# On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform

Authors: Saima Rashid; Sobia Sultana; Rehana Ashraf; Mohammed K. A. Kaabar
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

The Black-Scholes model is well known for determining the behavior of capital asset pricing models in the finance sector. The present article deals with the Black-Scholes model via the Caputo fractional derivative and Atangana-Baleanu fractional derivative operator in the Caputo sense, respectively. The Jafari transform is merged with the Adomian decomposition method and new iterative transform method. It is worth mentioning that the Jafari transform is the unification of several existing transforms. Besides that, the convergence and uniqueness results are carried out for the aforesaid model. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest its applicability for a variety of nonlinear evolutionary problems.


---

## Body

## 1. Introduction

Recently, the subject of fractional calculus has garnered considerable prominence. Several well-known mathematicians have contributed to this field by proposing numerous fractional operators in various texts. The conclusions of contemporary calculus are often substantially more precise than those of ancient ones. It has presented the dynamic behavior of a variety of real-world situations that take place between two integers. Additionally, fractional operators have more dimensionality than integer differential operators such as Caputo, Liouville, Hadamard, Coimbra, Davison, Riesz, Riemann and Liouville, Weyl, and Jumarie, Caputo and Fabrizio [1], Atangana and Baleanu [2], and Scherer et al. [3] are some well-known fractional derivative formulations, see [4–9]. Furthermore, the Liouville-Caputo and AB operators are considered to be the best fractional filters in this field of research.Several methods that assess fractional differential equations (FDEs) for their reliability and trustworthiness are actively considered in [10, 11]. Certain popular approximate-analytical approaches including Adomian decomposition method [12], new iterative transform method [13], homotopy perturbation method [14], Haar wavelet method [15], Padé approximation [16], reproducing kernel Hilbert space method [17], new Legendre wavelet decomposition method [18], Bäcklund transformation method [19], and Lie symmetry analysis [20].Fischer Black and Myron Scholes developed a mathematical model for the capital asset pricing model in 1973. The revolutionary Black–Scholes model (BSM) is the foundation of modern financial theory which is remarkable to discuss contemporary economics without referencing the innovative BSM.The purpose of this research is to obtain new solutions by employing both decomposition method and Jafari iterative transform method into a BSM. In banking and finance, the fractional formulation of BSM is represented by [16]:
(1)DςϑU+ϖ22S2∂2U∂S2+ζS∂U∂S−ζU=0,subject to the playoff mapping
(2)US,T=maxS−E,0,where US,ς represents the option’s value at S asset values at time ς and T is the expiry period. The fundamental stock price is indicated by the letter E. The risk-free mortgage to expiry is represented by the variable ζ. The volatility of a financial commodity is represented by the constant ϖ. We also include the necessary hypotheses: a constant risk-free interest rate u, no processing fees, the ability to purchase and sell an unlimited number of stocks, and no prohibitions on speculative trading. Finally, we include European options. Furthermore, it is remarkable that U0,ς=0 and US,T≈S as S↦∞. The BSM in (1) can then be represented as a parabolic diffusion equation. Introducing the subsequent transformations
(3)S=Eexpw1,ς=T−2τϖ2,U=EQw1,ς,then, Equation (1) reduces to
(4)DςϑQw1,ς=∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς,subject to initial conditions
(5)Qw1,0=maxexpw1−1,0,where η signifies the equilibrium between the free relationship in inflation and stock volatility. In [20], Cen and Le introduced the generalized fractional BSM. The BSM is described this way:
(6)DςϑQw1,ς=−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς,subject to initial conditions
(7)Qw1,0=maxw1−25exp−0.06,0.The fractional BSM with one commodity has been explored extensively [21, 22]. The fractional BSM is an extended variant of the classical BSM that extends the model’s limitations. Meng and Wang [23] used the BSM to investigate fractional opportunity valuation. The covered call price for bank international trade in China was calculated using the fractional BSM. Their findings suggest that when it comes to evaluating the influence of the pricing system, the fractional BSM outperforms the classical BSM [24]. Fall et al. [14] estimated the Black-Scholes option pricing equations via the homotopy perturbation method. Matadi and Zondi [25] contemplated the invariant solutions of BSM via theOrnstein-Uhlenbeck process. Kumar et al. [26] demonstrated numerical computation of fractional BSM arising in the financial market. Yavuz and Özdemir [27] proposed a diverse approach to the European option pricing model with a new fractional operator.Amidst Gorge Adomian’s massive boost in 1980, the Adomian decomposition method introduced a well-noted terminology. It has been intensively implemented for a diverse set of nonlinear PDEs, for instance, Fisher’s model [28] and Zakharov–Kuznetsov equation [29]. The ADM was determined to be significantly related to a variety of integral transforms, including Laplace, Swai, Mohand, Aboodh, and Elzaki. Very recently, Jafari [30] propounded a well-known integral transform which is known as the Jafari transform. The dominant feature of this transformation is that it has the ability to recapture several existing transformations, see Remark 8.In 2006, Yavuz and Özdemir [27] expounded a new iterative transform method (NITM) that is intensively employed by numerous researchers due to its frequent applicability in fractional ODES and PDEs. The recursive technique tends to the exact solution if it exists via successive approximations. A small proportion of estimates can be employed for analytical reasons with a reasonable level of accuracy for particular problems. The NITM does not need some restricted hypothesis for handling nonlinearity factors. For example, in [28], the authors employed NITM to find the numerical solution of the fifth- and sixth-order nonlinear boundary value problems, Rashid et al. [29] applied NITM to obtain the solution of the fractional Fornberg-Whitham equation, and Jafari [30] constructed the Laplace decomposition algorithm via NITM.Owing to the aforementioned trend, to obtain the explicit solution of the time-fractional BSM, we employ the Jafari transform decomposition technique (JDM) and Jafari iterative transform method (JITM). The Jafari transform merged the Adomian decomposition method and a new iterative method in an efficient manner to develop novel algorithmic approaches. The Jafari transform is the refinement of several existing transforms, see Remark8. Both projected schemes yield analytical solutions in a convergent series form. Mathematical characterizations of the BSM are illustrated via the AB fractional derivative operator in the Caputo sense. Simulation and tabulation studies depict a clear picture of the proposed approaches. The analytical solution, especially for fractional PDEs, is a useful mechanism for analyzing the behaviors of solutions that are challenging to numerically solve. The analytical solution can be used to investigate macroeconomic behaviors.This article’s entire content is divided into seven parts, which are described in the following order: Section2 summarises and presents the core concepts and terminology of the singular power law fractional derivative and nonsingular Atangana-Baleanu fractional derivative in the Caputo sense. In Section 3, two novel algorithms are developed via the new integral transform. In Section 4, convergence and uniqueness analyses are discussed and presented for the proposed model. Section 5 is the main part of the proposed work where we present a debate on the results and their interpretation. Finally, concluding remarks are presented in Section 7.


## 2. Preliminaries

In this section, we present some essential concepts, notions, and definitions concerning fractional derivative operators depending on power and Mittag-Leffler as a kernel, along with the detailed consequences of the Jafari transform.Definition 1 (see [1]).
The Caputo fractional derivativeCFD is described as follows:
(8)D0cςϑQς=1Γr−ϑ∫0ςQrw1ς−w1ϑ+1−rdw1,r−1<ϑ<r,drdςrQς,ϑ=r.Definition 2 (see [2]).
The Atangana-Baleanu fractional derivative operator in the Caputo formABC is stated as follows:
(9)η1ABCDςϑQς=Aϑ1−ϑ∫η1ςQ′ςEϑ−ϑς−w1ϑ1−ϑdw1,where Q∈H1a1,a2Sobolevspace,a1<a2,ϑ∈0,1 and Aϑ signifies a normalization function as Aϑ=A0=A1=1.Definition 3 (see [2]).
The fractional integral of theABC operator is described as follows:
(10)η1ABCIςϑQς=1−ϑAϑQς+ϑΓϑAϑ∫η1ςQw1ς−w1ϑ−1dw1.Definition 4 (see [30]).
Consider an integrable mappingQς defined on a set P, then
(11)P=Qς:∃M>0,κ>0,∣Qς∣<Mexpκς,ifς≥0.Definition 5 (see [30]).
Suppose the mappingsϕs,ψs:ℝ+↦ℝ+ such that ψs≠0∀s∈ℝ+. The Jafari transform of the mapping Qς presented by Qs is described as
(12)JQς,s=Qs=ϕs∫0∞Qςexp−ψsςdς.Theorem 6 (see [30]) (convolution property).
For Jafari transform, the subsequent holds true:(13)JQ1∗Q2=1ϕsQ1s∗Q2s.Definition 7.
The Jafari transform of the CFD operator is stated as follows:(14)J0cDςϑQς,s=ψϑsQs−ϕs∑κ=0ϑ−1ψϑ−κ−1sQκ0,r−1<ϑ<r,ϕ,ψ>0.Remark 8.
Definition7 leads to the following conclusions:
(1)
Takingϕs=1 and ψs=s, then we acquire the Laplace transform [35](2)
Takingϕs1/s and ψs=1/s, then we acquire the α-Laplace transform [36](3)
Takingϕs=1/s and ψs=1/s, then we acquire the Sumudu transform [37](4)
Takingϕs=1/s and ψs=1, then we acquire the Aboodh transform [38](5)
Takingϕs=s and ψs=s2, then we acquire the Pourreza transform [39, 40](6)
Takingϕs=s and ψs=1/s, then we acquire the Elzaki transform [41](7)
Takingϕs=w2 and ψs=s/w2, then we acquire the natural transform [42](8)
Takingϕs=s2 and ψs=s, then we acquire the Mohand transform [43](9)
Takingϕs=1/s2 and ψs=1/s, then we acquire the Sawi transform [44](10)
Takingϕs=1 and ψs=1/s, then we get the Kamal transform [45].(11)
Takingϕs=sα and ψs=1/s, then we acquire the G−transform [46, 47]Definition 9 (see [48]).
The Jafari transform of the ABC fractional derivative operator is described as(15)J0ABCDςϑQς,sϑ=Aϑψϑsϑ+1−ϑψϑsQs−ϕsψsQ0.Remark 10.
Definition9 leads to the following conclusions:
(1)
Takingϕs=1 and ψs=s, then we acquire the Laplace transform of ABC fractional derivative operator [2, 49](2)
Takingϕs=s and ψs=1/s, then we acquire the Elzaki transform of ABC fractional derivative operator [50](3)
Takingϕs=ψs=1/s, then we get the Sumudu transform of ABC fractional derivative operator [51](4)
Takingϕs=1 and ψs=s/w2, then we get the Shehu transform of ABC fractional derivative operator [51]Definition 11 (see [52]).
The Mittag-Leffler function for a single parameter is described as(16)Eϑz=∑κ=0∞zκΓκϑ+1,ϑ,z∈ℂ,Rϑ≥0.


## 3. New Semianalytical Approach for Nonlinear PDEs

Consider the generic fractional form of PDE:(17)DςϑQw1,ς+LQw1,ς+ℕQw1,ς=Fw1,ς,ς>0,0<ϑ≤1,with ICs
(18)Qw1,0=Gw1,where Dςϑ=∂ϑQw1,ς/∂ςϑ symbolizes the Caputo and ABC fractional derivative of order ϑ∈0,1 while L and ℕ denotes the linear and nonlinear factors, respectively. Also, Fw1,ς represents the source term.


### 3.1. Configuration of Jafari Decomposition Method

Taking into account the Jafari transform to (17), we acquire
(19)JDςϑQw1,ς+L¯Qw1,ς+ℕ¯Qw1,ς=JFw1,ς.Firstly, the differentiation rule of Jafari transform with respect to CFD was applied; then, we apply the ABC fractional derivative operator as follows:(20)ψϑςUw1,ς=ϕς∑q=0n−1ψϑ−1−qςQq0+JL¯Qw1,ς+ℕ¯Qw1,ς+JFw1,ς,(21)ψϑςAϑϑ+1−ϑψϑςUw1,ς=ϕςψςψϑςAϑϑ+1−ϑψϑςQ0+JL¯Qw1,ς+ℕ¯Qw1,ς+JFw1,ς.The inverse Jafari transform of (20) and (21), respectively, yields
(22)Qw1,ς=J−1ϕς∑q=0n−1ψςϑ−q−1Qq0+1ψϑςJFw1,ς−J−11ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς.Qw1,ς=J−1ϕςψςQ0+ϑ+1−ϑψϑςψϑςAϑJFw1,ς−J−1ϑ+1−ϑψϑςψϑςAϑJL¯Qw1,ς+ℕ¯Qw1,ς.Therefore, the Jafari decomposition method was utilized to derive the solution of (17) by satisfying the assumption that Qw1,ς has a solution of this equation which can be expressed as
(23)Qw1,ς=∑q=0∞Qqw1,ς.Thus, the nonlinear termℕ¯w1,ς can be evaluated by the Adomian decomposition method prescribed as
(24)ℕ¯Qw1,ς=∑q=0∞A~qQ0,Q1,⋯,q=0,1,⋯,where
(25)A~qQ0,Q1,⋯=1q!dqdϑqℕ¯∑j=0∞ϑjQjϑ=0,q>0.Inserting (23) and (24) into (26) and (27), respectively, we have
(26)∑q=0∞Qqw1,ς=Gw1+G~w1−J−11ψϑςJL¯Qw1,ς+∑q=0∞A~q,(27)∑q=0∞Qqw1,ς=Gw1+G~w1−J−1ϑ+1−ϑψϑςAϑψϑςJL¯Qw1,ς+∑q=0∞A~q.Consequently, the recursive technique for (26) and (18) are established as
(28)Q0w1,ς=Gw1+G~w1,q=0,Qq+1w1,ς=−J−11ψϑςJL¯Qqw1,ς+∑q=0∞A~q,q≥1,Qq+1w1,ς=−J−1ϑ+1−ϑψϑςAϑψϑςJL¯Qqw1,ς+∑q=0∞A~q,q≥1.


### 3.2. Construction of Jafari Iterative Transform Method

With the aid of Jafari transform to (17) along with the IC (18), we obtain
(29)JDςϑQw1,ς+L¯Qw1,ς+ℕ¯Qw1,ς=JFw1,ς.First, we apply the differentiation rule of Jafari transform for CFD, and then, we consider for ABC fractional derivative operator, respectively, we get(30)JQw1,ς=ϕςψϑς∑q=0n−1ψϑ−1−qςQqψ,0−1ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς+1ψϑςJFw1,ς,(31)JQw1,ς=ϕςψϑς∑q=0n−1ψϑ−1−qςQqψ,0−ψϑς1−ϑ+ϑAϑψϑςJL¯Qw1,ς+ℕ¯Qw1,ς+ψϑς1−ϑ+ϑAϑψϑςJFw1,ς.By the virtue of the inverse Jafari transform of (30) and (31), respectively, this yields
(32)Qw1,ς=Gψ+J−11ψϑςJFw1,ς−J−11ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς,(33)Qw1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςJFw1,ς−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Qw1,ς+ℕ¯Qw1,ς.Using the fact of an iterative process, we find(34)Qw1,ς=∑q=0∞Qqw1,ς.Also, the operatorL¯ is linear; therefore,
(35)L¯∑q=0∞Qqw1,ς=∑q=0∞L¯Qqw1,ς,and the nonlinearity ℕ¯ dealt by (see [27])
(36)ℕ¯∑q=0∞Qqw1,ς=ℕ¯Q0w1,ς+∑q=1∞ℕ¯∑κ=0qQqw1,ς−ℕ¯∑κ=0q−1Qqw1,ς=ℕ¯Q0+∑q=1∞Dq,where Dq=ℕ¯∑κ=0qQq−ℕ¯∑κ=0q−1Qq.Plugging (37), (39), and (36) into (32) and (33), respectively, we attain
(37)∑q=0∞Qqw1,ς=Gψ+J−11ψϑςJFw1,ς,−J−11ψϑςJL¯∑κ=0qQqw1,ς+ℕ¯Q0+∑κ=1qDq,(38)∑q=0∞Qqw1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςJFw1,ς,−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯∑q=0qQqw1,ς+ℕ¯Q0+∑q=1qDq.Finally, we derive the following iterative process for CFD:(39)Q0w1,ς=Gψ+J−11ψϑςLFw1,ς,Q1w1,ς=−J−11ψϑςJL¯Q0w1,ς+ℕ¯Q0w1,ς,⋮Qq+1w1,ς=−J−11ψϑςJL¯Qqw1,ς+Dq.Again, the iterative process for ABC fractional derivative operator is presented as follows:(40)Q0w1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςLFw1,ς,Q1w1,ς=−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Q0w1,ς+ℕ¯Q0w1,ς,⋮Qq+1w1,ς=−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Qqw1,ς+Dq.Finally, (37), (39), and (40) produce the q-term solution in series representation, stated as
(41)Qw1,ς¬Q0w1,ς+Q1w1,ς+Q2w1,ς+⋯+Qqw1,ς,q∈ℕ.


## 3.1. Configuration of Jafari Decomposition Method

Taking into account the Jafari transform to (17), we acquire
(19)JDςϑQw1,ς+L¯Qw1,ς+ℕ¯Qw1,ς=JFw1,ς.Firstly, the differentiation rule of Jafari transform with respect to CFD was applied; then, we apply the ABC fractional derivative operator as follows:(20)ψϑςUw1,ς=ϕς∑q=0n−1ψϑ−1−qςQq0+JL¯Qw1,ς+ℕ¯Qw1,ς+JFw1,ς,(21)ψϑςAϑϑ+1−ϑψϑςUw1,ς=ϕςψςψϑςAϑϑ+1−ϑψϑςQ0+JL¯Qw1,ς+ℕ¯Qw1,ς+JFw1,ς.The inverse Jafari transform of (20) and (21), respectively, yields
(22)Qw1,ς=J−1ϕς∑q=0n−1ψςϑ−q−1Qq0+1ψϑςJFw1,ς−J−11ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς.Qw1,ς=J−1ϕςψςQ0+ϑ+1−ϑψϑςψϑςAϑJFw1,ς−J−1ϑ+1−ϑψϑςψϑςAϑJL¯Qw1,ς+ℕ¯Qw1,ς.Therefore, the Jafari decomposition method was utilized to derive the solution of (17) by satisfying the assumption that Qw1,ς has a solution of this equation which can be expressed as
(23)Qw1,ς=∑q=0∞Qqw1,ς.Thus, the nonlinear termℕ¯w1,ς can be evaluated by the Adomian decomposition method prescribed as
(24)ℕ¯Qw1,ς=∑q=0∞A~qQ0,Q1,⋯,q=0,1,⋯,where
(25)A~qQ0,Q1,⋯=1q!dqdϑqℕ¯∑j=0∞ϑjQjϑ=0,q>0.Inserting (23) and (24) into (26) and (27), respectively, we have
(26)∑q=0∞Qqw1,ς=Gw1+G~w1−J−11ψϑςJL¯Qw1,ς+∑q=0∞A~q,(27)∑q=0∞Qqw1,ς=Gw1+G~w1−J−1ϑ+1−ϑψϑςAϑψϑςJL¯Qw1,ς+∑q=0∞A~q.Consequently, the recursive technique for (26) and (18) are established as
(28)Q0w1,ς=Gw1+G~w1,q=0,Qq+1w1,ς=−J−11ψϑςJL¯Qqw1,ς+∑q=0∞A~q,q≥1,Qq+1w1,ς=−J−1ϑ+1−ϑψϑςAϑψϑςJL¯Qqw1,ς+∑q=0∞A~q,q≥1.


## 3.2. Construction of Jafari Iterative Transform Method

With the aid of Jafari transform to (17) along with the IC (18), we obtain
(29)JDςϑQw1,ς+L¯Qw1,ς+ℕ¯Qw1,ς=JFw1,ς.First, we apply the differentiation rule of Jafari transform for CFD, and then, we consider for ABC fractional derivative operator, respectively, we get(30)JQw1,ς=ϕςψϑς∑q=0n−1ψϑ−1−qςQqψ,0−1ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς+1ψϑςJFw1,ς,(31)JQw1,ς=ϕςψϑς∑q=0n−1ψϑ−1−qςQqψ,0−ψϑς1−ϑ+ϑAϑψϑςJL¯Qw1,ς+ℕ¯Qw1,ς+ψϑς1−ϑ+ϑAϑψϑςJFw1,ς.By the virtue of the inverse Jafari transform of (30) and (31), respectively, this yields
(32)Qw1,ς=Gψ+J−11ψϑςJFw1,ς−J−11ψϑςJL¯Qw1,ς+ℕ¯Qw1,ς,(33)Qw1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςJFw1,ς−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Qw1,ς+ℕ¯Qw1,ς.Using the fact of an iterative process, we find(34)Qw1,ς=∑q=0∞Qqw1,ς.Also, the operatorL¯ is linear; therefore,
(35)L¯∑q=0∞Qqw1,ς=∑q=0∞L¯Qqw1,ς,and the nonlinearity ℕ¯ dealt by (see [27])
(36)ℕ¯∑q=0∞Qqw1,ς=ℕ¯Q0w1,ς+∑q=1∞ℕ¯∑κ=0qQqw1,ς−ℕ¯∑κ=0q−1Qqw1,ς=ℕ¯Q0+∑q=1∞Dq,where Dq=ℕ¯∑κ=0qQq−ℕ¯∑κ=0q−1Qq.Plugging (37), (39), and (36) into (32) and (33), respectively, we attain
(37)∑q=0∞Qqw1,ς=Gψ+J−11ψϑςJFw1,ς,−J−11ψϑςJL¯∑κ=0qQqw1,ς+ℕ¯Q0+∑κ=1qDq,(38)∑q=0∞Qqw1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςJFw1,ς,−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯∑q=0qQqw1,ς+ℕ¯Q0+∑q=1qDq.Finally, we derive the following iterative process for CFD:(39)Q0w1,ς=Gψ+J−11ψϑςLFw1,ς,Q1w1,ς=−J−11ψϑςJL¯Q0w1,ς+ℕ¯Q0w1,ς,⋮Qq+1w1,ς=−J−11ψϑςJL¯Qqw1,ς+Dq.Again, the iterative process for ABC fractional derivative operator is presented as follows:(40)Q0w1,ς=Gψ+J−1ψϑς1−ϑ+ϑAϑψϑςLFw1,ς,Q1w1,ς=−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Q0w1,ς+ℕ¯Q0w1,ς,⋮Qq+1w1,ς=−J−1ψϑς1−ϑ+ϑAϑψϑςJL¯Qqw1,ς+Dq.Finally, (37), (39), and (40) produce the q-term solution in series representation, stated as
(41)Qw1,ς¬Q0w1,ς+Q1w1,ς+Q2w1,ς+⋯+Qqw1,ς,q∈ℕ.


## 4. Convergence and Uniqueness Analyses of BSM via ABC Fractional Derivative Operator

The subsequent subsections will highlight how sufficient requirements guarantee the emergence of a unique solution. Our anticipated existence of solutions in the case of JDM is followed by [53].Theorem 12 (uniqueness theorem).
Equation (24) has a unique solution whenever 0<ε<1, where ε=K1+K2+K3/Aϑϑςϑ/Γϑ+1+1−ϑ.Proof.
Assuming all continuous functions on the Banach space are denoted byΩ=ℂI,∙. Also, suppose that I=0,T have the norm ∙. Now, we define a function V:Ω↦Ω such that
(42)Qℓ+1w1,ς=Qw1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJLQℓw1,ς+P¯Qℓw1,ς+ℕQℓw1,ς,ℓ≥0,where LQw1,ς≡∂3Qw1,ς/∂w12 and P¯Qw1,ς≡∂Qw1,ς/∂w1. Here, suppose that LQw1,ς and ℕQw1,ς are also Lipschitzian with ∣P¯Q−P¯Q^∣<K1∣Q−Q^∣ and ∣LQ−LQ^∣<K2∣Q−Q^∣, where K1 and K2 are Lipschitz constant, respectively, and Q,Q^ are distinct functional values.
(43)∥VQ−VQ^∥=maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJLQw1,ς+P¯Qw1,ς+ℕQw1,ς−J−1ψϑς1−ϑ+ϑAϑψϑςJLQ^w1,ς+P¯Q^w1,ς+ℕQ^w1,ς≤maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJLQw1,ς−LQ^w1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJP¯Qw1,ς−P¯Q^w1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJℕQw1,ς−ℕQ^w1,ς≤maxς∈IK1J−1ψϑς1−ϑ+ϑAϑψϑςJ∣Qw1,ς−Q^w1,ς∣+K2J−1ψϑς1−ϑ+ϑAϑψϑςJ∣Qw1,ς−Q^w1,ς∣+K3J−1ψϑς1−ϑ+ϑAϑψϑςJ∣Qw1,ς−Q^w1,ς∣≤maxς∈IK1+K2+K3J−1ψϑς1−ϑ+ϑAϑψϑςJ∣Qw1,ς−Q^w1,ς∣≤K1+K2+K3J−1ψϑς1−ϑ+ϑAϑψϑςJ∥Qw1,ς−Q^w1,ς∥=K1+K2+K3J−1ψϑς1−ϑ+ϑAϑψϑςϕsψs∥Qw1,ς−Q^w1,ς∥=K1+K2+K3AϑϑςϑΓϑ+1+1−ϑ∥Qw1,ς−Q^w1,ς∥.
Since0<ε<1, the mapping is contraction. Consequently, by the Banach contraction fixed point theorem, (17) has a unique. This gives the desired result.Theorem 13 (convergence analysis).
The general form solution of (17) will be convergent.Proof.
Assume thatW^ℓ be the nth partial sum, i.e., W^ℓ=∑m=0ℓQℓw1,ς. Here, we prove that a Cauchy sequence W^ℓ in Banach space U.
We acquire by considering a new form of Adomian polynomials.(44)R¯W^ℓ=H~ℓ+∑p=0ℓ−1H~p,ℕW^ℓ=H~ℓ+∑c=0ℓ−1H~c.
Now,(45)∥W^ℓ−W^q∥=maxς∈I∣W^ℓ−W^q∣=maxς∈I∣∑m=q+1ℓQ^w1,ς∣,m=1,2,3,⋯≤maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=q+1ℓLQℓ−1w1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=q+1ℓP¯Qℓ−1w1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=q+1ℓH~ℓ−1w1,ς=maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1LQℓw1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1P¯Qℓw1,ς+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1H~ℓw1,ς≤maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1LW^ℓ−1−LW^q−1+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1P¯W^ℓ−1−P¯W^q−1+J−1ψϑς1−ϑ+ϑAϑψϑςJ∑m=qℓ−1ℕW^ℓ−1−ℕW^q−1≤maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJLW^ℓ−1−LW^q−1+J−1ψϑς1−ϑ+ϑAϑψϑςJP¯W^ℓ−1−P¯W^q−1+J−1ψϑς1−ϑ+ϑAϑψϑςJℕW^ℓ−1−ℕW^q−1≤K1maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJW^ℓ−1−W^q−1+K2maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJW^ℓ−1−W^q−1+K3maxς∈IJ−1ψϑς1−ϑ+ϑAϑψϑςJW^ℓ−1−W^q−1=K1+K2+K3J−1ψϑς1−ϑ+ϑAϑψϑςϕsψϑsW^ℓ−1−W^q−1.=K1+K2+K3AϑϑςϑΓϑ+1+1−ϑW^ℓ−1−W^q−1.
Considern=q+1, then,
(46)∥W^q+1−W^q∥≤ε∥W^q−W^q−1∥≤ε2∥W^q−1−W^q−2∥≤⋯≤εq∥W^1−W^0∥,where ε=K1+K2+K3/Aϑϑςϑ/Γϑ+1+1−ϑ. Now, from triangular inequality we have
(47)∥W^ℓ−W^q∥≤∥W^q+1−W^q∥+∥W^q+2−W^q+1∥+⋯+∥W^ℓ−W^ℓ−1∥≤εq+εq+1+⋯+εℓ−1∥W^1−W^0∥≤εq1−εℓ−qε∥Q1∥,since 0<ε<1, we have 1−εℓ−q<1, then,
(48)∥W^ℓ−W^q∥≤εq1−εmaxς∈I∥Q1∥.
Therefore,∣Q1∣<∞ (since Qw1,ς is bounded). Furthermore, as q↦∞, then ∥W^ℓ−W^q∥↦0. Thus, W^1 is a Cauchy sequence in K. Consequently, the series ∑n=0∞Qℓ is convergent and this yields the immediate consequence.Theorem 14 (see [53]) (error estimate).
The absolute error of the series solution (17) to (24) is calculated as
(49)maxς∈I∣Qw1,ς−∑ℓ=1qQℓw1,ς∣≤εq1−εmaxς∈I∥Q1∥.


## 5. Physical Interpretation of Time-Fractional Black-Scholes Models

In this section, we compute the approximate analytical solution of BSM via the CFD and ABC fractional derivative operators by using the Jafari decomposition method.


### 5.1. Jafari Decomposition Method

Example 1 (see [16]).
Assume the time-fractional one-dimensional BSM (4) subject to IC (5).Case 1.
Firstly, we solve the (4) by using Caputo fractional derivative operator incorporating the Jafari decomposition method.Applying Jafari transform to both sides of (4), we have
(50)JDςϑQw1,ς=J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Using the differentiation rule of Jafari transform, we have(51)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.In view of (5), we get
(52)JQw1,ς=ϕςψϑςmaxexpw1−1,0+1ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the inverse Jafari transform on both sides yields(53)Qw1,ς=J−1ϕςψϑςmaxexpw1−1,0+J−11ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.With the help of Jafari decomposition method, we find(54)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form
(55)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−11ψϑςJ∑q=0∞Qw1,ςw1w1+η−1∑q=0∞Qw1,ςw1−η∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−11ψϑςJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0J−1ϕsψϑ+1s=ηmaxexpw1,0−ηmaxexpw1−1,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0ς3ϑΓ3ϑ+1.⋮.The approximate solution for Example1 is expressed as
(56)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηςϑΓϑ+1+η2ς2ϑΓ2ϑ+1−η3ς3ϑΓ3ϑ+1+⋯+maxexpw1,01−1+ηςϑΓϑ+1−η2ς2ϑΓ2ϑ+1+η3ς3ϑΓ3ϑ+1+⋯=maxexpw1−1,0Eϑ−ηςϑ+maxexpw1,0Eϑ1−ηςϑ.Case 2.
Now, we solve(4) by using the ABC fractional derivative operator incorporating the Jafari decomposition method.Considering (50) and using the differentiation rule of Jafari transform, we have
(57)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.In view of (5), we get
(58)JQw1,ς=ϕςψϑςmaxexpw1−1,0+ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the inverse Jafari transform on both sides of the above equation yields(59)Qw1,ς=J−1ϕςψϑςmaxexpw1−1,0+J−1ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.By the Jafari decomposition method, we find:(60)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(61)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ∑q=0∞Qw1,ςw1w1+η−1∑q=0∞Qw1,ςw1−η∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The approximate solution for Example1 is expressed as:
(62)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯+maxexpw1,0−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.The integer-order solution of Example1 can be obtained with the aid of Taylor series expansion and setting ϑ=1 as follows:
(63)Qw1,ς=maxexpw1−1,0exp−ης+maxexpw1,01−exp−ης.Example 2 (see [20]).
Assume the time-fractional one-dimensional BSM (6) subject to IC (7).Case 1.
Firstly, we solve the (6) in the sense of Caputo fractional derivative operator incorporating the Jafari decomposition method.Applying Jafari transform to both sides of (6), we have
(64)JDςϑQw1,ς=J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Using the differentiation rule of Jafari transform, we have(65)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.In view of (7), we get
(66)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+1ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the inverse Jafari transform on both sides yields(67)Qw1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0+J−11ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.By the Jafari decomposition method, we find(68)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(69)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−11ψϑςJ−0.082+sinw12w12∑q=0∞Qw1,ςw1w1−0.06w1∑q=0∞Qw1,ςw1+0.06∑q=0∞Qw1,ς,∀q=0,1,2,⋯,Q1w1,ς=J−11ψϑςJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0J−1ϕsψϑ+1s=−0.06w1+0.06maxw1−25exp−0.06,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−0.000216w1+0.00216maxw1−25exp−0.06,0ς3ϑΓ3ϑ+1,⋮.The approximate solution for Example2 is expressed as
(70)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06×1−1−0.06ςϑΓϑ+1−0.0036ς2ϑΓ2ϑ+1−0.000216ς3ϑΓ3ϑ+1+⋯=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06,01−Eϑ0.06ςϑ.Case 2.
Now, we solve the (6) by using ABC fractional derivative operator incorporating the Jafari decomposition method.Considering (64) and using the differentian rule of Jafari transform, we have
(71)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.In view of (7), we get
(72)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the inverse Jafari transform on both sides yields:(73)Qw1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0+J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.By the Jafari decomposition method, we find:(74)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(75)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∑q=0∞Qw1,ςw1w1−0.06w1∑q=0∞Qw1,ςw1+0.06∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−−0.000216w1+0.00216maxw1−25exp−0.06,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The approximate solution for Example2 is expressed as:
(76)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0−0.06w1−0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+ϑ−0.0036w1−0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.The integer-order solution of Example2 can be obtained with the aid of Taylor series expansion and setting ϑ=1 as follows:
(77)Qw1,ς=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06,01−exp−0.06ς.


### 5.2. Jafari Iterative Transform Method

Example 3 (see[16]).
Assume the time-fractional one-dimensional BSM (4) subject to IC (5).Case 1.
First, we surmise the Caputo fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (4) with IC (5), we have:
(78)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Correspondingly, we get(79)JQw1,ς=ϕςψϑςmaxexpw1−1,0+1ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the suggested analytical method, we obtain(80)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0,Q1w1,ς=J−11ψϑςJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0J−1ϕsψϑ+1s=ηmaxexpw1,0−ηmaxexpw1−1,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0ς3ϑΓ3ϑ+1.⋮.The series form solution is(81)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(82)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηςϑΓϑ+1+η2ς2ϑΓ2ϑ+1−η3ς3ϑΓ3ϑ+1+⋯+maxexpw1,01−1+ηςϑΓϑ+1−η2ς2ϑΓ2ϑ+1+η3ς3ϑΓ3ϑ+1+⋯=maxexpw1−1,0Eϑ−ηςϑ+maxexpw1,0Eϑ1−ηςϑ.Case 2.
Now, we consider the ABC fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (4) with IC (5), we have
(83)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Correspondingly, we get(84)JQw1,ς=ϕςψϑςmaxexpw1−1,0+ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the suggested analytical method, we obtain(85)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The series form solution is(86)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(87)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯+maxexpw1,0−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.Example 4 (see [20]).
Assume the time-fractional one-dimensional BSM (6) subject to IC (7).Case 1.
First, we surmise the Caputo fractional derivative operator for (6) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (6) with IC (7), we have
(88)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Correspondingly, we get:(89)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+1ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the suggested analytical method, we obtain(90)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0,Q1w1,ς=J−11ψϑςJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0J−1ϕsψϑ+1s=−0.06w1+0.06maxw1−25exp−0.06,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−0.000216w1+0.00216maxw1−25exp−0.06,0ς3ϑΓ3ϑ+1.⋮.The series form solution is(91)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(92)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06×1−1−0.06ςϑΓϑ+1−0.0036ς2ϑΓ2ϑ+1−0.000216ς3ϑΓ3ϑ+1+⋯=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.061−Eϑ0.06ςϑ.Case 2.
Now, we consider the ABC fractional derivative operator for (6) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (6) with IC (7), we have
(93)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Correspondingly, we get(94)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the suggested analytical method, we obtain(95)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−−0.000216w1+0.00216maxw1−25exp−0.06,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The series form solution is(96)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(97)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0−0.06w1−0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+ϑ−0.0036w1−0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.


## 5.1. Jafari Decomposition Method

Example 1 (see [16]).
Assume the time-fractional one-dimensional BSM (4) subject to IC (5).Case 1.
Firstly, we solve the (4) by using Caputo fractional derivative operator incorporating the Jafari decomposition method.Applying Jafari transform to both sides of (4), we have
(50)JDςϑQw1,ς=J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Using the differentiation rule of Jafari transform, we have(51)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.In view of (5), we get
(52)JQw1,ς=ϕςψϑςmaxexpw1−1,0+1ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the inverse Jafari transform on both sides yields(53)Qw1,ς=J−1ϕςψϑςmaxexpw1−1,0+J−11ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.With the help of Jafari decomposition method, we find(54)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form
(55)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−11ψϑςJ∑q=0∞Qw1,ςw1w1+η−1∑q=0∞Qw1,ςw1−η∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−11ψϑςJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0J−1ϕsψϑ+1s=ηmaxexpw1,0−ηmaxexpw1−1,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0ς3ϑΓ3ϑ+1.⋮.The approximate solution for Example1 is expressed as
(56)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηςϑΓϑ+1+η2ς2ϑΓ2ϑ+1−η3ς3ϑΓ3ϑ+1+⋯+maxexpw1,01−1+ηςϑΓϑ+1−η2ς2ϑΓ2ϑ+1+η3ς3ϑΓ3ϑ+1+⋯=maxexpw1−1,0Eϑ−ηςϑ+maxexpw1,0Eϑ1−ηςϑ.Case 2.
Now, we solve(4) by using the ABC fractional derivative operator incorporating the Jafari decomposition method.Considering (50) and using the differentiation rule of Jafari transform, we have
(57)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.In view of (5), we get
(58)JQw1,ς=ϕςψϑςmaxexpw1−1,0+ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the inverse Jafari transform on both sides of the above equation yields(59)Qw1,ς=J−1ϕςψϑςmaxexpw1−1,0+J−1ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.By the Jafari decomposition method, we find:(60)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(61)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ∑q=0∞Qw1,ςw1w1+η−1∑q=0∞Qw1,ςw1−η∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The approximate solution for Example1 is expressed as:
(62)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯+maxexpw1,0−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.The integer-order solution of Example1 can be obtained with the aid of Taylor series expansion and setting ϑ=1 as follows:
(63)Qw1,ς=maxexpw1−1,0exp−ης+maxexpw1,01−exp−ης.Example 2 (see [20]).
Assume the time-fractional one-dimensional BSM (6) subject to IC (7).Case 1.
Firstly, we solve the (6) in the sense of Caputo fractional derivative operator incorporating the Jafari decomposition method.Applying Jafari transform to both sides of (6), we have
(64)JDςϑQw1,ς=J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Using the differentiation rule of Jafari transform, we have(65)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.In view of (7), we get
(66)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+1ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the inverse Jafari transform on both sides yields(67)Qw1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0+J−11ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.By the Jafari decomposition method, we find(68)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(69)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−11ψϑςJ−0.082+sinw12w12∑q=0∞Qw1,ςw1w1−0.06w1∑q=0∞Qw1,ςw1+0.06∑q=0∞Qw1,ς,∀q=0,1,2,⋯,Q1w1,ς=J−11ψϑςJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0J−1ϕsψϑ+1s=−0.06w1+0.06maxw1−25exp−0.06,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−0.000216w1+0.00216maxw1−25exp−0.06,0ς3ϑΓ3ϑ+1,⋮.The approximate solution for Example2 is expressed as
(70)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06×1−1−0.06ςϑΓϑ+1−0.0036ς2ϑΓ2ϑ+1−0.000216ς3ϑΓ3ϑ+1+⋯=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06,01−Eϑ0.06ςϑ.Case 2.
Now, we solve the (6) by using ABC fractional derivative operator incorporating the Jafari decomposition method.Considering (64) and using the differentian rule of Jafari transform, we have
(71)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.In view of (7), we get
(72)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the inverse Jafari transform on both sides yields:(73)Qw1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0+J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.By the Jafari decomposition method, we find:(74)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0.Here, we surmise that the unknown functionQw1,ς can be written by an infinite series of the form:
(75)Qw1,ς=∑q=0∞Qqw1,ς,∑q=0∞Qq+1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∑q=0∞Qw1,ςw1w1−0.06w1∑q=0∞Qw1,ςw1+0.06∑q=0∞Qw1,ς,q=0,1,2,⋯,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−−0.000216w1+0.00216maxw1−25exp−0.06,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The approximate solution for Example2 is expressed as:
(76)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0−0.06w1−0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+ϑ−0.0036w1−0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.The integer-order solution of Example2 can be obtained with the aid of Taylor series expansion and setting ϑ=1 as follows:
(77)Qw1,ς=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06,01−exp−0.06ς.


## 5.2. Jafari Iterative Transform Method

Example 3 (see[16]).
Assume the time-fractional one-dimensional BSM (4) subject to IC (5).Case 1.
First, we surmise the Caputo fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (4) with IC (5), we have:
(78)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Correspondingly, we get(79)JQw1,ς=ϕςψϑςmaxexpw1−1,0+1ψϑςJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the suggested analytical method, we obtain(80)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0,Q1w1,ς=J−11ψϑςJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0J−1ϕsψϑ+1s=ηmaxexpw1,0−ηmaxexpw1−1,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0ς3ϑΓ3ϑ+1.⋮.The series form solution is(81)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(82)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηςϑΓϑ+1+η2ς2ϑΓ2ϑ+1−η3ς3ϑΓ3ϑ+1+⋯+maxexpw1,01−1+ηςϑΓϑ+1−η2ς2ϑΓ2ϑ+1+η3ς3ϑΓ3ϑ+1+⋯=maxexpw1−1,0Eϑ−ηςϑ+maxexpw1,0Eϑ1−ηςϑ.Case 2.
Now, we consider the ABC fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (4) with IC (5), we have
(83)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Correspondingly, we get(84)JQw1,ς=ϕςψϑςmaxexpw1−1,0+ϑ+1−ϑψϑςψϑςAϑJ∂2Qw1,ς∂w12+η−1∂Qw1,ς∂w1−ηQw1,ς.Employing the suggested analytical method, we obtain(85)Q0w1,ς=J−1ϕςψϑςmaxexpw1−1,0=maxexpw1−1,0,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ0w1,ςw1w1+η−1Q0w1,ςw1+ηQ0=ηmaxexpw1,0−ηmaxexpw1−1,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ1w1,ςw1w1+η−1Q1w1,ςw1+ηQ1=−η2maxexpw1,0+η2maxexpw1−1,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJQ2w1,ςw1w1+η−1Q2w1,ςw1+ηQ2=−η3maxexpw1,0+η3maxexpw1−1,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The series form solution is(86)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(87)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxexpw1−1,01−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯+maxexpw1,0−ηAϑϑςϑΓϑ+1+ϑ−η2A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.Example 4 (see [20]).
Assume the time-fractional one-dimensional BSM (6) subject to IC (7).Case 1.
First, we surmise the Caputo fractional derivative operator for (6) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (6) with IC (7), we have
(88)ψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Correspondingly, we get:(89)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+1ψϑςJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the suggested analytical method, we obtain(90)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0,Q1w1,ς=J−11ψϑςJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0J−1ϕsψϑ+1s=−0.06w1+0.06maxw1−25exp−0.06,0ςϑΓϑ+1,Q2w1,ς=J−11ψϑςJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0ς2ϑΓ2ϑ+1,Q3w1,ς=J−11ψϑςJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−0.000216w1+0.00216maxw1−25exp−0.06,0ς3ϑΓ3ϑ+1.⋮.The series form solution is(91)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(92)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.06×1−1−0.06ςϑΓϑ+1−0.0036ς2ϑΓ2ϑ+1−0.000216ς3ϑΓ3ϑ+1+⋯=maxw1−25exp−0.06,0+w1−maxw1−25exp−0.061−Eϑ0.06ςϑ.Case 2.
Now, we consider the ABC fractional derivative operator for (6) with the new iterative transform method via the Jafari transform.Implementing the Jafari transform to (6) with IC (7), we have
(93)ψϑςAϑϑ+1−ϑψϑςJQw1,ς=ϕς∑q=0q−1ψϑ−1−qςQq0+J−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Correspondingly, we get(94)JQw1,ς=ϕςψϑςmaxw1−25exp−0.06,0+ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12∂2Qw1,ς∂w12−0.06w1∂Qw1,ς∂w1+0.06Qw1,ς.Employing the suggested analytical method, we obtain(95)Q0w1,ς=J−1ϕςψϑςmaxw1−25exp−0.06,0=maxw1−25exp−0.06,0,Q1w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q0w1,ςw1w1−0.06w1Q0w1,ςw1−0.06Q0=−0.06w1+0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+1−ϑ,Q2w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q1w1,ςw1w1−0.06w1Q1w1,ςw1−0.06Q1=−0.0036w1+0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2,Q3w1,ς=J−1ϑ+1−ϑψϑςψϑςAϑJ−0.082+sinw12w12Q2w1,ςw1w1−0.06w1Q2w1,ςw1−0.06Q2=−−0.000216w1+0.00216maxw1−25exp−0.06,0A3ϑ×ϑ3ς3ϑΓ3ϑ+1+3ϑ21−ϑς2ϑΓ2ϑ+1+3ϑ1−ϑ2ςϑΓϑ+1+1−ϑ3.⋮.The series form solution is(96)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯.Consequently, we have(97)Qw1,ς=Q0w1,ς+Q1w1,ς+Q2w1,ς+Q3w1,ς+⋯,=maxw1−25exp−0.06,0−0.06w1−0.06maxw1−25exp−0.06,0AϑϑςϑΓϑ+1+ϑ−0.0036w1−0.0036maxw1−25exp−0.06,0A2ϑϑ2ς2ϑΓ2ϑ+1+2ϑ1−ϑςϑΓϑ+1+1−ϑ2−⋯.


## 6. Results and Discussion

Two distinctive techniques have been used to analyse the exact approximate solution of time-fractional BSM in this investigation. With MATLAB package 21, exact analytical solutions for the Black–Scholes option pricing models are presented via the CFD and ABC fractional derivative operators at any order for varying values of space and time.For the solution in Examples1 and 3, we conducted simulation studies for several Brownian movements with varied values of w1 and ς in Tables 1 and 2. Table 3 exibits a numerical comparative analysis of the Homotopy perturbation approach and the Jafari decomposition method in perspective of absolute error for (4) taking into consideration both fractional derivative operators.Table 1
The exact,JDMCFD, and JDMABC solutions of Example 1 at various fractional orders with varying values of w1 and ς.
w1ςϑ=0.7ϑ=0.8ϑ=0.9ϑ=1JDMCFDϑ=1JDMABCExact0.200.10.50384795970.48408953980.46745968600.45380011140.45380011140.453799422200.20.61422597720.59116764800.56971057600.55020606710.55020606710.550194046900.30.65856185670.63719313130.61610055390.59563192950.59563192950.595571903900.40.69566928980.69566928980.65765492720.63806909430.63806909430.637881937400.50.72816829040.71214116190.69543142600.67778238270.67778238270.67733155110.400.10.68012360610.65345252910.63100457450.61256607730.61256607730.612565147000.20.75021730290.72205379590.69584606890.67202320800.67202320800.672008526500.30.80436926850.77826944820.75250691600.72750648170.72750648170.727433166200.40.84969238960.82675002700.80326154220.77933935190.77933935190.779110757800.50.88938675860.86981117960.84940186200.82784527190.82784527190.82729462480.600.10.83070484820.79812872120.77071072750.77071072750.77071072750.748188760000.20.91631748270.88191849760.84990830770.82081099970.82081099970.820793067600.30.98245884280.95058045040.91911402250.88857842320.88857842320.888488875500.41.0378166281.0097947630.98110586300.95188723370.95188723370.951608028300.51.0862994401.0623897741.0374617771.0111324981.0111324981.0104599360.800.11.0146251930.97483662160.94134820840.91384120310.91384120310.913839815200.21.1191927011.0771776861.0380803511.0025408191.0025408191.00251891700.31.1999779401.1610415841.1226084021.0853121371.0853121371.08520276300.41.2675920921.2333661091.1983254071.1626376931.1626376931.16229667000.51.3268091321.2976058001.2671586761.2350000221.2350000221.2341785531.000.11.2392660091.1906681381.1497652981.1161681661.1161681661.11616647100.21.3669850521.3156677961.2679142041.2245061221.2245061221.22447937000.31.4656563661.4180993931.3711569991.3256032381.3256032381.32546964800.41.5482404771.5064367671.4636379571.4200488851.4200488851.41963235900.51.6205683331.5848993031.5477111011.5084324331.5084324331.507429089Table 2
The exact,JITMCFD, and JITMABC solutions of Example 3 at various fractional orders with varying w1 and ς.
w1ςϑ=0.75ϑ=0.85ϑ=0.95ϑ=1JITMCFDϑ=1JITMABCExact00.200.60.75086770040.73727809910.72276922960.71503661620.71503661620.714114127100.70.77923907840.76818666570.75645199880.75009661580.75009661580.748409973600.80.80599729340.79719412270.78815058240.78322720300.78322720300.780387209000.90.83166660490.82477313670.81821276710.81469319920.81469319920.810202585611.00.85666801380.85132903690.84696385450.84475942560.84475942560.838002258500.400.60.91711188040.90051350410.88279233080.87334769540.87334769540.872220964700.70.95176475980.93826531260.92393255790.91617007570.91617007570.914110006200.80.98444731750.97369510060.96264929540.95663586620.95663586620.953167089800.91.0157998851.0073801840.99936733070.99506852080.99506852080.989583673011.01.0463366751.0398156341.0344839881.0317914931.0317914931.0235382700.600.61.1201629801.0998896771.0782449871.0667092841.0667092841.06533309200.71.1624881021.1459998401.1284937741.1190126571.1190126571.11649648200.81.2024066681.1892738811.1757825041.1684376851.1684376851.16420091200.91.2407007811.2304169351.2206300141.2153794361.2153794361.20868022711.01.2779985011.2700336831.2635215961.2602329741.2602329741.2501524660.800.61.3681701531.3434082861.3169714021.3028816611.3028816611.30120077600.71.4198661751.3997273661.3783454081.3667651461.3667651461.36369188300.81.4686228211.4525823991.4361039941.4271330121.4271330121.42195820500.91.5153953561.5028346391.4908808661.4844677951.4844677951.47628536311.01.5609508941.5512226441.5432687621.5392520311.5392520311.5269396701.000.61.6710867991.6408425861.6085525021.5913432541.5913432541.58929021700.71.7342284621.7096308661.6835148841.6693707181.6693707181.66561702700.81.7937799651.7741881481.7540613791.7431041971.7431041971.73678367400.91.8509080681.8355663721.8209660011.8131330591.8131330591.80313901511.01.9065497271.8946676151.8849527231.8800466761.8800466761.865008325Table 3
The comparsion analysis among HPM [14], JDMCFD, and JDMABC of Example 1 for approximated results of Qw1,ς and absolute error Eabs=∥Eexact−Eapprox∥ at ϑ=1 with varying values of w1 and ς.
w1ς∥Exact−HPM∥∥Exact−JDMCFD∥∥Exact−JDMABC∥0.20.1768.9000000×10−7761.7000000×10−9761.7000000×10−90.213.099020×10−512.0202000×10−612.0202000×10−60.364.2248000×10−560.0256000×10−660.0256000×10−60.4189.107690×10−5187.1569000×10−6187.1569000×10−60.5455.4455000×10−5450.8316000×10−6450.8316000×10−60.40.1933.050000×10−8930.3000000×10−9930.3000000×10−90.215.7634000×10−514.6815000×10−614.6815000×10−60.375.9988000×10−573.3155000×10−673.3155000×10−60.4230.5679000×10−5228.5941000×10−6228.5941000×10−60.5554.5621000×10−5550.6471000×10−6550.6471000×10−60.60.12.0098000×10−51.1363000×10−61.1363000×10−60.216.4534000×10−517.9321000×10−617.9321000×10−60.390.6658000×10−589.5477000×10−689.5477000×10−60.4220.3489000×10−5279.2054000×10−6279.2054000×10−60.5660.9890000×10−5672.5620000×10−6672.5620000×10−60.80.12.0094000×10−51.3879000×10−61.3879000×10−60.223.0020000×10−521.9020000×10−621.9020000×10−60.3111.5690000×10−5109.3740000×10−6109.3740000×10−60.8345.9980000×10−5341.0230000×10−6341.0230000×10−60.5824.0000000×10−5821.4690000×10−6821.4690000×10−61.00.12.346789233×10−61.695175303×10−61.695175303×10−60.23.700000000×10−42.675000000×10−52.675000000×10−50.32.888600000×10−31.336000000×10−41.336000000×10−40.45.120000000×10−34.165000000×10−44.165000000×10−40.52.000000000×10−21.003000000×10−31.003000000×10−3The findings of a simulation study for the BSM considered in Examples2 and 4 are shown in Tables 4 and 5. The comparative analysis of the homotopy perturbation with the projected methods is represented in Table 6. This comparison clearly illustrates that the synthesised trajectories are more powerful and effective than the existing ones.Table 4
The exact,JDMCFD, and JDMABC solutions of Example 2 at various fractional-orders with varying w1 and ς.
w1ςϑ=0.7ϑ=0.8ϑ=0.9ϑ=1JDMCFDϑ=1JDMABCExact0.20.10.00261213750.00202939220.00156398150.00119640720.00119640720.0011964070.20.00422043530.00351725770.00290758800.00238565760.00238565760.00238565740.30.00557972080.00484499710.00417325290.00356779440.00356779440.0035677930.40.00679611940.00607549490.00538806750.00474286080.00474286080.00474285800.50.00791454290.00723655650.00656454690.00591090000.00591090000.00591089330.40.10.00522427500.00405878450.00312796300.00239281440.00239281440.00239281430.20.00844087060.00703451540.00581517600.00477131520.00477131520.00477131480.30.0111594410.00968999420.00834650590.00713558880.00713558880.00713558700.40.0135922380.0121509890.0107761350.00948572160.00948572160.00948571600.50.0158290850.0144731130.0131290930.0118218000.0118218000.01182178660.60.10.00783641250.00608817670.00469194450.00358922160.00358922160.00358922150.20.0126613050.0105517730.00872276410.00715697280.00715697280.00715697220.30.0167391620.0145349910.0125197580.0107033830.0107033830.01070338050.40.0203883580.0182264840.0161642020.0142285820.0142285820.01422857410.50.0237436280.0217096690.0196936400.0177327000.0177327000.01773267990.80.10.0104485500.00811756900.00625592610.00478562880.00478562880.00478562870.20.0168817410.0140690300.0116303520.00954263040.00954263040.00954262960.30.0223188830.0193799880.0166930110.0166930110.0166930110.01427117400.40.0271844770.0243019790.0215522700.0189714430.0189714430.01897143210.50.0316581710.0289462260.0262581870.0236436000.0236436000.02364357321.00.10.0130606870.0101469610.00781990760.00598203600.00598203600.0059820350.20.0211021760.0175862880.0145379400.0119282880.0119282880.0119282870.30.0278986040.0242249850.0208662640.0178389720.0178389720.0178389670.40.0339805970.0303774740.0269403370.0237143040.0237143040.0237142900.50.0395727140.0361827820.0328227340.0295545000.0295545000.029554466Table 5
The exact,JITMCFD, and JITMABC solutions of Example 4 at various fractional orders with varying w1 and ς.
w1ςϑ=0.75ϑ=0.85ϑ=0.97ϑ=1JITMCFDϑ=1JITMABCExact00.200.60.00865486780.00802823180.00739066610.00707195520.00707195520.007071941300.70.00968255550.00912210780.00852978210.00822606960.00822606960.008226043800.80.0106673970.0101857170.00965376100.00937328640.00937328640.009373242500.90.0116156670.0112228420.0107638530.0105136480.0105136480.010513578711.00.0125321070.0122363870.0118610420.0116472000.0116472000.011647093200.400.60.0173097350.0160564630.0147813320.0141439100.0141439100.014143882600.70.0193651110.0182442150.0170595640.0164521390.0164521390.016452087700.80.0213347950.0203714350.0193075220.0187465720.0187465720.018746485100.90.0232313340.0224456840.0215277070.0210272970.0210272970.021027157411.00.0250642140.0244727740.0237220840.0232944000.0232944000.02329418650.600.60.0259646030.0240846950.0221719980.0212158650.0212158650.021215823900.70.0290476660.0273663230.0255893460.0246782080.0246782080.024678131600.80.0320021920.0305571530.0289612830.0281198590.0281198590.028119727700.90.0348470020.0336685270.0322915600.0315409460.0315409460.03154073611.00.0375963210.0367091620.0355831260.0349416000.0349416000.03494127980.800.60.0346194710.0321129270.0295626640.0282878200.0282878200.028287765200.70.0387302220.0364884310.0341191280.0329042780.0329042780.03290417500.80.0426695900.0407428710.0386150440.0374931450.0374931450.037492970300.90.0464626690.0448913690.0430554140.0420545950.0420545950.04205431411.00.0501284280.0489455490.0474441680.0465888000.0465888000.04658837311.000.60.0432743390.0401411590.0369533300.0353597760.0353597760.03535970600.70.0484127770.0456105390.0426489100.0411303480.0411303480.04113021900.80.0533369880.0509285880.0482688050.0468664320.0468664320.04686621200.90.0580783360.0561142110.0538192670.0525682440.0525682440.05256789311.00.0626605350.0611819370.0593052100.0582360000.0582360000.058235466Table 6
The comparsion analysis among HPM [14], JDMCFD, and JDMABC of Example 2 for approximated results of Qw1,ς and absolute error Eabs=∥Eexact−Eapprox∥ at ϑ=1 with varying w1 and ς.
w1ς∥Exact−HPM∥∥Exact−JDMCFD∥∥Exact−JDMABC∥0.20.13.00450×10−102.00000×10−112.00000×10−110.22.00000×10−91.80000×10−101.80000×10−100.39.50000×10−98.80000×10−108.80000×10−100.43.89000×10−82.76000×10−92.76000×10−90.57.70000×10−86.70000×10−96.70000×10−90.40.15.780000×10−104.00000×10−114.00000×10−110.24.80000×10−93.60000×10−103.60000×10−100.32.98000×10−81.76000×10−91.76000×10−90.46.45000×10−85.52000×10−95.52000×10−90.52.47000×10−71.34000×10−81.34000×10−80.60.17.96600×10−106.00000×10−116.00000×10−110.26.785000×10−95.40000×10−105.40000×10−100.33.98000×10−82.64000×10−92.64000×10−90.49.31000×10−88.28000×10−98.28000×10−90.53.003000×10−72.01000×10−82.01000×10−80.80.19.890000×10−108.00000×10−118.00000×10−110.29.80000×10−97.20000×10−107.20000×10−100.34.94000×10−83.52000×10−93.52000×10−90.42.89000×10−71.10400×10−81.10400×10−80.53.60089×10−72.68000×10−82.68000×10−81.00.12.9900×10−91.00000×10−101.00000×10−100.211.00011×10−99.00000×10−109.00000×10−100.36.40000×10−84.40000×10−94.40000×10−90.42.87000×10−71.38000×10−81.38000×10−80.54.89000×10−73.35000×10−83.35000×10−8Figure1 depicts the behavior of the Jafari decomposition method’s outcome from Qw1,ς for Example 1. Figures 1(a) and 1(b) demonstrates the behavior of exact and approximate solutions of Black-Scholes option pricing models via the CFD operator, whereas Figures 2(a) and 2(b) depicts the absolute errors behavior and fractional-order variation of Qw1,ς. The different fractional orders at ϑ=0.7,0.8,0.9,1 behave analogously. In the same way, Figures 3(a) and 3(b) represent the two-dimensional options graphically of the obtained result Qw1,ς for (4) at ς=0.7 and ς=0.3.Figure 1
Three-dimensional illustration of the exact and approximate solution via CFD of Example1 when ϑ=1.
(a)(b)Figure 2
Three-dimensional illustration of the absolute error and multiple surface plots of Example1 via CFD for varying fractional orders.
(a)(b)Figure 3
Two-dimensional illustration of the exact, approximate solution via CFD, and ABC fractional derivative operator of Example1 with varying fractional orders at (a) ς=0.7 and (b) ς=0.3.
(a)(b)Continuing in the same manner, Figures4(a) and 4(b) illustrates the exact -approximate consequences Qw1,ς for (1.6) by the use of Jafari decomposition method the values of the options graphically. Figures 5(a) and 5(b) show the absolute error and response of acquired data for (6) with various standard and Brownian motions of 0.6,0.65,0.7,0.8,0.9, and 1, respectively. Thus, we conclude that the order of the profile pictures increases as the value of the time-dependent variable increases. It is worth mentioning that the fractional order has an acceleration effect on the diffusion processes.Figure 4
Three-dimensional illustration of the exact and approximate solution via CFD of Example1 when ϑ=1.
(a)(b)Figure 5
(a) Three-dimensional illustration of the absolute error plot of Example1 when ϑ=1. (b) Two-dimensional illustration of the of Example 1 for varying fractional orders when ς=0.7.
(a)(b)


## 7. Conclusions

This article has investigated the more general integral transform with the Adomian decomposition method and new iterative transform method. The Caputo and ABC fractional derivative operators have been implemented to deal with the BSM. Several distinct solutions have been proposed with the assumption of fractional orders. Various representations have been used to understand these solutions, which have clarified the significant properties of the fractional models in consideration. Without any restrictive assumptions, discretization, or linearization, the proposed methodology locates solutions. Elegance and originality have been invoked to describe our trajectory. Contrasting proposed findings to those acquired in earlier scholarly articles demonstrate the specificality of our solutions in the European option pricing model. The strategy’s powerful and successful implementation is explored and validated in order to demonstrate its applicability to additional nonlinear evolution equations that arise in banking and finance issues.



---

*Source: 7767848-2021-12-08.xml*