An Efficient Method for Solving Fractional Black-Scholes Model with Index and Exponential Decay Kernels

The Black-Scholes equation (BSe) is fascinating in the business world for predicting the performance of ﬁ nancial investment valuation systems. The Caputo fractional derivative (CFD) and Caputo-Fabrizio fractional derivative operators are used in this research to analyze the BSe. The Adomian decomposition method (ADM) and the new iterative transform (NIM) approach are combined alongside the Yang transform. In addition, the convergence and uniqueness results for the aforementioned framework have been calculated. The existence and uniqueness results have been established and frequently accompanied innovative aspects of the prospective system in ﬁ xed point terminologies. To provide additional insight into such concepts, a variety of illustrations and tabulations are used. Additionally, the provided techniques regulate and modify the obtained analytical results in a really productive fashion, allowing us to modify and regulate the converging domains of the series solution in a pragmatic manner.


Introduction
Recently, the investigation of modified derivatives and integrals has grown in prominence in recent decades, owing to its appealing implications in a wide range of disciplines, including Maxwell fluids [1,2], circuit theory [3], and epidemics [4,5]. As an outgrowth of conventional integer analysis, fractional calculus (FC) has been exploited to examine the implications and integrals of indefinite powers. Because integer-order derivative and integral operators are being used to simulate all real-world processes, numerous researchers have proposed multiple variations of fractional operators as a modification of the fractional formulations [6][7][8][9]. The interaction effect in FC has been utilized to represent numerous processes in thermodynamics, chemical engineering, biomechanics, and other disciplines, despite the fact that the analyzed formulae in FC are typically reluctant to analyze complicated phenomena [10][11][12][13][14]. In addition, fractional dif-ferential formulas have a higher granularity than integer differential operators. Illustrations comprise Katugampola, Weyl, Hadamard, Caputo, Riesz, Riemann, and Liouville, Weyl, Jumarie, Caputo and Fabrizio [15], and Grünwald and Letnikov [16]. Likewise, the Liouville-Caputo and Caputo-Fabrizio fractional filtrations are thought to be ideal.
In 1973, Fischer Black and Myron Scholes formulated a mathematical formula for passive investment valuation. The pioneering Black-Scholes equation (BSe) is at the core of contemporary financial economics, and it is indeed tough to communicate about mainstream capitalism without mentioning the revolutionary BSe.
The objective of this paper is to leverage the Yang decomposition method (YDM) and the Yang iterative transform method (YITM) to modify the results into a BSe. The fractional interpretation of BSe is characterized in financial services by [29]: subject to the playoff mapping where UðS, ϱÞ denotes the alternative means worth at S asset prices of the moment, ϱ, and T indicates the termination term. The symbol E symbolizes share value. The parameter ζ represents the uncertainty of borrowing until it matures. The continual ϖ indicates the unpredictability of a trading asset. The required assumptions are also entailed: a continuous uncertainty risk premium u, no operating charges, the capacity to transact an unrestricted quantity of inventory, and no restrictions on market manipulation. Ultimately, we provide European alternatives. It is also worth mentioning that Uð0, ϱÞ = 0 and UðS, T Þ ≈ S as S ↦ ∞: The parabolic diffusion problem can perhaps be described as the BSe in (1). Inducing the modifications that follow: then (1) diminishes to related initial condition (ICs) are where ζ designates the threshold when the direct connection involving wage growth and market instability coincides. Cen and Le proposed the generalized fractional BSe in [30]. The BSe is stated as follows: D δ ϱ U y 1 , ϱ ð Þ= −0:08 2 + sin y 1 ð Þ 2 y 2 1 ∂ 2 U y 1 , ϱ ð Þ ∂y 2 1 − 0:06y 1 ∂U y 1 , ϱ ð Þ ∂y 1 + 0:06U y 1 supplemented ICs The fractional BSe considering a particular resource has been widely explored ( [31,32]). The fractional BSe is a version of the classical BSe that expands its restrictions. The BSe was implemented by Meng and Wang [33] to analyze fractional potential assessment. The fractional BSe was used to determine the insured guarantee valuation for treasury foreign trade in China. Their results indicate that the fractional BSe surpasses the traditional BSe when it pertains to measuring the impact of the pricing system [34]. The Black-Scholes financial theory was calculated using the HPM by Fall et al. [35]. By adopting the Ornstein-Uhlenbeck Procedures, Matadi and Zondi [36] explored the consistent values of BSe. The computational estimation of fractional BSe emerging in the banking system was demonstrated by Kumar et al. [37]. Employing a novel fractional operator, Yavuz and Özdemir [38] suggested a novel strategy for the European efficient market hypothesis.
The ADM introduced a well-known concept during George Adomian's significant surge in 1980. For example, it has been frequently applied to deal with a variety of complex PDEs like the K(2,2) and K(3,3) models [19], biological population model [39], Swift-Hohenberg model [40], and henceforth. The ADM is essential because it overcomes the necessity for a smaller component in the considerations, eliminating the challenges that occur with classic Adomian approaches. The main objective of this research was to leverage the ADM to analyze fractional-order BSe using a recently designed integral transformation known as the "Yang transformation" [41].
Daftardar-Gejji and Jafari [42] proposed NITM in 2006, which is frequently adopted by scholars owing to its usefulness in fractional ODES and PDEs. If a precise result emerges, the iterative method leads to it through repeated estimates. For methodological concerns, a significant fraction of projections can be considered with a satisfactory amount of precision for specific issues. For managing nonlinearity components, the NITM sometimes does not require a restrictive assumption. For instance, researchers exploited NITM to develop analytical results for the fractional Schrödinger equation in [43], and Wang and Liu used NITM to address the fractional Fornberg-Whitham model in [44]. Widatalla and Liu used NITM to develop the Laplace decomposition algorithm in [22].
Due to the aforesaid tendency, we apply the YDM and the YITM to achieve the expressive result of the fractionalorder BSe. For renewability algorithmic techniques, the Yang transform efficiently integrates the ADM and NITM. The

Preliminaries
In this part, we address several key ideas, conceptions, and terminologies related to fractional derivative operators involving index and exponential decay as a kernel, as well as the Yang transform's specific repercussions.
Definition 1 (see [9]). The Caputo fractional derivative ðCFDÞ is described as follows: Definition 2 (see [15]). The Caputo fractional derivative operator is described as follows: where U ∈ H 1 ða 1 , a 2 ÞðSobolev spaceÞ, a 1 < a 2 , δ ∈ ½0, 1, and AðδÞ signifies a normalization function as AðδÞ = Að0Þ = Að1Þ = 1: Definition 3 (see [15]). The fractional integral of the Caputo-Fabrizio operator is defined as Definition 4 (see [41]). The Yang transform is described as follows: The Yang transform of a range of vital expressions is as follows: Definition 5 (see [41]). The Yang transform of the CFD operator is mentioned as Definition 6 (see [45]). The Yang transform of the Caputo-Fabrizio fractional derivative operator is stated as Definition 7 (see [46]). The Mittag-Leffler function for single parameter is defined as

Algorithmic Configuration for Nonlinear PDEs
Let us surmise the fractional version of nonlinear PDE: having ICs where D δ ϱ = ∂ δ Uðy 1 , ϱÞ/∂ϱ δ represents the Caputo-Fabrizio fractional derivative considering the order δ ∈ ð0, 1 whilst L and N indicates the linear and nonlinear functionals, respectively. Furthermore, Qðy 1 , ϱÞ indicates the source term.
3.1. Construction of Yang Decomposition Method. Incorporating the Yang transformation to (16), we obtain

Journal of Function Spaces
Initially, we implement the Yang transform differentiabilty criteria to CFD, and then further implement the Caputo-Fabrizio fractional derivative operator as described in the following: The inverse Yang transform of (19) and (20), respectively, gives The infinite series Uðy 1 , ϱÞ illustrates the result of the Yang decomposition approach: As a consequence, the nonlinear component Nðy 1 , ϱÞ can be assessed employing the Adomian decomposition approach, as follows: wherẽ Putting (20) and (24) into (21) and (22), respectively, we attain As a nutshell, the iterative approach for (26) and (27) is as follows: First, we apply the differentiation rule of Yang transform for CFD, and then we consider for Caputo-Fabrizio fractional derivative operator, respectively, and we get Using the fact of the inverse Yang transform of (30) and (31), respectively, produces Journal of Function Spaces Employing the recursive approach, we determine Moreover, utilizing the linearity L of the operator, thus we have and the nonlinearity N handled by (see [42]) where D p = Nð∑ p κ=0 U p Þ − Nð∑ p−1 κ=0 U p Þ: Inserting (37), (39), and (36) into (32) and (33), respectively, we observe Ultimately, for CFD, we develop appropriate analysis procedure: The exploratory procedure for the Caputo-Fabrizio fractional derivative operator is shown then: Eventually, the q-term result in series formulation is generated by (37), (39), and (40), and we have

Mathematical Formulations of BSM via Caputo-Fabrizio Fractional Derivative Operator
The coming parts will illustrate how well the adequate conditions ensure the formation of a unique solution. Our hypothesis of the existence of solutions in the scenario of YDM is developed by [47].
For 0 < ε < 1, the functional is contraction. As a result of the Banach contraction fixed point hypothesis, (16) has a fixed value. This produces the intended outcome.
Theorem 9 (Convergence analysis). Equation (16) has a generic type solution and will be convergent.
Proof. Surmise thatŜ ℓ be the nth partial sum, that is,Ŝ ℓ = ∑ ℓ m=0 U ℓ ðy 1 , ϱÞ: Further, we exhibit fŜ ℓ g is a Cauchy sequence in Banach space U: We do it by contemplating a novel kind of Adomian polynomials.

Mathematical Description of BSM Time-Fractional Systems
Here, we construct the estimated analytical solution of BSM considering the CFD and Caputo-Fabrizio fractional derivative operators utilizing the Yang decomposition approach.
To begin, we utilize the Caputo fractional derivative operator employing the Yang decomposition approach to analyze the (4). Implementing the Yang transform on (4), we get Utilizing the Yang transform's differentiation criteria gives Utilizing (5), we find Applying the inverse Yang transform produces To determine this, apply the Yang decomposition approach as follows: We predict that the unidentified mapping Uðy 1 , ϱÞ may be expressed as an infinite series of the pattern For Example 1, the series form solution is developed as 7 Journal of Function Spaces follows: Case 2. The Caputo-Fabrizio fractional derivative operator and the Yang decomposition approach are now used to solve equation (4). Assuming (50) and implementing the Yang transform's differentiation criteria, we obtain Utilizing (5), we obtain Employing the Yang decomposition approach produces We predict that the unidentified mapping Uðy 1 , ϱÞ may be expressed as an infinite series of the pattern For Example 1, the series form solution is developed as follows: Considering the Taylor series expansion and assigning 8 Journal of Function Spaces δ = 1, the exact findings of Example 1 can be determined as Example 2 (see [30]). Surmise the fractional-order BSM (6) supplemented with the (7).
To begin, we utilize the Caputo fractional derivative operator, employing the Yang decomposition approach to analyze the (6). Implementing the Yang transform on (6), we get Utilizing the Yang transform's differentiation criteria, gives Utilizing (7), we find Applying the inverse Yang transform produces To determine this, apply the Yang decomposition approach as follows: We predict that the unidentified mapping Uðy 1 , ϱÞ may be expressed as an infinite series of the pattern For Example 2, the series form solution is developed as follows: Case 2. The Caputo-Fabrizio fractional derivative operator and the Yang decomposition approach are now used to solve the (6).

Journal of Function Spaces
Assuming (65) and implementing the Yang transform's differentiation criteria, we obtain Utilizing (7), we obtain Employing the inverse Yang transform gives Employing the Yang decomposition approach, we obtain We predict the unidentified mapping Uðy 1 , ϱÞ may be expressed as an infinite series of the pattern

Journal of Function Spaces
For Example 2, the series form solution is developed as follows: Considering the Taylor series expansion and assigning δ = 1, the exact findings of Example 2 can be determined as
Case 1. To begin, we utilize the Caputo fractional derivative operator, employing the Yang decomposition approach to analyze the (4). Implementing the Yang transform on (4), we get It follows that In view of the proposed algorithm in Section 3.2, we find The result in series representation is Eventually, we have Case 2. The (4) is now addressed utilizing the Caputo-Fabrizio fractional derivative operator and the Yang iterative transform method. Assuming (4) and implementing the Yang transform's differentiation criteria, we obtain It follows that In view of the proposed algorithm in Section 3.2, we find The result in series representation is Eventually, we have Example 4 (see [30]). Surmise the fractional-order BSM (6) supplemented with the (7).
To begin, we utilize the Caputo fractional derivative operator employing the Yang iterative transform method to analyze the (6). Implementing the Yang transform on (6), we get It follows that In view of the proposed algorithm in Section 3.2, we have The result in series representation is Journal of Function Spaces Consequently, we have Case 2. The Caputo-Fabrizio fractional derivative operator and the Yang iterative transform method are now used to solve the (6). Assuming (6) and implementing the Yang transform's differentiation criteria, we obtain It follows that In view of the proposed algorithm in Section 3.2, we have The result in series representation is Consequently, we have 5.3. Results and Explanation. Throughout this investigation, two distinct methodologies are being employed to assess the precise analytical solutions of fractional-order BSe. For various spatial and temporal parameters, the CFD and Caputo-Fabrizio fractional derivative operators in MATLAB package 21 facilitate appropriate numerical findings for the BSe option revenue frameworks utilizing multiple orders. We built modeling tests for many Brownian deformations involving different y 1 parameters, and the results are shown in Table 1 for Examples 1 and 3, respectively. Table 2 illustrates a computational evaluation of the HPM [35] and the Yang decomposition technique for (4) in accordance with absolute error, considering both fractional derivative operators into account. Table 3 illustrates the results of a mathematical model for the BSe used in Examples 2 and 4. Table 4 reports the interpretation of an evaluation of the HPM [35] and predicted approaches. The synthetically produced profiles are significantly better reliable and pragmatic than the old ones, as evidenced by this analysis.
For Example 1, Figure 1 displays the evolution of the Yang decomposition technique's data from Uðy 1 , ϱÞ:    14 Journal of Function Spaces        Figure 6(b) refers to the dynamic of the two-dimensional alternatives of the analysis values Uðy 1 , ϱÞ for (6). Finally, we deduce that as the amount of the time-dependent component improves, the hierarchy of the feature images tends to rise as well. It is important to remember that the fractional order has a simulatory effect on the diffusion mechanism.

Conclusion
The Adomian decomposition approach and the new iterative transform procedure have been leveraged to analyze the Yang transform. To interact effectively with the BSe, the Caputo and Caputo-Fabrizio fractional derivative opera-tors have been constructed. Considering the supposition of fractional order, numerous new outcomes have been presented. To clarify the crucial aspects of the fractional frameworks under evaluation, diverse visualizations were attempted to explicate these results. The suggested scheme identifies the findings without any underlying limitations, deconvolution, or quantization. Our transformation has been described in terms of refinement and inventiveness. When comparing our results to those discovered in existing academic publications, it becomes clear that our approaches in the European Choice Valuation framework are exceptional. The schemes' effective and comprehensive execution is investigated and confirmed in an attempt to display that it may be applicable to other nonlinear evolutionary models that emerge in business and accountancy.

Data Availability
No data were used to support this study.

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