
Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.
Citation: Humaira Yasmin, Aljawhara H. Almuqrin. Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies[J]. AIMS Mathematics, 2024, 9(6): 16721-16752. doi: 10.3934/math.2024811
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Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.
Fractional calculus (FC) is a subfield of mathematics that deals with integrals and derivatives of complex or arbitrary orders [1,2,3,4,5], also being named non-Newtonian and extended calculus. L'Hospital questioned the implications of derivative of order 1/2 [6,7,8]. The answer from Leibniz on September 30, 1695, is recognized as the origin of the non-Newtonian calculus. Fractional calculus is beneficial in many scientific fields, including engineering, physics, signal and fiber optics, chemistry, biology, control theory, viscoelasticity, solid-state and stochastic finance, and economics, to mention a few [9,10,11,12]. Although there are numerous fractional definitions of derivatives, not all of them have gained widespread use. The Atangana-Baleanu fractional derivative, Caputo-Fabrizio fractional derivative, conformable operators, Riemann-Liouville (R-L) derivative, and Caputo fractional derivative are the most well-known [13,14,15]. When describing complex systems with high-order dynamics and sophisticated nonlinear processes, it may be better to use fractional than integer-order derivatives. Two primary characteristics are explain for this. First, fractional derivatives can have any order, not just integral [16,17,18]. Second, the non-integer order derivative is useful, as it allows the system to retain a long-lasting memory, incorporating information from both the past and the present [19,20].
In 1885, Adolf Fick described his laws of dispersion. Later, Fick's second law was dubbed the diffusion equation. Diffusion is the system by which atoms or molecules slowly shift from high to low chemical potential areas. Other researchers have extended the classical diffusion and wave equations to describe other physical systems slow diffusion, the classical wave equations, and the diffusion-wave hybrid [21]. Electrochemistry, acoustics, electromagnetic and phase phenomena, filtrations, organic chemistry, bacteriology, cosmology, and system dynamics are among the many areas where the diffusion equation is commonly used [22]. Ficks's second law of dispersion applies the spread of substances in a medium in response to chemical potential gradients. One or more variable's comparative value forms a gradient, which are expressed as measures of one or more variables, primarily independent ones [23]. A variety of factors may contribute to the formation of entropy in engineering systems. As investigated in an influential series of papers by Bejan and colleagues [24], mass transfer, heat transfer, viscous dissipation, thermal coupling, electrical conduction, and chemical reactions are the primary mechanisms by which entropy is produced in thermal systems. The collocation method (CM) [25], diffusion and Tsallis entropy [26], entropy production, symmetric fractional diffusion [27], and many other techniques have been used for the solution of the diffusion equation [28,29,30,31,32].
In 1822, the heat equation was formulated by Joseph Fourier. The relationship between the heat equation and Brownian motion. Particles in suspension in a gas or fluid generate wavelike motion when they collide with the fluid's fast-moving atoms. The transferred particles' dynamic energy is known as heat. When thermal energy is transferred from a warmer object to a cooler one, this causes the atoms in the latter to move faster and collide with its boundaries. This transfers some of this energy to the system's atoms, causing them to move faster [33]. In classical physics, the wave equation describes waves in terms of fluid dynamics, mechanical waves, and the electromagnetic wave equation, which is a significant second-order linear PDE in acoustics, electromagnetics, and fluid dynamics. Light, sound, gravity, and matter (including the Klein-Gordon equation in relativistic quantum physics) are all examples of phenomena governed by this equation. Alembert and Euler discovered the wave equation in one and three dimensions. Numerous mathematicians have contributed to solutions heat and wave equations, employing methods such as the optimal homotopy asymptotic method (OHAM) [34], the modified Adomian decomposition method (MADM) [35], and many others [36,37,38,39].
The Galerkin method was used by R. Mittal and P. Singhal [40] to find numerical solutions to Burgers' equation. In light of this, scientists have worked very hard to solve the fractional Burgers' equation analytically and/or numerically. A. Esen and O. Tasbozan [41] provided a numerical solution of the time-fractional Burgers' equation using cubic B-spline finite elements as one example. A. Esen et al. [42] used that homotopy analysis method (HAM) to solve the time-fractional Burgers' equation analytically. Additionally, E. Abdel Salam et al. [43] solved the space-time fractional Burgers' equation using the fractional Riccati expansion technique. M. Inc. solved the space-time-fractional Burgers' equations using a variational iteration approach [44].
Residual power series method (RPSM) was created in 2013 by Omar Abu Arqub [45]. It was formed by combining the Taylor series with the residual error function. For DEs, the solution is provided by an infinite convergence series [46]. Novel RPSM algorithms that seek to provide efficient and precise approximation solutions have been developed in response to a variety of DEs, such as Boussinesq DEs, fuzzy DEs, KdV-Burger's equation, and many more [47,48,49,50,51,52].
A novel approach for solving fractional-order differential equations (FDEs) was developed by combining two efficient techniques. Namely, the Sumudu transform has been combined with the Shehu transformation and the Adomian decomposition method [53], and the Laplace transform has been combined with RPSM [54,55,56], the natural transform [57], and the homotopy perturbation approach [58]. See [59,60] for more information on the combination of these methods
The fundamental idea of the new approach is to convert the original problem into an Aboodh transform space, identify many solutions to the transformed version of the equation, and then use the inverse Aboodh transform to solve the original equation. With power series expansion, the new approach may be utilized to solve linear and nonlinear PDEs without discretization, linearization, or perturbation. This method does not depend on matching coefficients of related terms or recursion relations, which sets it apart from the traditional power series methodology. In contrast to the RPSM, the proposed limit-based approach reveals series coefficients instead of fractional derivatives.
Two straightforward methods can be used for solving fractional differential equations, named ATIM and Aboodh residual power series method (ARPSM), as mentioned in [59,60]. Both methods do not only return a numerical solution free from any discretization or linearization of PDEs but also, at the same time, instantly and explicitly display a symbolic representation of the result. For example, these techniques were used to analyze the fractional Hirota-Satsuma coupled Korteweg-de Vries (KdV) problem and derive some highly accurate approximations. The results were effective, accurate, and stable [61]. These methods also succeeded in analyzing some of the most widespread evolution equations in plasma physics and fluid physics, called the Berger-type equations. Some analytical approximations for the damp Burger equation and some related equations were derived, and the accuracy of the derived approximations was tested in a number of different ways to verify the effectiveness of the suggested methods [62]. Moreover, these approaches were used for solving more complicated and strong nonlinear fractional physical equations, as illustrated in [63]. This study introduces ARPSM and ATIM to solve the time-fractional equations of heat, diffusion, and Burger's equations. Compared with other numerical techniques, these approaches provide numerical solutions that are more exact and accurate. The numerical results are also subjected to comparison analysis. The results of the proposed methodologies show good agreement with each other, indicating their effectiveness and dependability, with extra graphical significance for various values of fractional-order derivatives. As a result, the techniques are precise, simple to use, impervious to computational error cycles, and time-efficient. Other researchers will be able to solve various partial differential equations with ease because of the foundation this study provides.
Definition 2.1. [64] If ω(μ,τ) is an exponentially ordered piecewise continuous function, then the Aboodh transform (AT) for σ≥0 can be defined by
A[ω(μ,τ)]=Ψ(μ,ξ)=1ξ∫∞0ω(μ,τ)e−τξdτ, r1≤ξ≤r2. |
The following is one specific definition of the Aboodh inverse transform (AIT):
A−1[Ψ(μ,ξ)]=ω(μ,τ)=12πi∫u+i∞u−i∞Ψ(μ,τ)ξeτξdτ, |
where μ=(μ1,μ2,⋯,μp)∈R and p∈N.
Lemma 2.1. [65,66] Two piecewise continuous functions that are exponentially ordered are assumed to exist, ω1(μ,τ) and ω2(μ,τ) on [0,∞[. Let A[ω1(μ,τ)]=Ψ1(μ,τ),A[ω2(μ,τ)]=Ψ2(μ,τ) and χ1,χ2 are constants. For that reason, the following features are true:
(1) A[χ1ω1(μ,τ)+χ2ω2(μ,τ)]=χ1Ψ1(μ,ξ)+χ2Ψ2(μ,τ),
(2) A−1[χ1Ψ1(μ,τ)+χ2Ψ2(μ,τ)]=χ1ω1(μ,ξ)+χ2ω2(μ,τ),
(3) A[Jpτω(μ,τ)]=Ψ(μ,ξ)ξp,
(4) A[Dpτω(μ,τ)]=ξpΨ(μ,ξ)−∑r−1K=0ωK(μ,0)ξK−p+2, r−1<p≤r, r∈N.
Definition 2.2. [67] The fractional derivative of the function ω(μ,τ) is defined by the Caputo operator as follows, with respect to order p:
Dpτω(μ,τ)=Jm−pτω(m)(μ,τ), r≥0, m−1<p≤m, |
where μ=(μ1,μ2,⋯,μp)∈Rp and m,p∈R,Jm−pτ is the R-L integral of ω(μ,τ).
Definition 2.3. [68] The power series form reads
∞∑r=0ℏr(μ)(τ−τ0)rp=ℏ0(τ−τ0)0+ℏ1(τ−τ0)p+ℏ2(τ−τ0)2p+⋯, |
where μ=(μ1,μ2,⋯,μp)∈Rp and p∈N. The variable τ and the series coefficients in a multiple fractional power series (MFPS) are ℏr(μ)′s, and the series is about τ0.
Lemma 2.2. Let ω(μ,τ) be the exponential order function. The AT is defined as follows: A[ω(μ,τ)]=Ψ(μ,ξ). Hence,
A[Drpτω(μ,τ)]=ξrpΨ(μ,ξ)−r−1∑j=0ξp(r−j)−2Djpτω(μ,0),0<p≤1, | (2.1) |
where μ=(μ1,μ2,⋯,μp)∈Rp and p∈N and Drpτ=DpτDpτ⋯Dpτ(r−times).
Proof. We may prove Eq (2.1) via induction. Using r=1 in Eq (2.1) yields the following results:
A[D2pτω(μ,τ)]=ξ2pΨ(μ,ξ)−ξ2p−2ω(μ,0)−ξp−2Dpτω(μ,0). |
Lemma 2.1, states that Eq (2.1) is true for r=1. For r=2, the result is given by Eq (2.1).
A[D2prω(μ,τ)]=ξ2pΨ(μ,ξ)−ξ2p−2ω(μ,0)−ξp−2Dpτω(μ,0). | (2.2) |
The left-hand side (L.H.S) of Eq (2.2) allows us to get
L.H.S=A[D2pτω(μ,τ)]. | (2.3) |
The expressions of Eq (2.3) read
L.H.S=A[Dpτ(Dpτω(μ,τ))]. | (2.4) |
Let
z(μ,τ)=Dpτω(μ,τ). | (2.5) |
Thus, Eq (2.4) becomes
L.H.S=A[Dpτz(μ,τ)]. | (2.6) |
Equation (2.6) is changed due to the utilization of the Caputo derivative
L.H.S=A[J1−pz′(μ,τ)]. | (2.7) |
The R-L integral for AT, which is given in Eq (2.7), may be used to get
L.H.S=A[z′(μ,τ)]ξ1−p. | (2.8) |
Using the AT's derivative property, then Eq (2.8) may be changed into the following format:
L.H.S=ξpZ(μ,ξ)−z(μ,0)ξ2−p. | (2.9) |
From Eq (2.5), we obtain
Z(μ,ξ)=ξpΨ(μ,ξ)−ω(μ,0)ξ2−p, |
where A[z(μ,τ)]=Z(μ,ξ). Therefore, Eq (2.9) is transformed into the following form
L.H.S=ξ2pΨ(μ,ξ)−ω(μ,0)ξ2−2p−Dpτω(μ,0)ξ2−p, | (2.10) |
when r=K. Eqs (2.1) and (2.10) are compatible. Assuming that for r=K, the Eq (2.1) the hold true, let us proceed, we put r=K in Eq (2.1):
A[DKpτω(μ,τ)]=ξKpΨ(μ,ξ)−K−1∑j=0ξp(K−j)−2DjpτDjpτω(μ,0), 0<p≤1. | (2.11) |
The next step is to illustrate Eq (2.1) for the value of r=K+1. Using Eq (2.1), we can write
A[D(K+1)pτω(μ,τ)]=ξ(K+1)pΨ(μ,ξ)−K∑j=0ξp((K+1)−j)−2Djpτω(μ,0). | (2.12) |
After examining the left-hand side of Eq (2.12), we get
L.H.S=A[DKpτ(DKpτ)]. | (2.13) |
Let
DKpτ=g(μ,τ). |
By Eq (2.13), we drive
L.H.S=A[Dpτg(μ,τ)]. | (2.14) |
Applying the Caputo derivative and R-L integral to Eq (2.14) yields the following result
L.H.S=ξpA[DKpτω(μ,τ)]−g(μ,0)ξ2−p. | (2.15) |
Equation (2.11) is used to get
L.H.S=ξrpΨ(μ,ξ)−r−1∑j=0ξp(r−j)−2Djpτω(μ,0). | (2.16) |
Moreover, the following outcome is obtained using Eq (2.16):
L.H.S=A[Drpτω(μ,0)]. |
For r=K+1, the Eq (2.1) so holds. Therefore, Eq (2.1) is true for all positive integers via mathematical induction.
A novel interpretation of multiple fractional Taylor's series (MFTS) is given in the following lemma. This formula will be useful for the ARPSM (explained below).
Lemma 2.3. Assume that ω(μ,τ) is the exponential order function. The equation A[ω(μ,τ)]=Ψ(μ,ξ) is the AT of ω(μ,τ). The MFTS notation for AT reads
Ψ(μ,ξ)=∞∑r=0ℏr(μ)ξrp+2,ξ>0, | (2.17) |
where, μ=(s1,μ2,⋯,μp)∈Rp, p∈N.
Proof. Let us investigate Taylor's series' fractional order expression:
ω(μ,τ)=ℏ0(μ)+ℏ1(μ)τpΓ[p+1]++ℏ2(μ)τ2pΓ[2p+1]+⋯ | (2.18) |
The following equation represents the equality that is achieved when Eq (2.18) is processed using the AT:
A[ω(μ,τ)]=A[ℏ0(μ)]+A[ℏ1(μ)τpΓ[p+1]]+A[ℏ1(μ)τ2pΓ[2p+1]]+⋯ |
The AT's features allow for this to be accomplished.
A[ω(μ,τ)]=ℏ0(μ)1ξ2+ℏ1(μ)Γ[p+1]Γ[p+1]1ξp+2+ℏ2(μ)Γ[2p+1]Γ[2p+1]1ξ2p+2⋯ |
As a result, (2.17) is obtained, which is a special version of Taylor's s eries in the AT.
Lemma 2.4. To illustrate the MFPS in its new form, as stated in Taylor's series (2.17), let A[ω(μ,τ)]=Ψ(μ,ξ).
ℏ0(μ)=limξ→∞ξ2Ψ(μ,ξ)=ω(μ,0). | (2.19) |
Proof. The novel formulation of Taylor's series allows us to infer the preceding:
ℏ0(μ)=ξ2Ψ(μ,ξ)−ℏ1(μ)ξp−ℏ2(μ)ξ2p−⋯ | (2.20) |
The necessary solution may be obtained by putting limξ→∞ into Eq (2.19) and doing a quick computation, as shown in Eq (2.20).
Theorem 2.1. This is the MFPS notation for the function A[ω(μ,τ)]=Ψ(μ,ξ).
Ψ(μ,ξ)=∞∑0ℏr(μ)ξrp+2, ξ>0, |
where μ=(μ1,μ2,⋯,μp)∈Rp and p∈N. Then we have
ℏr(μ)=Drprω(μ,0), |
where Drpτ=Dpτ.Dpτ.⋯.Dpτ(r−times).
Proof. The new Taylor's series is as follows:
ℏ1(μ)=ξp+2Ψ(μ,ξ)−ξpℏ0(μ)−ℏ2(μ)ξp−ℏ3(μ)ξ2p−⋯ | (2.21) |
limξ→∞, is applied to Eq (2.21), we get
ℏ1(μ)=limξ→∞(ξp+2Ψ(μ,ξ)−ξpℏ0(μ))−limξ→∞ℏ2(μ)ξp−limξ→∞ℏ3(μ)ξ2p−⋯ |
We get the following equality by taking the limit:
ℏ1(μ)=limξ→∞(ξp+2Ψ(μ,ξ)−ξpℏ0(μ)). | (2.22) |
The result of applying Lemma 2.2 into Eq (2.22) reads
ℏ1(μ)=limξ→∞(ξ2A[Dpτω(μ,τ)](ξ)). | (2.23) |
Moreover, it is transformed into by using Lemma 2.3 to the Eq (2.23)
ℏ1(μ)=Dpτω(μ,0). |
Again, assuming limit ξ→∞ and using the new form of Taylor's series, we get
ℏ2(μ)=ξ2p+2Ψ(μ,ξ)−ξ2pℏ0(μ)−ξpℏ1(μ)−ℏ3(μ)ξp−⋯ |
The outcome is given by Lemma 2.3
ℏ2(μ)=limξ→∞ξ2(ξ2pΨ(μ,ξ)−ξ2p−2ℏ0(μ)−ξp−2ℏ1(μ)). | (2.24) |
Equation (2.24) is transformed into the following form using Lemmas 2.2 and 2.4
ℏ2(μ)=D2pτω(μ,0). |
The subsequent results are acquired through the application of the identical procedure to the new Taylor's series:
ℏ3(μ)=limξ→∞ξ2(A[D2pτω(μ,p)](ξ)). |
Lemma 2.4 is employed to figure out the final equation.
ℏ3(μ)=D3pτω(μ,0). |
In general,
ℏr(μ)=Drpτω(μ,0). |
Consequently, the proof is concluded.
The laws regulating the convergence of Taylor's series in its modified form are delineated and validated in the following theorem.
Theorem 2.2. The new form of the formula for multiple fractional Taylor's, as presented in Lemma 2.3, is as follows: A[ω(μ,τ)]=Ψ(μ,ξ). When |ξaA[D(K+1)pτω(μ,τ)]|≤T, for all 0<ξ≤s and 0<p≤1, the residual RK(μ,ξ) of the newly developed MFTS satisfies the inequality:
|RK(μ,ξ)|≤Tξ(K+1)p, 0<ξ≤s. |
Proof. Assume A[Drpτω(μ,τ)](ξ) is defined on 0<ξ≤s for r=0,1,2,⋯,K+1. Proceed with the assumption that |ξ2A[DτK+1ω(μ,tau)]|≤T, on 0<ξ≤s. The following relationship utilizing the new Taylor's series:
RK(μ,ξ)=Ψ(μ,ξ)−K∑r=0ℏr(μ)ξrp+2. | (2.25) |
Equation (2.25) is transformed into the following form using Theorem 2.1
RK(μ,ξ)=Ψ(μ,ξ)−K∑r=0Drpτω(μ,0)ξrp+2. | (2.26) |
To solve Eq (2.26), simply multiply both sides by ξ(K+1)a+2, which leads to
ξ(K+1)p+2RK(μ,ξ)=ξ2(ξ(K+1)pΨ(μ,ξ)−K∑r=0ξ(K+1−r)p−2Drpτω(μ,0)). | (2.27) |
Applying Lemma 2.2 to Eq (2.27) produces the following outcome:
ξ(K+1)p+2RK(μ,ξ)=ξ2A[D(K+1)pτω(μ,τ)]. | (2.28) |
Taking absolute of Eq (2.28) yields
|ξ(K+1)p+2RK(μ,ξ)|=|ξ2A[D(K+1)pτω(μ,τ)]|. | (2.29) |
The outcome resulting from the implementation of the conditions outlined in Eq (2.29) is as follows:
−Tξ(K+1)p+2≤RK(μ,ξ)≤Tξ(K+1)p+2. | (2.30) |
The required result is derived from Eq (2.30)
|RK(μ,ξ)|≤Tξ(K+1)p+2. |
Thus, an innovative series convergence condition is established.
Our overall model solution was constructed using an ARPSM set of rules, which is elaborated in this section.
Step 1. As the general equation is simplified, we obtain
Dqpτω(μ,τ)+ϑ(μ)N(ω)−δ(μ,ω)=0. | (3.1) |
Step 2. Equation (3.1) is subjected to the AT on both sides in order to obtain:
A[Dqpτω(μ,τ)+ϑ(μ)N(ω)−δ(μ,ω)]=0. | (3.2) |
By employing Lemma 2.2, Eq (3.2) can be transformed into the following form
Ψ(μ,s)=q−1∑j=0Djτω(μ,0)sqp+2−ϑ(μ)Y(s)sqp+F(μ,s)sqp, | (3.3) |
where A[δ(μ,ω)]=F(μ,s),A[N(ω)]=Y(s).
Step 3. The solution to Eq (3.3) is expressed in the following form:
Ψ(μ,s)=∞∑r=0ℏr(μ)srp+2, s>0. |
Step 4. The necessary procedures are as follows:
ℏ0(μ)=lims→∞s2Ψ(μ,s)=ω(μ,0). |
The outcome is as follows when Theorem 2.2 is applied
ℏ1(μ)=Dpτω(μ,0), |
ℏ2(μ)=D2pτω(μ,0), |
⋮ |
ℏw(μ)=Dwpτω(μ,0). |
Step 5. The formula to determine the Ψ(μ,s) series after the Kth truncation is as follows:
ΨK(μ,s)=K∑r=0ℏr(μ)srp+2, s>0, |
ΨK(μ,s)=ℏ0(μ)s2+ℏ1(μ)sp+2+⋯+ℏw(μ)swp+2+K∑r=w+1ℏr(μ)srp+2. |
Step 6. For the following results, it is necessary to consider the Aboodh residual function (ARF) from Eq (3.3) and the Kth-truncated ARF independently:
ARes(μ,s)=Ψ(μ,s)−q−1∑j=0Djτω(μ,0)sjp+2+ϑ(μ)Y(s)sjp−F(μ,s)sjp, |
and
AResK(μ,s)=ΨK(μ,s)−q−1∑j=0Djτω(μ,0)sjp+2+ϑ(μ)Y(s)sjp−F(μ,s)sjp. | (3.4) |
Step 7. Equation (3.4) should be modified by placing ΨK(μ,s) rather than its expansion form
AResK(μ,s)=(ℏ0(μ)s2+ℏ1(μ)sp+2+⋯+ℏw(μ)swp+2+K∑r=w+1ℏr(μ)srp+2)−q−1∑j=0Djτω(μ,0)sjp+2+ϑ(μ)Y(s)sjp−F(μ,s)sjp. | (3.5) |
Step 8. To determine the solution to Eq (3.5), multiply each side by sKp+2
sKp+2AResK(μ,s)=sKp+2(ℏ0(μ)s2+ℏ1(μ)sp+2+⋯+ℏw(μ)swp+2+K∑r=w+1ℏr(μ)srp+2−q−1∑j=0Djτω(μ,0)sjp+2+ϑ(μ)Y(s)sjp−F(μ,s)sjp). | (3.6) |
Step 9. Under the assumption that lims→∞, we compute Eq (3.6) to obtain:
lims→∞sKp+2AResK(μ,s)=lims→∞sKp+2(ℏ0(μ)s2+ℏ1(μ)sp+2+⋯+ℏw(μ)swp+2+K∑r=w+1ℏr(μ)srp+2−q−1∑j=0Djτω(μ,0)sjp+2+ϑ(μ)Y(s)sjp−F(μ,s)sjp). |
Step 10. One may get the value of ℏK(μ) by figuring out the equation given
lims→∞(sKp+2AResK(μ,s))=0, |
where K=w+1,w+2,⋯
Step 11. To obtain the K-approximate solution of Eq (3.3), substitute the values of ℏK(μ) with a Ψ(μ,s) series that has been truncated.
Step 12. In order to get the K-approximate solution ωK(μ,τ), solve ΨK(μ,s) using the AIT.
Consider the general PDE of the space-time-fractional order
Dpτω(μ,τ)=Φ(ω(μ,τ),Dτμω(μ,τ),D2τμω(μ,τ),D3τμω(μ,τ)), 0<p,τ≤1, | (3.7) |
which is subjected to the initial condition (IC)
ω(k)(μ,0)=hk, k=0,1,2,⋯,m−1, | (3.8) |
The function ω(μ,τ) is unknown, while Φ(ω(μ,τ),Dτμω(μ,τ),D2τμω(μ,τ),D3τμω(μ,τ)) may be a nonlinear operator or linear of ω(μ,τ),Dτμω(μ,τ),D2τμω(μ,τ) and D3τμω(μ,τ). The subsequent equation can be derived by performing the AT operation on both of the sides of Eq (3.7)
A[ω(μ,τ)]=1sp(m−1∑k=0ω(k)(μ,0)s2−p+k+A[Φ(ω(μ,τ),Dτμω(μ,τ),D2τμω(μ,τ),D3τμω(μ,τ))]). | (3.9) |
When the problem is resolved utilizing the AIT yields
ω(μ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,0)s2−p+k+A[Φ(ω(μ,τ),Dτμω(μ,τ),D2τμω(μ,τ),D3τμω(μ,τ))])]. | (3.10) |
The outcome of employing the ATIM is denoted by an infinite series
ω(μ,τ)=∞∑i=0ωi. | (3.11) |
Since Φ(ω,Dτμω,D2τμω,D3τμω) is either a nonlinear or linear operator, which can be decomposed as follows:
Φ(ω,Dτμω,D2τμω,D3τμω)=Φ(ω0,Dτμω0,D2τμω0,D3τμω0)+∞∑i=0(Φ(i∑k=0(ωk,Dτμωk,D2τμωk,D3τμωk))−Φ(i−1∑k=1(ωk,Dτμωk,D2τμωk,D3τμωk))). | (3.12) |
When the values of (3.11) and (3.12) are substituted for Eq (3.10), the subsequent equation is generated.
∞∑i=0ωi(μ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,0)s2−p+k+A[Φ(ω0,Dτμω0,D2τμω0,D3τμω0)])]+A−1[1sp(A[∞∑i=0(Φi∑k=0(ωk,Dτμωk,D2τμωk,D3τμωk))])]−A−1[1sp(A[(Φi−1∑k=1(ωk,Dτμωk,D2τμωk,D3τμωk))])], | (3.13) |
ω0(μ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,0)s2−p+k)],ω1(μ,τ)=A−1[1sp(A[Φ(ω0,Dτμω0,D2τμω0,D3τμω0)])],⋮ωm+1(μ,τ)=A−1[1sp(A[∞∑i=0(Φi∑k=0(ωk,Dτμωk,D2τμωk,D3τμωk))])]−A−1[1sp(A[(Φi−1∑k=1(ωk,Dτμωk,D2τμωk,D3τμωk))])], m=1,2,⋯ | (3.14) |
For the analytical approximation of the solution to Eq (3.7)'s m-term, the subsequent expression may be employed
ω(μ,τ)=m−1∑i=0ωi. | (3.15) |
Example 3.1. Consider the time-fractional diffusion model to investigate by ARPSM [32]:
Dpτω(μ,ψ,τ)−∂2ω(μ,ψ,τ)∂μ2−∂2ω(μ,ψ,τ)∂ψ2=0, | (3.16) |
which is subjected to the IC
ω(μ,ψ,0)=eμ+ψ. | (3.17) |
The exact solution to Example 3.1 for the integer case, i.e., for p=1, reads
ω(μ,ψ,τ)=eμ+ψ+2τ. | (3.18) |
By applying AT to Eq (3.16) and with the help of Eq (3.17), we get
ω(μ,ψ,s)−eμ+ψs2−1sp[∂2ω(μ,ψ,s)∂μ2]−1sp[∂2ω(μ,ψ,s)∂ψ2]=0. | (3.19) |
Thus, the term series that are required is generally given as
ω(μ,ψ,s)=eμ+ψs2+k∑r=1fr(μ,ψ,s)srp+1, r=1,2,3,4,⋯ | (3.20) |
The ARFs are given by
AτRes(μ,ψ,s)=ω(μ,ψ,s)−eμ+ψs2−1sp[∂2ω(μ,ψ,s)∂μ2]−1sp[∂2ω(μ,ψ,s)∂ψ2]=0, | (3.21) |
and the kth-LRFs as
AτResk(μ,ψ,s)=ωk(μ,ψ,s)−eμ+ψs2−1sp[∂2ωk(μ,ψ,s)∂μ2]−1sp[∂2ωk(μ,ψ,s)∂ψ2]=0. | (3.22) |
In finding fr(μ,ψ,s), for r=1,2,3,.... We solve the expression lims→∞(srp+1) by multiplying the final equation by srp+1, substituting the rth-ARF Eq (3.22) for the rth-truncated series Eq (3.20). AτResω,r(μ,ψ,s))=0, and r=1,2,3,⋯. Following are its few terms:
f1(μ,ψ,s)=2eμ+ψ, | (3.23) |
f2(μ,ψ,s)=4eμ+ψ, | (3.24) |
f3(μ,ψ,s)=8eμ+ψ, | (3.25) |
and so on.
Putting the values of fr(μ,s), r=1,2,3,⋯, in Eq (3.20), we get
ω(μ,ψ,s)=eμ+ψs2+2eμ+ψsp+1+4eμ+ψs2p+1+8eμ+ψs3p+1+⋯ | (3.26) |
AIT is applied to get
ω(μ,ψ,τ)=eμ+ψ(1+2τpΓ(p+1)+4τ2pΓ(2p+1)+8τ3pΓ(3p+1)+⋯). | (3.27) |
Now, we proceed to apply ATIM for analyzing Example 3.1. The result obtained by applying the AT on both sides of Eq (3.16) reads
A[Dpτω(μ,ψ,τ)]=1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2]). | (3.28) |
The result obtained by applying the AIT on both sides of Eq (3.28) reads
ω(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2])]. | (3.29) |
By iteratively applying the AT, the following equation is obtained:
ω0(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k)]=A−1[ω(μ,ψ,0)s2]=eμ+ψ. |
The equivalent form was obtained by substituting the RL integral into Eq (3.16)
ω(μ,ψ,τ)=eμ+ψ−A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2]. | (3.30) |
The subsequent terms are attainable through the utilization of the ATIM procedure:
ω0(μ,ψ,τ)=eμ+ψ,ω1(μ,ψ,τ)=2τpeμ+ψΓ(p+1),ω2(μ,ψ,τ)=4τ2peμ+ψΓ(2p+1),ω3(μ,ψ,τ)=8τ3peμ+ψΓ(3p+1). | (3.31) |
The following constitutes the final solution of the ATIM procedure:
ω(μ,ψ,τ)=ω0(μ,ψ,τ)+ω1(μ,ψ,τ)+ω2(μ,ψ,τ)+ω3(μ,ψ,τ)+⋯ | (3.32) |
ω(μ,ψ,τ)=eμ+ψ(1+2τpΓ(p+1)+4τ2pΓ(2p+1)+8τ3pΓ(3p+1)+⋯). | (3.33) |
The approximation (3.27) using ARPSM has the same form as the approximation (3.33) using ATIM. Since these approximations are entirely symmetrical, we will analyze only one of them in the numerical analysis. The numerical and graphical analysis of the approximations (3.27)/(3.33) is demonstrated in Figures 1 and 2 and Table 1. Figure 1 examines the impact of the fractional parameter p on the wave behavior as defined by the approximations (3.27)/(3.33). It is observed that the behavior of these waves is greatly influenced by the fractional parameter p, as demonstrated in Figure 1. This is one of the most important goals of studying the behavior of non-linear phenomena described by fractional differential equations, because their solutions reveal the ambiguity about these phenomena' behavior and properties that non-fractal differential equations cannot. Also, we assessed the accuracy and stability of the approximations (3.27)/(3.33) by comparing them with the exact solution (3.18) for the integer case. This verification is evident in Figure 2. Furthermore, we conducted a numerical analysis of these approximations and estimated the absolute error compared to the exact solution for the integer case, as depicted in Table 1. Table 1 helps to obtain a deeper and more precise understanding of the analysis results of the test function wave solutions computed by two different solution methods. This table quantitatively compares several values of p and analyzes how these methods are efficient in solving the test Example 3.1.
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.0 | 1.39879 | 1.20765 | 1.12750 | 1.12750 | 1.480958×10−6 |
0.2 | 1.70848 | 1.47503 | 1.37713 | 1.37713 | 1.808846×10−6 |
0.4 | 2.08675 | 1.80161 | 1.68203 | 1.68203 | 2.209330×10−6 |
0.6 | 2.54876 | 2.20049 | 2.05443 | 2.05443 | 2.698482×10−6 |
0.8 | 3.11306 | 2.68768 | 2.50929 | 2.50929 | 3.295933×10−6 |
1.0 | 3.80230 | 3.28274 | 3.06485 | 3.06485 | 4.025662×10−6 |
1.2 | 4.64414 | 4.00955 | 3.74342 | 3.74342 | 4.916955×10−6 |
1.4 | 5.67237 | 4.89728 | 4.57222 | 4.57223 | 6.005583×10−6 |
1.6 | 6.92824 | 5.98155 | 5.58452 | 5.58453 | 7.335235×10−6 |
1.8 | 8.46218 | 7.30588 | 6.82095 | 6.82096 | 8.959277×10−6 |
2.0 | 10.3357 | 8.92343 | 8.33113 | 8.33114 | 1.094290×10−5 |
Example 3.2. Consider the time-fractional heat model to investigate by ARPSM [34]:
Dpτω(μ,ψ,τ)−∂2ω(μ,ψ,τ)∂μ2−∂2ω(μ,ψ,τ)∂ψ2=0, | (3.34) |
which is subjected to the IC
ω(μ,ψ,0)=sin(μ)cos(ψ). | (3.35) |
The exact solution to Example 3.2 for the integer case, i.e., for p=1, reads
ω(μ,ψ,τ)=e−2τsin(μ)cos(ψ). | (3.36) |
The result obtained by applying the AT on both sides of Eq (3.35) reads
ω(μ,ψ,s)−sin(μ)cos(ψ)s2−1sp[∂2ω(μ,ψ,s)∂μ2]−1sp[∂2ω(μ,ψ,s)∂ψ2]=0. | (3.37) |
Thus, the term series that are required is generally given as
ω(μ,ψ,s)=sin(μ)cos(ψ)s2+k∑r=1fr(μ,ψ,s)srp+1, r=1,2,3,4,⋯ | (3.38) |
The following is ARFs:
AτRes(μ,ψ,s)=ω(μ,ψ,s)−sin(μ)cos(ψ)s2−1sp[∂2ω(μ,ψ,s)∂μ2]−1sp[∂2ω(μ,ψ,s)∂ψ2]=0, | (3.39) |
and the kth-LRFs as
AτResk(μ,ψ,s)=ωk(μ,ψ,s)−sin(μ)cos(ψ)s2−1sp[∂2ωk(μ,ψ,s)∂μ2]−1sp[∂2ωk(μ,ψ,s)∂ψ2]=0. | (3.40) |
In finding fr(μ,ψ,s), for r=1,2,3,.... We solve the expression lims→∞(srp+1) by multiplying the final equation by srp+1, substituting the rth-ARF Eq (3.40) for the rth-truncated series Eq (3.38). AτResω,r(μ,ψ,s))=0, and r=1,2,3,⋯. Following are its few terms:
f1(μ,ψ,s)=−2sin(μ)cos(ψ), | (3.41) |
f2(μ,ψ,s)=4sin(μ)cos(ψ), | (3.42) |
f3(μ,ψ,s)=−8sin(μ)cos(ψ), | (3.43) |
and so on.
Putting the values of fr(μ,s), r=1,2,3,⋯, in Eq (3.38), we deduce
ω(μ,ψ,s)=sin(μ)cos(ψ)s2−2sin(μ)cos(ψ)sp+1+4sin(μ)cos(ψ)s2p+1−8sin(μ)cos(ψ)s3p+1+⋯ | (3.44) |
The following constitutes the final solution of the ARPSM procedure:
ω(μ,ψ,τ)=ω0(μ,ψ,τ)+ω1(μ,ψ,τ)+ω2(μ,ψ,τ)+ω3(μ,ψ,τ)+⋯ | (3.45) |
ω(μ,ψ,τ)=sin(μ)cos(ψ)(1−2τpΓ(p+1)+4τ2pΓ(2p+1)−8τ3pΓ(3p+1)+⋯). | (3.46) |
Now, we proceed to apply ATIM for analyzing Example 3.2. The result obtained by applying the AT on both sides of Eq (3.34) reads
A[Dpτω(μ,ψ,τ)]=1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2]). | (3.47) |
The result obtained by applying the AIT on both sides of Eq (3.47) is as follows:
ω(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2])]. | (3.48) |
By iteratively applying the AT, the following equation is obtained:
ω0(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k)]=A−1[ω(μ,ψ,0)s2]=sin(μ)cos(ψ). |
The equivalent form was obtained by substituting the RL integral into Eq (3.34)
ω(μ,ψ,τ)=sin(μ)cos(ψ)−A[∂2ω(μ,ψ,τ)∂μ2+∂2ω(μ,ψ,τ)∂ψ2]. | (3.49) |
The subsequent terms are attainable through the utilization of the ATIM procedure:
ω0(μ,ψ,τ)=sin(μ)cos(ψ),ω1(μ,ψ,τ)=−2τpsin(μ)cos(ψ)Γ(p+1),ω2(μ,ψ,τ)=4τ2psin(μ)cos(ψ)Γ(2p+1),ω3(μ,ψ,τ)=−8τ2psin(μ)cos(ψ)Γ(2p+1). | (3.50) |
The following constitutes the final solution of the ATIM procedure:
ω(μ,ψ,τ)=ω0(μ,ψ,τ)+ω1(μ,ψ,τ)+ω2(μ,ψ,τ)+ω3(μ,ψ,τ)+⋯ | (3.51) |
ω(μ,ψ,τ)=sin(μ)cos(ψ)(1−2τpΓ(p+1)+4τ2pΓ(2p+1)−8τ3pΓ(3p+1)+⋯). | (3.52) |
It is clear from the approximation (3.46) that using ARPSM is the same approximation (3.51) using ATIM. Since these approximations are completely symmetrical, we will analyze only one of them in the numerical analysis. Figures 3 and 4 and Table 2 exhibit the numerical and graphical examination of the approximations (3.46)/(3.51). Figure 3 examines the influence of the fractional parameter p on the profile of approximations (3.46)/(3.51). It is shown that the behavior of these waves is significantly impacted by changing the value of the fractional parameter p. Moreover, the approximations (3.46)/(3.51) are compared to the exact solution (3.36) in order to confirm their high level of accuracy, as shown in Figure 4. Additionally, we performed a quantitative study of these approximations and calculated the absolute error compared to the exact solution (3.36) for the integer case, as illustrated in Table 2. This table facilitates a more comprehensive and accurate comprehension of the analysis findings from the test function wave solutions computed using two distinct solution approaches. This Table provides a quantitative comparison of several values of p and examines the efficiency of different strategies for the test problem. Additionally, we performed a quantitative study of these approximations and calculated the absolute error compared to the exact solution (3.36) for the integer case, as illustrated in Table 2. This Table provides a quantitative comparison of several values of p and examines the efficiency of the used approaches for the current test Example 3.2.
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.2 | 0.098056 | 0.130488 | 0.161831 | 0.161844 | 1.26684×10−5 |
0.4 | 0.192204 | 0.255773 | 0.317211 | 0.317236 | 2.48317×10−5 |
0.6 | 0.278688 | 0.370861 | 0.459945 | 0.459981 | 3.60051×10−5 |
0.8 | 0.354062 | 0.471165 | 0.584342 | 0.584387 | 4.57431×10−5 |
1.0 | 0.415321 | 0.552684 | 0.685443 | 0.685496 | 5.36574×10−5 |
1.2 | 0.460022 | 0.612170 | 0.759217 | 0.759277 | 5.94326×10−5 |
1.4 | 0.486384 | 0.647251 | 0.802724 | 0.802787 | 6.28384×10−5 |
1.6 | 0.493355 | 0.656527 | 0.814229 | 0.814293 | 6.37391×10−5 |
1.8 | 0.480658 | 0.639630 | 0.793274 | 0.793336 | 6.20986×10−5 |
2.0 | 0.448798 | 0.597233 | 0.740693 | 0.740751 | 5.79825×10−5 |
Example 3.3. Consider the time-fractional Burger's model to investigate by ARPSM [44]:
Dpτω(μ,ψ,τ)−∂2ω(μ,ψ,τ)∂μ−∂2ω(μ,ψ,τ)∂ψ−ω(μ,ψ,τ)∂ω(μ,ψ,τ)∂μ=0, | (3.53) |
which is subjected to the IC:
ω(μ,ψ,0)=μ+ψ. | (3.54) |
The exact solution to Example 3.3 for the integer case, i.e., for p=1, reads
ω(μ,ψ,τ)=μ+ψ1−τ. | (3.55) |
The result obtained by applying the AT on both sides of Eq (3.54) reads
ω(μ,ψ,s)−μ+ψsp−1sp[∂2ω(μ,ψ,s)∂μ]−1sp[∂2ω(μ,ψ,s)∂ψ]−1spAτ[A−1τω(μ,ψ,s)×∂A−1τω(μ,ψ,s)∂μ]=0. | (3.56) |
Thus, the term series that are required is generally given as
ω(μ,ψ,s)=μ+ψs2+k∑r=1fr(μ,ψ,s)srp+1, r=1,2,3,4⋯ | (3.57) |
The ARFs are given by
AτRes(μ,ψ,s)=ω(μ,ψ,s)−μ+ψsp−1sp[∂2ω(μ,ψ,s)∂μ]−1sp[∂2ω(μ,ψ,s)∂ψ]−1spAτ[A−1τω(μ,ψ,s)×∂A−1τω(μ,ψ,s)∂μ]=0, | (3.58) |
and the kth-LRFs as
AτResk(μ,ψ,s)=ωk(μ,ψ,s)−μ+ψsp−1sp[∂2ωk(μ,ψ,s)∂μ]−1sp[∂2ωk(μ,ψ,s)∂ψ]−1spAτ[A−1τωk(μ,ψ,s)×∂A−1τωk(μ,ψ,s)∂μ]=0. | (3.59) |
In finding fr(μ,ψ,s), for r=1,2,3,.... We solve the expression lims→∞(srp+1) by multiplying the final equation by srp+1, substituting the rth-ARF Eq (3.59) for the rth-truncated series Eq (3.57). AτResω,r(μ,ψ,s))=0, and r=1,2,3,⋯. Following are its few terms:
f1(μ,ψ,s)=μ+ψ, | (3.60) |
f2(μ,ψ,s)=2(μ+ψ), | (3.61) |
f3(μ,ψ,s)=4(μ+ψ). | (3.62) |
and so on.
Putting the values of fr(μ,s), r=1,2,3,⋯, in Eq (3.57), we deduce
ω(μ,ψ,s)=μ+ψs2+μ+ψsp+1+2(μ+ψ)s2p+1+4(μ+ψ)s3p+1+⋯ | (3.63) |
AIT is applied to get
ω(μ,ψ,τ)=(μ+ψ)(1+τpΓ(p+1)+2τ2pΓ(2p+1)+4τ3pΓ(3p+1)+⋯). | (3.64) |
Now, we proceed to apply ATIM for analyzing Example 3.3. The result obtained by applying the AT on both sides of Eq (3.53) reads
A[Dpτω(μ,ψ,τ)]=1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ+∂2ω(μ,ψ,τ)∂ψ+ω(μ,ψ,τ)∂ω(μ,ψ,τ)∂μ]). | (3.65) |
The result obtained by applying the AIT on both sides of (3.65) reads
ω(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k+A[∂2ω(μ,ψ,τ)∂μ+∂2ω(μ,ψ,τ)∂ψ+ω(μ,ψ,τ)∂ω(μ,ψ,τ)∂μ])]. | (3.66) |
By applying the AT to the above equation, we get
ω0(μ,ψ,τ)=A−1[1sp(m−1∑k=0ω(k)(μ,ψ,0)s2−p+k)]=A−1[ω(μ,ψ,0)s2]=μ+ψ. |
The equivalent form was obtained by substituting the RL integral into Eq (3.53)
ω(μ,ψ,τ)=μ+ψ−A[∂2ω(μ,ψ,τ)∂μ+∂2ω(μ,ψ,τ)∂ψ+ω(μ,ψ,τ)∂ω(μ,ψ,τ)∂μ]. | (3.67) |
The subsequent terms are attainable through the utilization of the ATIM procedure:
ω0(μ,ψ,τ)=μ+ψ,ω1(μ,ψ,τ)=τp(μ+ψ)Γ(p+1),ω2(μ,ψ,τ)=τ2p(μ+ψ)(τpΓ(2p+1)2Γ(p+1)2Γ(3p+1)+2)Γ(2p+1),ω3(μ,ψ,τ)=(τ4p(μ+ψ)((√π41−3pτ2pΓ(p+1)2Γ(2p+1)2Γ(5p+1))/(Γ(3p+12))+(τ3pΓ(2p+1)4Γ(6p+1))/(Γ(7p+1))+(2τpΓ(p+1)Γ(3p+1)(2Γ(3p+1)Γ(p+1)3+Γ(2p+1)3)Γ(4p+1))/(Γ(5p+1))+(√π41−2pΓ(p+1)3Γ(3p+1)3)/(Γ(2p+12))))/(Γ(p+1)4Γ(2p+1)2Γ(3p+1)2). | (3.68) |
The following constitutes the final solution using the ATIM procedure:
ω(μ,ψ,τ)=ω0(μ,ψ,τ)+ω1(μ,ψ,τ)+ω2(μ,ψ,τ)+ω3(μ,ψ,τ)+⋯ | (3.69) |
ω(μ,ψ,τ)=μ+ψ+τp(μ+ψ)Γ(p+1)+τ2p(μ+ψ)(τpΓ(2p+1)2Γ(p+1)2Γ(3p+1)+2)Γ(2p+1)+(τ4p(μ+ψ)((√π41−3pτ2pΓ(p+1)2Γ(2p+1)2Γ(5p+1))/(Γ(3p+12))+(τ3pΓ(2p+1)4Γ(6p+1))/(Γ(7p+1))+(2τpΓ(p+1)Γ(3p+1)(2Γ(3p+1)Γ(p+1)3+Γ(2p+1)3)Γ(4p+1))/(Γ(5p+1))+(√π41−2pΓ(p+1)3Γ(3p+1)3)/(Γ(2p+12))))/(Γ(p+1)4Γ(2p+1)2Γ(3p+1)2)+⋯ | (3.70) |
Figure 5 demonstrates the impact of the fractional parameter on the behavior of the approximation (3.63) using ARPSM of Example 3.3. Figure 6 presents a visual comparison between the approximation (3.63) using ARPSM and the exact solution (3.55) for the integer case. In addition, we investigated the approximation (3.69) using ATIM, which is illustrated in Figures 7 and 8. The approximations (3.63)/(3.69) and the exact solution (3.55) for the integer case show complete agreement (see Figures 6 and 8), which enhances the strength, accuracy, and efficiency of these methods in analyzing more complicated evolution equations. By comparing and analyzing Figures 5–8, we can evaluate the performance of the ARPSM/ATIM method in solving Burger's model's constraints. Tables 3 and 4 provide a detailed analysis of different p values for the approximations (3.63) and (3.69) using ARPSM and ATIM, respectively. In addition, Table 5 compares the absolute error for the two approximations (3.63) and (3.69) as compared to the exact solution (3.55).
μ | ARPSMp=0.5 | ARPSMp=0.7 | ARPSMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113585 | 0.104648 | 0.101010 | 0.101010 | 3.434343×10−8 |
0.2 | 0.340754 | 0.313944 | 0.303030 | 0.303030 | 1.030303×10−7 |
0.4 | 0.567923 | 0.523240 | 0.505050 | 0.505050 | 1.717171×10−7 |
0.6 | 0.795093 | 0.732536 | 0.707070 | 0.707070 | 2.404040×10−7 |
0.8 | 1.022260 | 0.941832 | 0.909090 | 0.909090 | 3.090909×10−7 |
1.0 | 1.249430 | 1.151130 | 1.111111 | 1.111111 | 3.777777×10−7 |
1.2 | 1.476600 | 1.360420 | 1.313130 | 1.313130 | 4.464646×10−7 |
1.4 | 1.703770 | 1.569720 | 1.515150 | 1.515150 | 5.151515×10−7 |
1.6 | 1.930940 | 1.779020 | 1.717170 | 1.717170 | 5.838383×10−7 |
1.8 | 2.158110 | 1.988310 | 1.919190 | 1.919190 | 6.525252×10−7 |
2.0 | 2.385280 | 2.197610 | 2.121210 | 2.121210 | 7.212121×10−7 |
μ | ATIMp=0.5 | ATIMp=0.7 | ATIMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113413 | 0.104641 | 0.10101 | 0.101010 | 6.717342×10−8 |
0.2 | 0.340240 | 0.313924 | 0.30303 | 0.303030 | 2.015202×10−7 |
0.4 | 0.567067 | 0.523206 | 0.50505 | 0.505051 | 3.358671×10−7 |
0.6 | 0.793894 | 0.732489 | 0.70707 | 0.707071 | 4.702139×10−7 |
0.8 | 1.020720 | 0.941772 | 0.90909 | 0.909091 | 6.045608×10−7 |
1.0 | 1.247550 | 1.151050 | 1.11111 | 1.111110 | 7.389076×10−7 |
1.2 | 1.474380 | 1.360340 | 1.31313 | 1.313130 | 8.732545×10−7 |
1.4 | 1.701200 | 1.569620 | 1.51515 | 1.515150 | 1.007601×10−6 |
1.6 | 1.928030 | 1.778900 | 1.71717 | 1.717170 | 1.141948×10−6 |
1.8 | 2.154860 | 1.988180 | 1.91919 | 1.919190 | 1.276295×10−6 |
2.0 | 2.381680 | 2.197470 | 2.12121 | 2.121210 | 1.410641×10−6 |
μ | ARPSMp=1 | ATIMp=1 | Exact | ErrorARPSM | ErrorATIM |
0.0 | 0.10101 | 0.10101 | 0.10101 | 3.434343×10−8 | 6.717342×10−8 |
0.2 | 0.30303 | 0.30303 | 0.30303 | 1.030303×10−7 | 2.015202×10−7 |
0.4 | 0.50505 | 0.50505 | 0.50505 | 1.717171×10−7 | 3.358671×10−7 |
0.6 | 0.70707 | 0.70707 | 0.70707 | 2.404040×10−7 | 4.702139×10−7 |
0.8 | 0.90909 | 0.90909 | 0.90909 | 3.090909×10−7 | 6.045608×10−7 |
1.0 | 1.11111 | 1.11111 | 1.11111 | 3.777777×10−7 | 7.389076×10−7 |
1.2 | 1.31313 | 1.31313 | 1.31313 | 4.464646×10−7 | 8.732545×10−7 |
1.4 | 1.51515 | 1.51515 | 1.51515 | 5.151515×10−7 | 1.007601×10−6 |
1.6 | 1.71717 | 1.71717 | 1.71717 | 5.838383×10−7 | 1.141948×10−6 |
1.8 | 1.91919 | 1.91919 | 1.91919 | 6.525252×10−7 | 1.276295×10−6 |
2.0 | 2.12121 | 2.12121 | 2.12121 | 7.212121×10−7 | 1.410641×10−6 |
Our study has provided new insights into the dynamics of some physical time-fractional differential equations, including heat, diffusion, and Burger equations. Both the Aboodh residual power series approach and the Aboodh transform iterative method have been applied to investigate time-fractional equations inside the Caputo operator framework. These approaches have proven to be efficient and accurate tools for solving fractional equations and offering insights into the behavior and characteristics of fractional systems. By employing these methods, we have not only captured the non-local and memory-dependent characteristics inherent to time-fractional dynamics but also paved the way for potential applications in understanding and predicting the complex behavior of these systems. The obtained outcomes highlight the importance of fractional calculus in modeling different nonlinear phenomena associated with various physical and engineering systems and solving more complicated differential equations related to other natural applications. The analytical approximations of the Aboodh techniques facilitate future studies and applications in several scientific and engineering fields, including physics, biology, finance, and many others. In summary, our study advances the field of fractional calculus by offering novel approaches for analyzing time-fractional equations. This enriches our understanding of the underlying dynamics and opens new avenues for future investigations and practical implementations.
Given the favorable outcomes achieved via the Aboodh techniques, we can employ these techniques to examine various evolutionary equations that can be utilized to depict many nonlinear structures (waves) that emerge and propagate in diverse plasma systems and various physical systems. For instance, the fluid equations for various plasma models can be reduced to different fractional evolution (wave) equations. These equations are employed to model and describe the properties of different nonlinear structures, such as solitary waves, shock waves, and cnoidal waves that propagate in diverse plasma systems, including KdV-type equations [69,70], Kawahara-type equations [71,72] Burgers-type equations [73]. On the other hand, using one of the reductive perturbation methods, the fluid equations for many different plasma models can also be reduced to the nonlinear Schodinger-type equations [74,75], which are used to describe many modulated envelope structures that propagate at group velocity, such as dark-, bright-, gray-solitons, rogue waves and breathers, and modulated cnoidal waves. Therefore, Aboodh's methods can be applied to analyze these equations.
All authors contributed equally and approved the final version of the current manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R2), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA263).
The authors declare that they have no conflicts of interest.
The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R2), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA263).
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1. | Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari, Adel Almarashi, Lump-type kink wave phenomena of the space-time fractional phi-four equation, 2024, 9, 2473-6988, 34372, 10.3934/math.20241637 | |
2. | Yousef Jawarneh, Humaira Yasmin, Ali M. Mahnashi, A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation, 2024, 9, 2473-6988, 35308, 10.3934/math.20241678 |
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.0 | 1.39879 | 1.20765 | 1.12750 | 1.12750 | 1.480958×10−6 |
0.2 | 1.70848 | 1.47503 | 1.37713 | 1.37713 | 1.808846×10−6 |
0.4 | 2.08675 | 1.80161 | 1.68203 | 1.68203 | 2.209330×10−6 |
0.6 | 2.54876 | 2.20049 | 2.05443 | 2.05443 | 2.698482×10−6 |
0.8 | 3.11306 | 2.68768 | 2.50929 | 2.50929 | 3.295933×10−6 |
1.0 | 3.80230 | 3.28274 | 3.06485 | 3.06485 | 4.025662×10−6 |
1.2 | 4.64414 | 4.00955 | 3.74342 | 3.74342 | 4.916955×10−6 |
1.4 | 5.67237 | 4.89728 | 4.57222 | 4.57223 | 6.005583×10−6 |
1.6 | 6.92824 | 5.98155 | 5.58452 | 5.58453 | 7.335235×10−6 |
1.8 | 8.46218 | 7.30588 | 6.82095 | 6.82096 | 8.959277×10−6 |
2.0 | 10.3357 | 8.92343 | 8.33113 | 8.33114 | 1.094290×10−5 |
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.2 | 0.098056 | 0.130488 | 0.161831 | 0.161844 | 1.26684×10−5 |
0.4 | 0.192204 | 0.255773 | 0.317211 | 0.317236 | 2.48317×10−5 |
0.6 | 0.278688 | 0.370861 | 0.459945 | 0.459981 | 3.60051×10−5 |
0.8 | 0.354062 | 0.471165 | 0.584342 | 0.584387 | 4.57431×10−5 |
1.0 | 0.415321 | 0.552684 | 0.685443 | 0.685496 | 5.36574×10−5 |
1.2 | 0.460022 | 0.612170 | 0.759217 | 0.759277 | 5.94326×10−5 |
1.4 | 0.486384 | 0.647251 | 0.802724 | 0.802787 | 6.28384×10−5 |
1.6 | 0.493355 | 0.656527 | 0.814229 | 0.814293 | 6.37391×10−5 |
1.8 | 0.480658 | 0.639630 | 0.793274 | 0.793336 | 6.20986×10−5 |
2.0 | 0.448798 | 0.597233 | 0.740693 | 0.740751 | 5.79825×10−5 |
μ | ARPSMp=0.5 | ARPSMp=0.7 | ARPSMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113585 | 0.104648 | 0.101010 | 0.101010 | 3.434343×10−8 |
0.2 | 0.340754 | 0.313944 | 0.303030 | 0.303030 | 1.030303×10−7 |
0.4 | 0.567923 | 0.523240 | 0.505050 | 0.505050 | 1.717171×10−7 |
0.6 | 0.795093 | 0.732536 | 0.707070 | 0.707070 | 2.404040×10−7 |
0.8 | 1.022260 | 0.941832 | 0.909090 | 0.909090 | 3.090909×10−7 |
1.0 | 1.249430 | 1.151130 | 1.111111 | 1.111111 | 3.777777×10−7 |
1.2 | 1.476600 | 1.360420 | 1.313130 | 1.313130 | 4.464646×10−7 |
1.4 | 1.703770 | 1.569720 | 1.515150 | 1.515150 | 5.151515×10−7 |
1.6 | 1.930940 | 1.779020 | 1.717170 | 1.717170 | 5.838383×10−7 |
1.8 | 2.158110 | 1.988310 | 1.919190 | 1.919190 | 6.525252×10−7 |
2.0 | 2.385280 | 2.197610 | 2.121210 | 2.121210 | 7.212121×10−7 |
μ | ATIMp=0.5 | ATIMp=0.7 | ATIMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113413 | 0.104641 | 0.10101 | 0.101010 | 6.717342×10−8 |
0.2 | 0.340240 | 0.313924 | 0.30303 | 0.303030 | 2.015202×10−7 |
0.4 | 0.567067 | 0.523206 | 0.50505 | 0.505051 | 3.358671×10−7 |
0.6 | 0.793894 | 0.732489 | 0.70707 | 0.707071 | 4.702139×10−7 |
0.8 | 1.020720 | 0.941772 | 0.90909 | 0.909091 | 6.045608×10−7 |
1.0 | 1.247550 | 1.151050 | 1.11111 | 1.111110 | 7.389076×10−7 |
1.2 | 1.474380 | 1.360340 | 1.31313 | 1.313130 | 8.732545×10−7 |
1.4 | 1.701200 | 1.569620 | 1.51515 | 1.515150 | 1.007601×10−6 |
1.6 | 1.928030 | 1.778900 | 1.71717 | 1.717170 | 1.141948×10−6 |
1.8 | 2.154860 | 1.988180 | 1.91919 | 1.919190 | 1.276295×10−6 |
2.0 | 2.381680 | 2.197470 | 2.12121 | 2.121210 | 1.410641×10−6 |
μ | ARPSMp=1 | ATIMp=1 | Exact | ErrorARPSM | ErrorATIM |
0.0 | 0.10101 | 0.10101 | 0.10101 | 3.434343×10−8 | 6.717342×10−8 |
0.2 | 0.30303 | 0.30303 | 0.30303 | 1.030303×10−7 | 2.015202×10−7 |
0.4 | 0.50505 | 0.50505 | 0.50505 | 1.717171×10−7 | 3.358671×10−7 |
0.6 | 0.70707 | 0.70707 | 0.70707 | 2.404040×10−7 | 4.702139×10−7 |
0.8 | 0.90909 | 0.90909 | 0.90909 | 3.090909×10−7 | 6.045608×10−7 |
1.0 | 1.11111 | 1.11111 | 1.11111 | 3.777777×10−7 | 7.389076×10−7 |
1.2 | 1.31313 | 1.31313 | 1.31313 | 4.464646×10−7 | 8.732545×10−7 |
1.4 | 1.51515 | 1.51515 | 1.51515 | 5.151515×10−7 | 1.007601×10−6 |
1.6 | 1.71717 | 1.71717 | 1.71717 | 5.838383×10−7 | 1.141948×10−6 |
1.8 | 1.91919 | 1.91919 | 1.91919 | 6.525252×10−7 | 1.276295×10−6 |
2.0 | 2.12121 | 2.12121 | 2.12121 | 7.212121×10−7 | 1.410641×10−6 |
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.0 | 1.39879 | 1.20765 | 1.12750 | 1.12750 | 1.480958×10−6 |
0.2 | 1.70848 | 1.47503 | 1.37713 | 1.37713 | 1.808846×10−6 |
0.4 | 2.08675 | 1.80161 | 1.68203 | 1.68203 | 2.209330×10−6 |
0.6 | 2.54876 | 2.20049 | 2.05443 | 2.05443 | 2.698482×10−6 |
0.8 | 3.11306 | 2.68768 | 2.50929 | 2.50929 | 3.295933×10−6 |
1.0 | 3.80230 | 3.28274 | 3.06485 | 3.06485 | 4.025662×10−6 |
1.2 | 4.64414 | 4.00955 | 3.74342 | 3.74342 | 4.916955×10−6 |
1.4 | 5.67237 | 4.89728 | 4.57222 | 4.57223 | 6.005583×10−6 |
1.6 | 6.92824 | 5.98155 | 5.58452 | 5.58453 | 7.335235×10−6 |
1.8 | 8.46218 | 7.30588 | 6.82095 | 6.82096 | 8.959277×10−6 |
2.0 | 10.3357 | 8.92343 | 8.33113 | 8.33114 | 1.094290×10−5 |
μ | ARPSM/ATIM | ARPSM/ATIM | ARPSM/ATIM | ||
p=0.5 | p=0.7 | p=1.0 | Exact | Errorp=1.0 | |
0.2 | 0.098056 | 0.130488 | 0.161831 | 0.161844 | 1.26684×10−5 |
0.4 | 0.192204 | 0.255773 | 0.317211 | 0.317236 | 2.48317×10−5 |
0.6 | 0.278688 | 0.370861 | 0.459945 | 0.459981 | 3.60051×10−5 |
0.8 | 0.354062 | 0.471165 | 0.584342 | 0.584387 | 4.57431×10−5 |
1.0 | 0.415321 | 0.552684 | 0.685443 | 0.685496 | 5.36574×10−5 |
1.2 | 0.460022 | 0.612170 | 0.759217 | 0.759277 | 5.94326×10−5 |
1.4 | 0.486384 | 0.647251 | 0.802724 | 0.802787 | 6.28384×10−5 |
1.6 | 0.493355 | 0.656527 | 0.814229 | 0.814293 | 6.37391×10−5 |
1.8 | 0.480658 | 0.639630 | 0.793274 | 0.793336 | 6.20986×10−5 |
2.0 | 0.448798 | 0.597233 | 0.740693 | 0.740751 | 5.79825×10−5 |
μ | ARPSMp=0.5 | ARPSMp=0.7 | ARPSMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113585 | 0.104648 | 0.101010 | 0.101010 | 3.434343×10−8 |
0.2 | 0.340754 | 0.313944 | 0.303030 | 0.303030 | 1.030303×10−7 |
0.4 | 0.567923 | 0.523240 | 0.505050 | 0.505050 | 1.717171×10−7 |
0.6 | 0.795093 | 0.732536 | 0.707070 | 0.707070 | 2.404040×10−7 |
0.8 | 1.022260 | 0.941832 | 0.909090 | 0.909090 | 3.090909×10−7 |
1.0 | 1.249430 | 1.151130 | 1.111111 | 1.111111 | 3.777777×10−7 |
1.2 | 1.476600 | 1.360420 | 1.313130 | 1.313130 | 4.464646×10−7 |
1.4 | 1.703770 | 1.569720 | 1.515150 | 1.515150 | 5.151515×10−7 |
1.6 | 1.930940 | 1.779020 | 1.717170 | 1.717170 | 5.838383×10−7 |
1.8 | 2.158110 | 1.988310 | 1.919190 | 1.919190 | 6.525252×10−7 |
2.0 | 2.385280 | 2.197610 | 2.121210 | 2.121210 | 7.212121×10−7 |
μ | ATIMp=0.5 | ATIMp=0.7 | ATIMp=1.0 | Exact | Errorp=1.0 |
0.0 | 0.113413 | 0.104641 | 0.10101 | 0.101010 | 6.717342×10−8 |
0.2 | 0.340240 | 0.313924 | 0.30303 | 0.303030 | 2.015202×10−7 |
0.4 | 0.567067 | 0.523206 | 0.50505 | 0.505051 | 3.358671×10−7 |
0.6 | 0.793894 | 0.732489 | 0.70707 | 0.707071 | 4.702139×10−7 |
0.8 | 1.020720 | 0.941772 | 0.90909 | 0.909091 | 6.045608×10−7 |
1.0 | 1.247550 | 1.151050 | 1.11111 | 1.111110 | 7.389076×10−7 |
1.2 | 1.474380 | 1.360340 | 1.31313 | 1.313130 | 8.732545×10−7 |
1.4 | 1.701200 | 1.569620 | 1.51515 | 1.515150 | 1.007601×10−6 |
1.6 | 1.928030 | 1.778900 | 1.71717 | 1.717170 | 1.141948×10−6 |
1.8 | 2.154860 | 1.988180 | 1.91919 | 1.919190 | 1.276295×10−6 |
2.0 | 2.381680 | 2.197470 | 2.12121 | 2.121210 | 1.410641×10−6 |
μ | ARPSMp=1 | ATIMp=1 | Exact | ErrorARPSM | ErrorATIM |
0.0 | 0.10101 | 0.10101 | 0.10101 | 3.434343×10−8 | 6.717342×10−8 |
0.2 | 0.30303 | 0.30303 | 0.30303 | 1.030303×10−7 | 2.015202×10−7 |
0.4 | 0.50505 | 0.50505 | 0.50505 | 1.717171×10−7 | 3.358671×10−7 |
0.6 | 0.70707 | 0.70707 | 0.70707 | 2.404040×10−7 | 4.702139×10−7 |
0.8 | 0.90909 | 0.90909 | 0.90909 | 3.090909×10−7 | 6.045608×10−7 |
1.0 | 1.11111 | 1.11111 | 1.11111 | 3.777777×10−7 | 7.389076×10−7 |
1.2 | 1.31313 | 1.31313 | 1.31313 | 4.464646×10−7 | 8.732545×10−7 |
1.4 | 1.51515 | 1.51515 | 1.51515 | 5.151515×10−7 | 1.007601×10−6 |
1.6 | 1.71717 | 1.71717 | 1.71717 | 5.838383×10−7 | 1.141948×10−6 |
1.8 | 1.91919 | 1.91919 | 1.91919 | 6.525252×10−7 | 1.276295×10−6 |
2.0 | 2.12121 | 2.12121 | 2.12121 | 7.212121×10−7 | 1.410641×10−6 |