques questions de geometrie et de mecanique, et sur un nouveaugenre de calcul pour resoudre ces questions, J. Ecole Polytech., 1832, 1–69.
    [9] W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709–726.
    [10] Y. L. Li, F. W. Liu, I. W. Turner, T. Li, Time-fractional diffusion equation for signal smoothing, Appl. Math. Comput., 326 (2018), 108–116. https://doi.org/10.1016/j.amc.2018.01.007 doi: 10.1016/j.amc.2018.01.007
    [11] H. Nasrolahpour, A note on fractional electrodynamics, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2589–2593. https://doi.org/10.1016/j.cnsns.2013.01.005 doi: 10.1016/j.cnsns.2013.01.005
    [12] P. Veeresha, D. G. Prakasha, H. M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method, Math. Sci., 13 (2019), 115–128. https://doi.org/doi:10.1007/s40096-019-0284-6 doi: 10.1007/s40096-019-0284-6
    [13] D. G. Prakasha, P. Veeresha, H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 134 (2019), 241. http://doi.org/10.1140/epjp/i2019-12590-5 doi: 10.1140/epjp/i2019-12590-5
    [14] L. Akinyemi, K. S. Nisar, C. A. Saleel, H. Rezazadeh, P. Veeresha, M. M. Khater, et al., Novel approach to the analysis of fifth-order weakly nonlocal fractional Schrodinger equation with Caputo derivative, Results Phys., 31 (2021), 104958. https://doi.org/10.1016/j.rinp.2021.104958 doi: 10.1016/j.rinp.2021.104958
    [15] L. M. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstr. Appl. Anal., 2013 (2013), 465160. https://doi.org/10.1155/2013/465160 doi: 10.1155/2013/465160
    [16] D. Ntiamoah, W. Ofori-Atta, L. Akinyemi, The higher-order modified Korteweg-de Vries equation: its soliton, breather and approximate solutions, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.06.042 doi: 10.1016/j.joes.2022.06.042
    [17] K. Nonlaopon, M. Naeem, A. M. Zidan, R. Shah, A. Alsanad, Numerical investigation of the time-fractional Whitham-Broer-Kaup equation involving without singular kernel operators, Complexity, 2021, (2021), 7979365. https://doi.org/10.1155/2021/7979365 doi: 10.1155/2021/7979365
    [18] D. Baleanu, G. C. Wu, S. D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals, 102 (2017), 99–105. https://doi.org/10.1016/j.chaos.2017.02.007 doi: 10.1016/j.chaos.2017.02.007
    [19] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A, 284 (2000), 376–384. https://doi.org/10.1016/S0378-4371(00)00255-7 doi: 10.1016/S0378-4371(00)00255-7
    [20] M. M. Al-Sawalha, A. S. Alshehry, K. Nonlaopon, R. Shah, O. Y. Ababneh, Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm, AIMS Mathematics, 7 (2022), 19739–19757. https://doi.org/10.3934/math.20221082 doi: 10.3934/math.20221082
    [21] J. M. Cruz-Duarte, J. Rosales-Garcia, C. R. Correa-Cely, A. Garcia-Perez, J. G. Avina-Cervantes, A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications, Commun. Nonlinear Sci. Numer. Simul., 61 (2018), 138–148. https://doi.org/10.1016/j.cnsns.2018.01.020 doi: 10.1016/j.cnsns.2018.01.020
    [22] M. M. Al-Sawalha, K. Nonlaopon, I. Khan. Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel, AIMS Mathematics, 8 (2023), 3730–3746. https://doi.org/10.3934/math.2023186 doi: 10.3934/math.2023186
    [23] M. M. Al-Sawalha, O. Y. Ababneh, R. Shah, A. Khan, K. Nonlaopon, Numerical analysis of fractional-order Whitham-Broer-Kaup equations with non-singular kernel operators, AIMS Mathematics, 8 (2023), 2308–2336. https://doi.org/10.3934/math.2023120 doi: 10.3934/math.2023120
    [24] M. M. Al-Sawalha, N. Amir, R. Shah, M. Yar, Novel analysis of fuzzy fractional Emden-Fowler equations within new iterative transform method, J. Funct. Spaces, 2022 (2022), 7731135. https://doi.org/10.1155/2022/7731135 doi: 10.1155/2022/7731135
    [25] M. M. Al-Sawalha, A. S. Alshehry, K. Nonlaopon, R. Shah, O. Y. Ababneh, Fractional view analysis of delay differential equations via numerical method, AIMS Mathematics, 7 (2022), 20510–20523. https://doi.org/10.3934/math.20221123 doi: 10.3934/math.20221123
    [26] Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 741–749. http://doi.org/10.1515/IJNSNS.2009.10.6.741 doi: 10.1515/IJNSNS.2009.10.6.741
    [27] S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488–494. https://doi.org/10.1016/j.amc.2005.11.025 doi: 10.1016/j.amc.2005.11.025
    [28] G. C. Wu, A fractional variational iteration method for solving fractional nonlinear differential equations, Comput. Math. Appl., 61 (2011), 2186–2190. https://doi.org/10.1016/j.camwa.2010.09.010 doi: 10.1016/j.camwa.2010.09.010
    [29] M. K. Alaoui, K. Nonlaopon, A. M. Zidan, A. Khan, R. Shah, Analytical investigation of fractional-order cahn-hilliard and gardner equations using two novel techniques, Mathematics, 10 (2022), 1643. https://doi.org/10.3390/math10101643 doi: 10.3390/math10101643
    [30] N. A. Shah, Y. S. Hamed, K. M. Abualnaja, J. D. Chung, R. Shah, A. Khan, A comparative analysis of fractional-order kaup-kupershmidt equation within different operators, Symmetry, 14 (2022), 986. https://doi.org/10.3390/sym14050986 doi: 10.3390/sym14050986
    [31] M. Alqhtani, K. M. Saad, R. Shah, W. Weera, W. M. Hamanah, Analysis of the fractional-order local poisson equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
    [32] D. G. Prakasha, P. Veeresha, M. S. Rawashdeh, Numerical solution for (2+1)-dimensional time-fractional coupled Burger equations using fractional natural decomposition method, Math. Methods Appl. Sci., 42 (2019), 3409–3427. https://doi.org/10.1002/mma.5533 doi: 10.1002/mma.5533
    [33] N. H. Aljahdaly, R. Shah, M. Naeem, M. A. Arefin, A comparative analysis of fractional space-time advection-dispersion equation via semi-analytical methods, J. Funct. Spaces, 2022 (2022), 4856002. https://doi.org/10.1155/2022/4856002 doi: 10.1155/2022/4856002
    [34] O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52. http://doi.org/10.5373/jaram.1447.051912 doi: 10.5373/jaram.1447.051912
    [35] V. N. Kovalnogov, R. V. Fedorov, D. A. Generalov, E. V. Tsvetova, T. E. Simos, C. Tsitouras, On a new family of Runge-Kutta-Nystrom pairs of orders 6(4), Mathematics, 10 (2022), 875. https://doi.org/10.3390/math10060875 doi: 10.3390/math10060875
    [36] K. Liu, Z. X. Yang, W. F. Wei, B. Gao, D. L. Xin, C. M. Sun, et al., Novel detection approach for thermal defects: study on its feasibility and application to vehicle cables, High Volt., 2022. https://doi.org/10.1049/hve2.12258 doi: 10.1049/hve2.12258
    [37] X. Gong, L. X. Wang, Y. Y. Mou, H. L. Wang, X. Q. Wei, W. F. Zheng, et al., Improved four-channel PBTDPA control strategy using force feedback bilateral teleoperation system, Internat. J. Control, 20 (2022), 1002–1017. http://doi.org/10.1007/s12555-021-0096-y doi: 10.1007/s12555-021-0096-y
    [38] L. Liu, J. Wang, L. C. Zhang, S. Zhang, Multi-AUV dynamic maneuver countermeasure algorithm based on interval information game and fractional-order DE, Fractal Fract., 6 (2022), 235. https://doi.org/10.3390/fractalfract6050235 doi: 10.3390/fractalfract6050235
    [39] O. A. Arqub, A. El-Ajou, S. Momani, Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, J. Comput. Phys., 293 (2015), 385–399. https://doi.org/10.1016/j.jcp.2014.09.034 doi: 10.1016/j.jcp.2014.09.034
    [40] O. A. Arqub, A. El-Ajou, A. S. Bataineh, I. Hashim, A representation of the exact solution of generalized Lane-Emden equations using a new analytical method, Abstr. Appl. Anal., 2013 (2013), 378593. https://doi.org/10.1155/2013/378593 doi: 10.1155/2013/378593
    [41] O. A. Arqub, Z. Abo-Hammour, R. Al-Badarneh, S. Momani, A reliable analytical method for solving higher-order initial value problems, Discrete Dyn. Nat. Soc., 2013 (2013), 673829. https://doi.org/10.1155/2013/673829 doi: 10.1155/2013/673829
    [42] A. El-Ajou, O. A. Arqub, S. Momani, D. Baleanu, A. Alsaedi, A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput., 257 (2015), 119–133. https://doi.org/10.1016/j.amc.2014.12.121 doi: 10.1016/j.amc.2014.12.121
    [43] S. Mukhtar, R. Shah, S. Noor, The numerical investigation of a fractional-order multi-dimensional model of Navier-Stokes equation via novel techniques, Symmetry, 14 (2022), 1102. https://doi.org/10.3390/sym14061102 doi: 10.3390/sym14061102
    [44] M. M. Al-Sawalha, R. P. Agarwal, R. Shah, O.Y. Ababneh, W. Weera, A reliable way to deal with fractional-order equations that describe the unsteady flow of a polytropic gas, Mathematics, 10 (2022), 2293. https://doi.org/10.3390/math10132293 doi: 10.3390/math10132293
    [45] N. A. Shah, H. A. Alyousef, S. A. El-Tantawy, R. Shah, J. D. Chung, Analytical investigation of fractional-order Korteweg-de-Vries-type equations under Atangana-Baleanu-Caputo operator: modeling nonlinear waves in a plasma and fluid, Symmetry, 14 (2022), 739. https://doi.org/10.3390/sym14040739 doi: 10.3390/sym14040739
    [46] A. El-Ajou, Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, Eur. Phys. J. Plus, 136 (2021), 229.
    [47] A. Roozi, E. Alibeiki, S. S. Hosseini, S. M. Shafiof, M. Ebrahimi, Homotopy perturbation method for special nonlinear partial differential equations, J. King Saud Univ. Sci., 23 (2011), 99–103. https://doi.org/10.1016/j.jksus.2010.06.014 doi: 10.1016/j.jksus.2010.06.014
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