A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

1  Introduction

Several real-life occurrences in thermodynamics, molecular biology, operations research, and other disciplines of materials research can be lucratively modelled using fractional derivatives [1–6]. The principal motivation behind this is that realistic modelling of a core challenge requires not only the precise moment but also the preceding sequential schedule, which can be efficiently accomplished by utilizing fractional calculus [7,8]. However, numerous applied science researchers have concentrated on fractional partial differential equations (PDEs) in designing procedures for interaction problems and discussing physical phenomena. Aside from that, estimated and analytical strategies for FDE solutions have been investigated [9,10]. The subject of fractional initial value problems (IVPs) has captured the attention of academic researchers because it has the functionality of describing several capabilities of real-life manifestations within a more believable methodology than conventional PDEs. Many accomplishments have been attributed to the assumption of strategy presence and consistency in the fractional IVP framework [11]. For additional scientific articles on fractional ordinary and PDEs emerging in various fields of scientific research, see [12–14].

In a given situation, the level of flexibility of the nonlinear system in contemporary calculus (such as conventional calculus) is greater than that of the local differential equations operator. Authors [15,16] encompass the following applications of computation. As a consequence, intellectuals place an elevated significance on the investigation of non-integer order differentiation and integration. Geometrically, the arbitrarily defined order derivatives, which are predominantly predefined integrals, describe the complete function’s concentration, or the entire global integration variety [17,18]. The practise of academics has greatly boosted the efficiency of differential equations as well as quantitative and quantifiable scientific studies. It is worth noting that the following derivative operators were developed using conclusive essential methodologies. It is a well-established fact that there is currently no underlying solution to this problem. Consequently, the power law kernel has multiple interpretations. The Caputo fractional-order derivative (CFD) [19] is perhaps the most appealing underlying conceptualization. Dynamical formulae are notorious for being hard to address quantitatively or precisely. As a result, computational intelligence methodologies for evaluating the foregoing formulas have been constructed. Numerous intellectuals have mainly investigated computational perspectives to investigate fractional PDE under CFD [20,21].

Furthermore, owing to the quantitative intricacies of the fractional operators involved, figuring out the numerical method for fractional IVP processes can be occasionally challenging. In this context, computational and analytical strategies have been created and energized in order to explore the outcomes of various types of linear/nonlinear fractional IVP mechanisms. To reference a couple different ones: the Adomian decomposition method, Legendre polynomial, Lie symmetry analysis, Haar wavelet method, spectral collocation method, homotopy perturbation method, homotopy analysis transform method, reproducing kernel Hilbert space method, Bernoulli polynomials, B-spline functions, Chebyshev polynomials and the residue power series method, see [22–24].

In 2013, Omar Abu Arqub, a Jordanian mathematician [25], invented the residual power series method (RPSM). However, RPSM is an analytical procedure for tackling ordinary, partial, and fuzzy DEs, as well as fractional-order integro-DEs, which correlates with the Taylor’s series having the residual error function. It offers linear and nonlinear DE series strategies in the context of convergence series. For its inaugural moment, RPSM was used to develop solutions to fuzzy DEs in 2013. Arqub et al. [26] applied an efficient technique for addressing the solution of higher-order IVP. Arqub et al. [27] used the RPSM to consider numerous findings for dynamical fractional-order boundary value problems. El-Ajou et al. [28] expounded the novel recursive approach RPSM for establishing the solutions of the nonlinear fractional KdV-Burgers model. Later on, this dynamical scheme merged with several integral transforms to make it more comprehensive. It is a useful meta-heuristic algorithm because it applies with the help of closed-form functional information. In the scenario of nonlinear challenges, obtaining a solution in closed form is unattainable, and determining the series coefficients is a tough challenge. To address the shortcomings of the classic PSM, an optimized version of the PSM is introduced that treats the coefficient values as transmogrified operations that pursue a set of regulations and are ascertained by recurrence connections. For more details on RPSM, see [29–31].

In 1780, the French mathematician and physicist P. S. Laplace [32,33] proposed the integral transform. In 1822, J. Fourier [34] invented the Fourier transform. Laplace and Fourier transforms are the cornerstone of operations and maintenance interpretation, a strand of mathematical concepts with enormously potent implementations not just in mathematical modeling but also in different scientific fields such as thermodynamics, technology, cosmology, and so forth. This research is based on the implementation of the ARA Ts (ARA Ts), an innovative integral transform, introduced by Saadeh et al. [35]. This transform is an influential and multi-functional generalization that consolidates several configurations of the conventional Laplace transform, including the Sumudu transform [36], the Elzaki transform [37], the Natural transform [38], the Yang transform [39] and the Shehu transform [40].

Numerous publications have been written to explain dynamic processes that can be induced and propagated in a variety of concentrations and configurations. The majority of academics have concentrated on minimizing the fundamental formulae of varying concentration models to evolution problems in the pattern of PDES such as the Swift-Hohenberg model (KdV) equation, Burger equation, Black-Scholes model, Boussinesq equation and so on [41–44].

Another development of the RPSM is assembled in this article by acclimating the ARA Ts [35,45] to the RPSM technique [25,26]. In this article, the new framework, ARA-residual power series method (ARARPSM), is used to effectively resolve fractional-order PDEs. Furthermore, the detailed explanation of the nonlinear fractional-order PDEs is outlined below:

•   The ARARPSM is an efficacious approach and a novel method for obtaining numerical approximations to dynamical fractional PDEs in series pattern. The series coefficients can be ascertained quickly by employing the notion of limit at ∞, which also helps in saving time and resources when compared to earlier traditional methods.

•   Five problems are analyzed statistically to identify the reliability and robustness of the suggested technique. Furthermore, analytical findings are also compared with the existing results and are in agreement with the exact findings and several other techniques.

•   Diagrammatically, relevance is indeed discovered for multiple fractional-order derivative attributes and the statistical performances of the mean absolute deviation, mean deviation, Theil’s inequality coefficients, and semi-interquartile range. Hence, the methodology is accurate, easy to employ, not influenced by supercomputing iterations of inconsistencies, and doesn’t necessitate an enormous amount of memory storage or time.

•   This technique, unlike the conventional power series technique, somehow doesn’t entail identifying the coefficient values of the commensurate terms or the application of a recursion connection. The suggested restriction concept-based methodology shows series coefficients but not fractional derivatives, similar to the RPSM. Unlike RPSM, which also demands numerous computations to quantify multiple fractional derivatives during the completion of the task successfully, only a very few computations are required to evaluate the coefficients.

2  Preliminaries

This section provides a number of interpretations, characteristics, and some helpful findings that form the foundation of the novel methodology. The ARA Ts is derived using the classic Laplace integral. In order to simplify the method for solving ordinary and partial DEs in the temporal domain, Saadeh et al. [35] proposed the ARA Ts in 2020. ARA is the identifier of the proposed transform; the term is not an acronym. It has some interesting properties, such as the ability to generate multiple transforms by varying the significance of the index m, which was also initiated in [45], a duality with the Laplace transform, and the ability to navigate the singularity at time zero.

Definition 2.1. ([22]) For δ>0, the Caputo derivative of the mapping Ψ(u,t) is described as

𝒟tδΨ(u,t)=Jtn−δ𝒟tnΨ(u,t),δ∈(n−1,n),m∈N,u∈𝒥,t>0,

where 𝒥 signifies an interval and Jtδ is the time-fractional Riemann-Liouville integral operator order δ>0 stated as

JtδΨ(u,t)={1Γ(δ)∫0t(t−ζ)δ−1Ψ(u,ζ)dζ, 0<ζ<t,Ψ(u,ζ),δ=0.

Definition 2.2. ([35]) The ARA Ts of order m of the continuous mapping Ψ(u,t) on the interval 𝒥×[0,∞) for t, is stated by

𝒢m[Ψ(u,t)]=s∫0∞tm−1exp⁡(−st)Ψ(u,t)dt,s>0.

In the assertions that follow, we list a few ARA Ts ation fundamentals [35] that are crucial to our studies.

Suppose that there are two continuous mappings Ψ(u,t) and ℏ(u,t) defined on 𝒥×[0,∞) for which the ARA Ts exists, then we have

(i) 𝒢m[a¯Ψ(u,t)+b¯ℏ(u,t)]=a¯𝒢m[Ψ(u,t)]+b¯𝒢m[ℏ(u,t)], where a¯ and b¯ are nonzero constants,

(ii) lims↦∞𝒢1[Ψ(u,t)]=Ψ(u,0),u∈𝒥,s>0,

(iii) 𝒢2[tδ]=Γ(δ+2)sδ+1,δ>0,s>0,

(iv) 𝒢1[𝒟tδΨ(u,t)]=sδ𝒢2[Ψ(u,t)]−sδΨ(u,0),δ∈(0,1],u∈𝒥,s>0,

(v) 𝒢2[𝒟tδΨ(u,t)]=sδ𝒢2[Ψ(u,t)]−δsδ−1𝒢1[Ψ(u,t)]+(δ−1)sδ−1Ψ(u,0),δ∈(0,1],u∈𝒥,s>0,

(vi) 𝒢2[𝒟t2δΨ(u,t)]=s2δ𝒢2[Ψ(u,t)]−2δs2δ−1𝒢1[Ψ(u,t)]+(2δ−1)s2δ−1Ψ(u,0)+(δ−1)sδ−1𝒟tδΨ(u,0),δ∈(0,1],u∈𝒥,s>0,

(vii) lims↦∞𝒢2[Ψ(u,t)]=Ψ(u,0),u∈𝒥, s>0.

Theorem 2.3. ([25]) Assume there is a mapping Ψ(u,t) with fractional power series (FPS) representation at time t=0, is defined as follows:

Ψ(u,t)=∑m=0∞a¯m(u)tmδ,δ∈(n−1,n],n=1,2,...,t∈[0,α].

For continuous mappings Ψ(u,t) and 𝒟tmδΨ(u,t) defined on 𝒥×[0,∞), then the coefficients a¯m(u) have the following formulation:

a¯m(u)=𝒟tmδΨ(u,0)Γ(mδ+1),for m=0,1,2,...,

where 𝒟tmδ=𝒟tδ.𝒟tδ...𝒟tδ⏟m−times.

Theorem 2.4. ([46]) Suppose there is a continuous mapping Ψ(u,t) defined on 𝒥×[0,α] for which the ARA Ts for the variable t occurs and is expressed by the FPS form

𝒢2[Ψ(u,t)]=∑m=0∞ℏm(u)smδ+1,δ∈(0,1],u∈𝒥 and s>0.(1)

Then

ℏm(u)=(mδ+1)𝒟tmδΨ(u,0).(2)

Remark 1. (a) The ℓth truncated series of the series interpretation (1) is stated as follows:

𝒢2[Ψ(u,t)]ℓ=∑m=0ℓℏmsmδ+1.(3)

(b) For the ARA Ts of order two of the mapping Ψ(u,t) has the series interpretations (1), then the ARA Ts of order one can be written as follows:

𝒢1[Ψ(u,t)]=∑m=0∞ℏm(mδ+1)smδ.(4)

and the ℓth truncated series is stated as follows:

p1𝒢1[Ψ(u,t)]ℓ=∑m=0ℓℏm(mδ+1)smδ.(5)

(c) The inverse of the ARA Ts of order two for the FPS (1) is presented as follows:

Ψ(u,t)=𝒢2−1(∑m=0∞ℏmsmδ+1)(t)=∑m=0∞𝒟tmδΨ(u,0)Γ(mδ+1)tmδ.(6)

Theorem 2.5. ([46]) Assume that there is a continuous mapping Ψ(u,t) defined on 𝒥×[0,α] for which the ARA Ts for the variable t holds. Also, suppose that 𝒢1 has the subsequent series formulation:

𝒢1[Ψ(u,t)]=∑m=0∞Cm(u)smδ,

where Cm(u)=𝒟tmδΨ(u,0).

If |𝒢1[𝒟t(m+1)δΨ(u,t)]|≤ℳ on s∈(0,d1], then the remainder R¯m(u,s) holds the subsequent variant:

|R¯m(u,s)|≤ℳ(u)s(m+1)δ,u∈𝒥, s∈(0,d1].

3  Configuring Series Findings of FPDEs

In this section, the new framework, ARARPSM, is used to effectively resolve fractional-order PDEs of the form:

𝒟tδΨ(u,t)=∂Υ(u,t)∂u,δ∈(0,1], u∈𝒥, t>0(7)

subject to the initial settings

Ψ(u,0)=x¯(u),𝒟tδΨ(u,0)=b¯(u),(8)

where Υu signifies the nonlinear term relative to u of order r1, while 𝒟tδ denotes the CFD of order δ and Ψ(u,t) is the known function depending on variable u and t, respectively.

This part explains the ARA-RPS approach to addressing time-fractional PDEs. The proposed method relies on Taylor’s expansion to generate solitary solutions after applying the ARA Ts to the governing formulation.

Taking the initial value problem (IVP) (7) and (8), we implement the ARARPSM.

Apply the ARA Ts of order two 𝒢2 on both sides of the equation with respect to the variable t on (7)

𝒢2[𝒟tδΨ(u,t)]=𝒢2(𝒩u[Ψ(u,t)]).(9)

In view of assertion (vi) and the initial conditions (8), then (9) reduces to

𝒢2[Ψ(u,t)]−2δs𝒢1[Ψ(u,t)]+2δ−1sa¯(u)+δ−1sδ+1b¯(u)−1s2δ𝒢2[𝒩u(𝒢2−1[𝒢2[Ψ(u,t)]])]=0.(10)

Suppose that the respective series conceptions correspond to the ARA-RPS solution of formula (9) has the following form:

𝒢1[Ψ(u,t)]=∑m=0∞ℏm(u)(mδ+1)smδ,(11)

and

𝒢2[Ψ(u,t)]=∑m=0∞ℏm(u)smδ+1.(12)

Making the use of assertion (vii), we have

lims↦∞s𝒢2[Ψ(u,t)]=Ψ(u,0).(13)

Since ℏ0(u)=a¯(u). So that, the series expression (12) reduces to

𝒢2[Ψ(u,t)]=a1(u)s+ℏ(u)sδ+1+∑m=2∞ℏm(u)smδ+1.(14)

To obtain ℏ(u), taking product on both sides of (14) by sδ+1 and applying the limit as s↦∞, we have

lims↦∞sδ+1𝒢2[Ψ(u,t)]=lims↦∞sδa¯(u)+lims↦∞sδℏm(u)smδ+1.

or accordingly

ℏ1(u)=lims↦∞(sδ𝒢2[Ψ(u,t)]−sδ−1a¯(u)).

Now, assertion (v) provides that

ℏ1(u)=lims↦∞s(𝒢2[𝒟tδΨ(u,t)]+δsδ−1𝒢1[Ψ(u,t)]−δsδ−1a¯(u))=lims↦∞s𝒢2[𝒟tδΨ(u,t)]+lims↦∞δ(sδ𝒢1[𝒟tδΨ(u,t)]−sδa¯(u)).

Employing assertion (iv), we attain

ℏ1(u)=lims↦∞s𝒢2[𝒟tδΨ(u,t)]+δlims↦∞𝒢1[𝒟tδΨ(u,t)].

Assertions (ii) and (vii) lead to

ℏ1(u)=(δ+1)𝒟tδb¯(u).

Therefore, the ARA-RPS findings of (10) has the series formulations:

𝒢1[Ψ(u,t)]=a¯(u)+b¯(u)sδ+∑m=2∞ℏm(u)(mδ+1)smδ,(15)

𝒢2[Ψ(u,t)]=a¯(u)s+(δ+1)b¯(u)sδ+1+∑m=2∞ℏm(u)smδ+1,(16)

and the ℓth truncated series expansion of (15) and (16) have the following interpretations:

𝒢1[Ψ(u,t)]ℓ=a¯(u)+b¯(u)sδ+∑m=2ℓℏm(u)(mδ+1)smδ,(17)

𝒢2[Ψ(u,t)]ℓ=a¯(u)s+(δ+1)b¯(u)sδ+1+∑m=2ℓℏm(u)smδ+1.(18)

To determine the parameter estimates of series developments in (17) and (18), we describe the ARA-residual component of (10), as shown

𝒢2 Res(u,s)=𝒢2[Ψ(u,t)]−2δs𝒢1[Ψ(u,t)]+2δ−1sa¯(u)+δ−1sδ+1b¯(u)+1s2δ𝒢2[𝒩u(𝒢2−1[𝒢2[Ψ(u,t)]])],(19)

and the truncated ℓth residue function is

𝒢2 Resℓ(u,s)=𝒢2[Ψ(u,t)]ℓ−2δs𝒢1[Ψ(u,t)]ℓ+2δ−1sa¯(u)+δ−1sδ+1b¯(u)+1s2δ𝒢2[𝒩u(𝒢2−1[𝒢2[Ψ(u,t)]ℓ])],ℓ=2,3,...(20)

Multiplying both sides of (18) by sℓδ+1,ℓ=2,3,... and applying the limit as s↦∞ will enable you to identify the coefficients ℏm(u), m≥2 in the series expansion (20). After that, we have

lims↦∞sℓδ+1𝒢2 Resℓ(u,s)=0,ℓ=2,3,...

The ARA-RPS solution can be found by releasing the evidence below:

(a1) 𝒢2 Res(u,s)=0,u∈𝒥,s>0,(b1) limℓ↦∞𝒢2 Resℓ(u,s)=𝒢2 Res(u,s),u∈𝒥,s>0,(c1) lims↦∞s𝒢2 Res(u,s)=0 and lims↦∞𝒢2 Resℓ(u,s)=0,u∈𝒥,s>0,(d1) lims↦∞sℓδ+1𝒢2 Res(u,s)=lims↦∞sℓδ+1𝒢2 Resℓ(u,s)=0,u∈𝒥,s>0.

In order to achieve the solution of the IVP (7) and (8) in the feature space, the achieved coefficients ℏm(u) are supplemented in the series findings (12), and then the inverse ARA Ts of order two 𝒢2−1 is used.

4  Test Examples

Here, we take into consideration three well-known and significant time fractional PDEs with varying coefficients challenges in order to illustrate the effectiveness and appropriateness of ARARPSM.

Example 1. Assume the subsequent nonlinear time fractional (1+1) wave like equation [47]:

𝒟t2δΨ(u,t)−u2∂∂uΨ(u,t)∂2∂u2Ψ(u,t)+u2(∂2∂u2Ψ(u,t))2+Ψ(u,t)=0,(21)

where δ∈(0,1],u∈R and t≥0 supplemented with initial conditions

Ψ(u,0)=0,𝒟tδΨ(u,0)=u2.(22)

Proof. Implementing the ARA Ts of order two 𝒢2 on (21), we have

𝒢2[𝒟t2δΨ(u,t)]−𝒢2[u2∂∂uΨ(u,t)∂2∂u2Ψ(u,t)]+𝒢2[u2(∂2∂u2Ψ(u,t))2]+𝒢2[Ψ(u,t)]=0.(23)

It follows that

s2δ𝒢2[Ψ(u,t)]−2δs2δ−1𝒢1[Ψ(u,t)]+(2δ−1)s2δ−1Ψ(u,0)+(δ−1)sδ−1𝒟tδΨ(u,0)−𝒢2[u2∂u(𝒢2−1[𝒢2[Ψ(u,t)]])∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+𝒢2[u2∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+𝒢2[(𝒢2−1[𝒢2[Ψ(u,t)]])]=0.(24)

After simplification, (24) reduces to

𝒢2[Ψ(u,t)]−2δs𝒢1[Ψ(u,t)]+2δ−1sΨ(u,0)+δ−1sδ+1𝒟tδΨ(u,0)−1s2δ𝒢2[u2∂u(𝒢2−1[𝒢2[Ψ(u,t)]])∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+1s2δ𝒢2[u2∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+1s2δ𝒢2[(𝒢2−1[𝒢2[Ψ(u,t)]])]=0.(25)

Suppose that the ARA-RPS result of (25) has the subsequent series expression:

𝒢1[Ψ(u,t)]=∑m=0∞ℏm(u)(mδ+1)smδ,(26)

𝒢2[Ψ(u,t)]=∑m=0∞ℏm(u)smδ+1,(27)

and the ℓth truncated series of the expansion (26) and (27) are

𝒢1[Ψ(u,t)]ℓ=∑m=0ℓℏm(u)(mδ+1)smδ,(28)

𝒢2[Ψ(u,t)]ℓ=∑m=0ℓℏm(u)smδ+1.(29)

Conducting product both sides of (29) by s and applying the limit as s↦∞, yields

lims↦∞s𝒢2[Ψ(u,t)]ℓ=ℏ0(u)+lims↦ℓℏm(u)smδ,

In view of the following assumption, we have

lims↦∞s𝒢2[Ψ(u,t)]ℓ=Ψ(u,0)

and the initial settings mentioned in (22), we deduce that ℏ0(u)=Ψ(u,0). Therefore, the series expression stated in (29) reduces to

𝒢2[Ψ(u,t)]ℓ=Ψ(u,0)+ℏ1(u)sδ+1+∑m=2ℓℏm(u)smδ+1.(30)

In order to evaluate ℏ(u), taking product on both sides of (31) by sδ+1 and applying the limit as s↦∞, to find

lims↦∞sδ+1𝒢2[Ψ(u,t)]ℓ=lims↦∞sδΨ(u,0)+ℏ1(u)+lims↦∞∑m=2ℓℏm(u)s(m−1)δ.

Therefore, we have

lims↦∞sδ+1𝒢2[Ψ(u,t)]ℓ=lims↦∞sδ+1Ψ(u,0)+ℏ1(u).

It follows that

ℏ1(u)=lims↦∞s(sδ𝒢2[Ψ(u,t)]ℓ−sδ−1Ψ(u,0)).

Considering assertion (v) provides that

ℏ(u)=lims↦∞s[𝒢2[𝒟tδΨ(u,t)]]+δsδ−1𝒢2[Ψ(u,t)]−δsδ−1Ψ(u,0)=lims↦∞s𝒢2[𝒟tδΨ(u,t)]+lims↦∞δ[sδ𝒢1[Ψ(u,t)]−sδΨ(u,0)].

Making the use of assertion (iv), gives

ℏ1(u)=lims↦∞s[𝒢2[𝒟tδΨ(u,t)]]+δlims↦∞𝒢1[𝒟tδΨ(u,t)].

Making the use of assertion (ii) and (vii) lead us

ℏ1(u)=(δ+1)u2.

Therefore, the ARA-RPS findings of (25) has the subsequent series formulations:

𝒢1[Ψ(u,t)]=u2+∑m=2∞ℏm(u)(mδ+1)smδ,(31)

𝒢2[Ψ(u,t)]=(δ+1)u2+∑m=2∞ℏm(u)smδ+1,(32)

and the ℓth truncated series of the expansions (31) and (32) have the formulation

𝒢1[Ψ(u,t)]ℓ=u2+∑m=2ℓℏm(u)(mδ+1)smδ,(33)

𝒢2[Ψ(u,t)]ℓ=(δ+1)u2+∑m=2ℓℏm(u)smδ+1.(34)

Furthermore, we classify the ARA-residue function of (25), then

𝒢2 Res(u,s)=𝒢2[Ψ(u,t)]−2δs𝒢1[Ψ(u,t)]+2δ−1sΨ(u,0)+δ−1sδ+1u2−1s2δ𝒢2[u2∂u(𝒢2−1[𝒢2[Ψ(u,t)]])∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+1s2δ𝒢2[u2∂u2(𝒢2−1[𝒢2[Ψ(u,t)]])]+1s2δ𝒢2[(𝒢2−1[𝒢2[Ψ(u,t)]])](35)

and the ℓth ARA-residual mapping of (35) is

𝒢2 Resℓ(u,s)=𝒢2[Ψ(u,t)]ℓ−2δs𝒢1[Ψ(u,t)]ℓ+2δ−1sΨ(u,0)+δ−1sδ+1u2−1s2δ𝒢2[u2∂u(𝒢2−1[𝒢2[Ψ(u,t)]ℓ])∂u2(𝒢2−1[𝒢2[Ψ(u,t)]ℓ])]+1s2δ𝒢2[u2∂u2(𝒢2−1[𝒢2[Ψ(u,t)]ℓ])]+1s2δ𝒢2[(𝒢2−1[𝒢2[Ψ(u,t)]ℓ])].(36)

Utilizing the fact that

𝒢2 Res(u,s)=0,lims↦∞𝒢2 Resℓ(u,s)=0,

lims↦∞sℓδ+1𝒢2 Res(u,s)=lims↦∞sℓδ+1𝒢2 Resℓ(u,s)=0,ℓ=2,3,... .

In order to evaluate the second unknown coefficient ℏ2(u) by inserting the second truncated series 𝒢1[Ψ(u,t)]2 and 𝒢2[Ψ(u,t)]2 into the second ARA-residual function 𝒢2Res2(s) to find

𝒢2 Res2(s)=𝒢2[Ψ(u,t)]2−2δs𝒢1[Ψ(u,t)]2+2δ−1sΨ(u,0)+δ−1sδ+1u2−1s2δ𝒢2[u2∂u(𝒢2−1[𝒢2[Ψ(u,t)]2])∂u2(𝒢2−1[𝒢2[Ψ(u,t)]2])]+1s2δ𝒢2[u2∂u2(𝒢2−1[𝒢2[Ψ(u,t)]2])]+1s2δ𝒢2[(𝒢2−1[𝒢2[Ψ(u,t)]2])]=0.(37)