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

Authors: Saima Rashid; Saad Ihsan Butt; Zakia Hammouch; Ebenezer Bonyah
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The Black-Scholes equation (BSe) is fascinating in the business world for predicting the performance of financial 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 fixed 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.


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## Body

## 1. 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 [6–9]. 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 [10–14]. In addition, fractional differential 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.It is imperative to address fractional-order nonlinear partial differential equations (NPDEs) that regulate the foregoing experimental results in order to successfully comprehend such occurrences. There is, however, no universal comprehensive principle that applies to NPDEs in an attempt to obtain a numerical approach. Researchers have determined successful approaches to derive meaningful numerical methods for NPDEs in recent times, including the inverse scattering transform Sin-Cos method (SCM) [17], homotopy perturbation method (HPM) [18], Adomian decomposition method (ADM) [19, 20], new iterative transform method (NITM) [21], variation iteration method (VIM) [22], G/G′ expansion method [23], Lie symmetry analysis (LSA) [24], Haar wavelet method [25], inverse scattering transform [26], simple equation method [27], Bäcklund transformation [28], and henceforth.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]:
(1)DϱδU+ϖ22S2∂2U∂S2+ζS∂U∂S−ζU=0,subject to the playoff mapping
(2)US,T=maxS−E,0,where US,ϱ 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 U0,ϱ=0 and US,T≈S as S↦∞. The parabolic diffusion problem can perhaps be described as the BSe in (1). Inducing the modifications that follow:
(3)S=Eexpy1,ϱ=T−2τϖ2,U=EUy1,ϱ,then (1) diminishes to
(4)DϱδUy1,ϱ=∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ,related initial condition (ICs) are
(5)Uy1,0=maxexpy1−1,0,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:
(6)DϱδUy1,ϱ=−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ,supplemented ICs
(7)Uy1,0=maxy1−25exp−0.06,0.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 theK(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 fractional-order BSe. For renewability algorithmic techniques, the Yang transform efficiently integrates the ADM and NITM. The Yang transform is a combination of a few different transforms. Both these proposed techniques produce interpretive findings in the sense of a convergent series. The Caputo-Fabrizio fractional derivative operator is used to explain quantitative categorizations of the BSe. The offered methodologies are well demonstrated in modeling and enumeration investigations. The exact-analytical findings are a valuable way to analyze the dynamics of systems that are problematic to computationally analyze, notably for fractional PDEs. Financial and monetary phenomena can be investigated using this approximate expression.


## 2. 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 derivativeCFD is described as follows:
(8)D0cϱδUϱ=1Γr−δ∫0ϱUry1ϱ−y1δ+1−rdy1,r−1<δ<r,drdϱrUϱ,δ=r.Definition 2 (see [15]).
The Caputo fractional derivative operator is described as follows:(9)CFDϱδUϱ=2−δAδ21−δ∫0ϱexp−δϱ−y11−δU′ϱdϱ,where U∈H1a1,a2Sobolevspace,a1<a2,δ∈0,1, and Aδ signifies a normalization function as Aδ=A0=A1=1.Definition 3 (see [15]).
The fractional integral of the Caputo-Fabrizio operator is defined as(10)CFIϱδUϱ=21−δ2−δAδUϱ+2δ2−δAδ∫0ϱUy1dy1.Definition 4 (see [41]).
The Yang transform is described as follows:(11)YUφ=Ys1=∫0∞Uφexp−φs1dφ,φ>0.
The Yang transform of a range of vital expressions is as follows:(12)Y1=s1,Yφ=s12,⋮YφδΓδ+1=s1δ+1.Definition 5 (see [41]).
The Yang transform of the CFD operator is mentioned as(13)YD0cϱδUϱ,s=φ−δQs−∑κ=0δ−1φ1−δ−κsUκ0,r−1<δ<r,φ>0.Definition 6 (see [45]).
The Yang transform of the Caputo-Fabrizio fractional derivative operator is stated as(14)YD0CFϱδUφ,s1=YUφ−s1U01+δs1−1.Definition 7 (see [46]).
The Mittag-Leffler function for single parameter is defined as(15)Eδz=∑κ=0∞z1κΓκδ+1,δ,z1∈ℂ,Rδ≥0.


## 3. Algorithmic Configuration for Nonlinear PDEs

Let us surmise the fractional version of nonlinear PDE:(16)DϱδUy1,ϱ+LUy1,ϱ+NUy1,ϱ=Qy1,ϱ,ϱ>0,0<δ≤1having ICs
(17)Uy1,0=Gy1,where Dϱδ=∂δUy1,ϱ/∂ϱδ represents the Caputo-Fabrizio fractional derivative considering the order δ∈0,1 whilst L and N indicates the linear and nonlinear functionals, respectively. Furthermore, Qy1,ϱ indicates the source term.


### 3.1. Construction of Yang Decomposition Method

Incorporating the Yang transformation to (16), we obtain
(18)YDϱδUy1,ϱ+L¯Uy1,ϱ+N¯Uy1,ϱ=YQy1,ϱ.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:(19)φ−δUy1,ϱ=∑p=0n−1φ1−δ−pUp0+YL¯Uy1,ϱ+N¯Uy1,ϱ+YQy1,ϱ,(20)11+δφ−1Uy1,ϱ=φ1+δφ−1U0+YL¯Uy1,ϱ+N¯Uy1,ϱ+YQy1,ϱ.The inverse Yang transform of (19) and (20), respectively, gives
(21)Uy1,ϱ=Y−1∑p=0n−1φ1−δ−pUp0+φδYQy1,ϱ−Y−1φδYL¯Uy1,ϱ+N¯Uy1,ϱ,(22)Uy1,ϱ=Y−1φU0+1+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ.The infinite seriesUy1,ϱ illustrates the result of the Yang decomposition approach:
(23)Uy1,ϱ=∑p=0∞Upy1,ϱ.As a consequence, the nonlinear componentN¯y1,ϱ can be assessed employing the Adomian decomposition approach, as follows:
(24)N¯Uy1,ϱ=∑p=0∞A~pU0,U1,⋯,p=0,1,⋯,where
(25)A~pU0,U1,⋯=1q!dpdδpN¯∑j=0∞δjUjδ=0,q>0.Putting (20) and (24) into (21) and (22), respectively, we attain
(26)∑p=0∞Upy1,ϱ=Gy1+G~y1−Y−1φδYL¯Uy1,ϱ+∑p=0∞A~p,(27)∑p=0∞Upy1,ϱ=Gy1+G~y1−Y−11+δφ−1YL¯Uy1,ϱ+∑p=0∞A~p.As a nutshell, the iterative approach for (26) and (27) is as follows:
(28)U0y1,ϱ=Gy1+G~y1,p=0,Uq+1y1,ϱ=−Y−1φδYL¯Upy1,ϱ+∑p=0∞A~p,q≥1,Uq+1y1,ϱ=−Y−11+δφ−1YL¯Upy1,ϱ+∑p=0∞A~p,q≥1.


### 3.2. Formation of Yang Iterative Transform Method

Implementing the Yang transform to (16) incorporates the ICs (17), ones obtain
(29)YDϱδUy1,ϱ+L¯Uy1,ϱ+N¯Uy1,ϱ=YQy1,ϱ.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(30)YUy1,ϱ=φδ∑p=0n−1φ1−δ−pϱUpφ,0−φδYL¯Uy1,ϱ+N¯Uy1,ϱ+φδYQy1,ϱ,(31)YUy1,ϱ=φδ∑p=0n−1φ1−δ−pϱUpφ,0−1+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ+1+δφ−1YQy1,ϱ.Using the fact of the inverse Yang transform of (30) and (31), respectively, produces
(32)Uy1,ϱ=Gφ+Y−1φδYQy1,ϱ−Y−1φδYL¯Uy1,ϱ+N¯Uy1,ϱ,(33)Uy1,ϱ=Gφ+Y−11+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ.Employing the recursive approach, we determine(34)Uy1,ϱ=∑p=0∞Upy1,ϱ.Moreover, utilizing the linearityL¯ of the operator, thus we have
(35)L¯∑p=0∞Upy1,ϱ=∑p=0∞L¯Upy1,ϱ,and the nonlinearity N¯ handled by (see [42])
(36)N¯∑p=0∞Upy1,ϱ=N¯U0y1,ϱ+∑q=1∞N¯∑κ=0pUpy1,ϱ−N¯∑κ=0p−1Upy1,ϱ=N¯U0+∑q=1∞Dp,where Dp=N¯∑κ=0pUp−N¯∑κ=0p−1Up.Inserting (37), (39), and (36) into (32) and (33), respectively, we observe
(37)∑p=0∞Upy1,ϱ=Gφ+Y−1φδYQy1,ϱ−Y−1φδYL¯∑κ=0pUpy1,ϱ+N¯U0+∑κ=1pDp,(38)∑p=0∞Upy1,ϱ=Gφ+Y−11+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯∑p=0pUpy1,ϱ+N¯U0+∑q=1pDp.Ultimately, for CFD, we develop appropriate analysis procedure:(39)U0y1,ϱ=Gφ+Y−1φδLQy1,ϱ,U1y1,ϱ=−Y−1φδYL¯U0y1,ϱ+N¯U0y1,ϱ,⋮Uq+1y1,ϱ=−Y−1φδYL¯Upy1,ϱ+Dp.The exploratory procedure for the Caputo-Fabrizio fractional derivative operator is shown then:(40)U0y1,ϱ=Gφ+Y−11+δφ−1LQy1,ϱ,U1y1,ϱ=−Y−11+δφ−1YL¯U0y1,ϱ+N¯U0y1,ϱ,⋮Uq+1y1,ϱ=−Y−11+δφ−1YL¯Upy1,ϱ+Dp.Eventually, theq-term result in series formulation is generated by (37), (39), and (40), and we have
(41)Uy1,ϱ≊U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+⋯+Upy1,ϱ,q∈ℕ.


## 3.1. Construction of Yang Decomposition Method

Incorporating the Yang transformation to (16), we obtain
(18)YDϱδUy1,ϱ+L¯Uy1,ϱ+N¯Uy1,ϱ=YQy1,ϱ.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:(19)φ−δUy1,ϱ=∑p=0n−1φ1−δ−pUp0+YL¯Uy1,ϱ+N¯Uy1,ϱ+YQy1,ϱ,(20)11+δφ−1Uy1,ϱ=φ1+δφ−1U0+YL¯Uy1,ϱ+N¯Uy1,ϱ+YQy1,ϱ.The inverse Yang transform of (19) and (20), respectively, gives
(21)Uy1,ϱ=Y−1∑p=0n−1φ1−δ−pUp0+φδYQy1,ϱ−Y−1φδYL¯Uy1,ϱ+N¯Uy1,ϱ,(22)Uy1,ϱ=Y−1φU0+1+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ.The infinite seriesUy1,ϱ illustrates the result of the Yang decomposition approach:
(23)Uy1,ϱ=∑p=0∞Upy1,ϱ.As a consequence, the nonlinear componentN¯y1,ϱ can be assessed employing the Adomian decomposition approach, as follows:
(24)N¯Uy1,ϱ=∑p=0∞A~pU0,U1,⋯,p=0,1,⋯,where
(25)A~pU0,U1,⋯=1q!dpdδpN¯∑j=0∞δjUjδ=0,q>0.Putting (20) and (24) into (21) and (22), respectively, we attain
(26)∑p=0∞Upy1,ϱ=Gy1+G~y1−Y−1φδYL¯Uy1,ϱ+∑p=0∞A~p,(27)∑p=0∞Upy1,ϱ=Gy1+G~y1−Y−11+δφ−1YL¯Uy1,ϱ+∑p=0∞A~p.As a nutshell, the iterative approach for (26) and (27) is as follows:
(28)U0y1,ϱ=Gy1+G~y1,p=0,Uq+1y1,ϱ=−Y−1φδYL¯Upy1,ϱ+∑p=0∞A~p,q≥1,Uq+1y1,ϱ=−Y−11+δφ−1YL¯Upy1,ϱ+∑p=0∞A~p,q≥1.


## 3.2. Formation of Yang Iterative Transform Method

Implementing the Yang transform to (16) incorporates the ICs (17), ones obtain
(29)YDϱδUy1,ϱ+L¯Uy1,ϱ+N¯Uy1,ϱ=YQy1,ϱ.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(30)YUy1,ϱ=φδ∑p=0n−1φ1−δ−pϱUpφ,0−φδYL¯Uy1,ϱ+N¯Uy1,ϱ+φδYQy1,ϱ,(31)YUy1,ϱ=φδ∑p=0n−1φ1−δ−pϱUpφ,0−1+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ+1+δφ−1YQy1,ϱ.Using the fact of the inverse Yang transform of (30) and (31), respectively, produces
(32)Uy1,ϱ=Gφ+Y−1φδYQy1,ϱ−Y−1φδYL¯Uy1,ϱ+N¯Uy1,ϱ,(33)Uy1,ϱ=Gφ+Y−11+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯Uy1,ϱ+N¯Uy1,ϱ.Employing the recursive approach, we determine(34)Uy1,ϱ=∑p=0∞Upy1,ϱ.Moreover, utilizing the linearityL¯ of the operator, thus we have
(35)L¯∑p=0∞Upy1,ϱ=∑p=0∞L¯Upy1,ϱ,and the nonlinearity N¯ handled by (see [42])
(36)N¯∑p=0∞Upy1,ϱ=N¯U0y1,ϱ+∑q=1∞N¯∑κ=0pUpy1,ϱ−N¯∑κ=0p−1Upy1,ϱ=N¯U0+∑q=1∞Dp,where Dp=N¯∑κ=0pUp−N¯∑κ=0p−1Up.Inserting (37), (39), and (36) into (32) and (33), respectively, we observe
(37)∑p=0∞Upy1,ϱ=Gφ+Y−1φδYQy1,ϱ−Y−1φδYL¯∑κ=0pUpy1,ϱ+N¯U0+∑κ=1pDp,(38)∑p=0∞Upy1,ϱ=Gφ+Y−11+δφ−1YQy1,ϱ−Y−11+δφ−1YL¯∑p=0pUpy1,ϱ+N¯U0+∑q=1pDp.Ultimately, for CFD, we develop appropriate analysis procedure:(39)U0y1,ϱ=Gφ+Y−1φδLQy1,ϱ,U1y1,ϱ=−Y−1φδYL¯U0y1,ϱ+N¯U0y1,ϱ,⋮Uq+1y1,ϱ=−Y−1φδYL¯Upy1,ϱ+Dp.The exploratory procedure for the Caputo-Fabrizio fractional derivative operator is shown then:(40)U0y1,ϱ=Gφ+Y−11+δφ−1LQy1,ϱ,U1y1,ϱ=−Y−11+δφ−1YL¯U0y1,ϱ+N¯U0y1,ϱ,⋮Uq+1y1,ϱ=−Y−11+δφ−1YL¯Upy1,ϱ+Dp.Eventually, theq-term result in series formulation is generated by (37), (39), and (40), and we have
(41)Uy1,ϱ≊U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+⋯+Upy1,ϱ,q∈ℕ.


## 4. 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].Theorem 8 (Uniqueness theorem).
For0<ε<1, then system (24) has a unique solution, where ε=K1+K2+K31+δϱ−1.Proof.
Surmise that there is a set of continuous mappings in the Banach spaceΩ=ℂI,. Considering I=0,T, present the norm . To continue this, suppose a mapping V:Ω↦Ω such that
(42)Uℓ+1y1,ϱ=Uy1,ϱ+Y−11+δφ−1YLUℓy1,ϱ+P¯Uℓy1,ϱ+NUℓy1,ϱ,ℓ≥0,where LUy1,ϱ≡∂3Uy1,ϱ/∂y12 and P¯Uy1,ϱ≡∂Uy1,ϱ/∂y1. Here, suppose that LUy1,ϱ and NUy1,ϱ are also Lipschitzian with P¯U−P¯U^<K1U−U^ and LU−LU^<K2U−U^, where K1 and K2 are Lipschitz constant, respectively, and U,U^ are distinct functional values.
(43)VU−VU^=maxϱ∈IY−11+δφ−1YLUy1,ϱ+P¯Uy1,ϱ+NUy1,ϱ−Y−11+δφ−1YLU^y1,ϱ+P¯U^y1,ϱ+NU^y1,ϱ≤maxϱ∈IY−11+δφ−1YLUy1,ϱ−LU^y1,ϱ+Y−11+δφ−1YP¯Uy1,ϱ−P¯U^y1,ϱ+Y−11+δφ−1YNUy1,ϱ−NU^y1,ϱ≤maxϱ∈IK1Y−11+δφ−1YUy1,ϱ−U^y1,ϱ+K2Y−11+δφ−1YUy1,ϱ−U^y1,ϱ+K3Y−11+δφ−1YUy1,ϱ−U^y1,ϱ≤maxϱ∈IK1+K2+K3Y−11+δφ−1YUy1,ϱ−U^y1,ϱ≤K1+K2+K3Y−11+δφ−1YUy1,ϱ−U^y1,ϱ=K1+K2+K3Y−11+δφ−1φUy1,ϱ−U^y1,ϱ=K1+K2+K31+δϱ−1Uy1,ϱ−U^y1,ϱ.
For0<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 thatS^ℓ be the nth partial sum, that is, S^ℓ=∑m=0ℓUℓy1,ϱ. Further, we exhibit S^ℓ is a Cauchy sequence in Banach space U.
We do it by contemplating a novel kind of Adomian polynomials.(44)R¯S^ℓ=H~ℓ+∑p=0ℓ−1H~p,NS^ℓ=H~ℓ+∑c=0ℓ−1H~c.
Now(45)S^ℓ−S^p=maxϱ∈IS^ℓ−S^p=maxϱ∈I∑m=q+1ℓU^y1,ϱ,m=1,2,3,⋯≤maxϱ∈IY−11+δφ−1Y∑m=q+1ℓLUℓ−1y1,ϱ+Y−11+δφ−1Y∑m=q+1ℓP¯Uℓ−1y1,ϱ+Y−11+δφ−1Y∑m=q+1ℓH~ℓ−1y1,ϱ=maxϱ∈IY−11+δφ−1Y∑m=qℓ−1LUℓy1,ϱ+Y−11+δφ−1Y∑m=qℓ−1P¯Uℓy1,ϱ+Y−11+δφ−1Y∑m=qℓ−1H~ℓy1,ϱ≤maxϱ∈IY−11+δφ−1Y∑m=qℓ−1LS^ℓ−1−LS^q−1+Y−11+δφ−1Y∑m=qℓ−1P¯S^ℓ−1−P¯S^q−1+Y−11+δφ−1Y∑m=qℓ−1NS^ℓ−1−NS^q−1≤maxϱ∈IY−11+δφ−1YLS^ℓ−1−LS^q−1+Y−11+δφ−1YP¯S^ℓ−1−P¯S^q−1+Y−11+δφ−1YNS^ℓ−1−NS^q−1≤K1maxϱ∈IY−11+δφ−1YS^ℓ−1−S^q−1+K2maxϱ∈IY−11+δφ−1YS^ℓ−1−S^q−1+K3maxϱ∈IY−11+δφ−1YS^ℓ−1−S^q−1=K1+K2+K3Y−11+δφ−1φS^ℓ−1−S^q−1.=K1+K2+K31+δϱ−1S^ℓ−1−S^q−1.
Assumen=q+1; then
(46)S^q+1−S^p≤εS^p−S^q−1≤ε2S^q−1−S^q−2≤⋯≤εpS^1−S^0,where ε=1+δϱ−1. In view of triangular variant, we have
(47)S^ℓ−S^p≤S^q+1−S^p+S^q+2−S^q+1+⋯+S^ℓ−S^ℓ−1≤εp+εq+1+⋯+εℓ−1S^1−S^0≤εp1−εℓ−pεU1,since 0<ε<1, we have 1−εℓ−p<1, and then
(48)S^ℓ−S^p≤εp1−εmaxϱ∈IU1.
Thus,U1<∞ (since Uy1,ϱ is bounded). Also, as q↦∞, then S^ℓ−S^p↦0. Therefore, S^1 is a Cauchy sequence in K. Ultimately, ∑n=0∞Uℓ is convergent, and the direct result is obtained.Theorem 10 (see [47]) (Error estimate).
The absolute inaccuracy of the (16) through (24) sum is determined as
(49)maxϱ∈IUy1,ϱ−∑ℓ=1pUℓy1,ϱ≤εp1−εmaxϱ∈IU1.


## 5. 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.


### 5.1. Yang Decomposition Method

Example 1 (see [29]).
Surmise the fractional-order BSM (4) supplemented with the (5).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
(50)YDϱδUy1,ϱ=Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing the Yang transform’s differentiation criteria gives(51)φ−δϱYUy1,ϱ=ϕϱ∑p=0p−1φ1−δ−pUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing (5), we find
(52)YUy1,ϱ=φmaxexpy1−1,0+φδY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Applying the inverse Yang transform produces(53)Uy1,ϱ=Y−1φ.maxexpy1−1,0+Y−1φδY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
To determine this, apply the Yang decomposition approach as follows:(54)U0y1,ϱ=Y−1φ.maxexpy1−1,0=maxexpy1−1,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(55)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−1φδY∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(56)U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1φδ+1=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−11φδϱYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−11φδϱYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
For Example1, the series form solution is developed as follows:
(57)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.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
(58)11+δ1−φYUy1,ϱ=11+δ1−φ∑p=0p−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing (5), we obtain
(59)YUy1,ϱ=φmaxexpy1−1,0+1+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ,Uy1,ϱ=Y−1φ.maxexpy1−1,0+Y−11+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Employing the Yang decomposition approach produces(60)Q0y1,ϱ=Y−1φmaxexpy1−1,0=maxexpy1−1,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(61)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−11+δφ−1Y∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(62)U1y1,ϱ=Y−11+δφ−1YU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,01+δϱ−1,U2y1,ϱ=Y−11+δφ−1YU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,01−δ2+2ϱδ1−δ+ρ2δ22,U3y1,ϱ=Y−11+δφ−1YU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0A3δ×1−δ3+3ϱ2δ21−δρ22+3ϱδ1−δ2+ϱ3δ33,⋮
For Example1, the series form solution is developed as follows:
(63)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯,=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.
Considering the Taylor series expansion and assigningδ=1, the exact findings of Example 1 can be determined as
(64)Uy1,ϱ=maxexpy1−1,0exp−ζϱ+maxexpy1,01−exp−ζϱ.Example 2 (see [30]).
Surmise the fractional-order BSM (6) supplemented with the (7).Case 1.
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
(65)YDϱδUy1,ϱ=Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing the Yang transform’s differentiation criteria, gives(66)φ−δYUy1,ϱ=∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ρ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing (7), we find
(67)YUy1,ϱ=φ.maxy1−25exp−0.06,0+φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Applying the inverse Yang transform produces(68)Uy1,ϱ=Y−1φ.maxy1−25exp−0.06,0+Y−1φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
To determine this, apply the Yang decomposition approach as follows:(69)Q0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(70)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−1φδY∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(71)U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1φδ+1=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−11φδϱYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−11φδϱYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
For Example2, the series form solution is developed as follows:
(72)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,0×1−1−0.06ϱδΓδ+1−0.0036ϱ2δΓ2δ+1−0.000216ϱ3δΓ3δ+1+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,01−Eδ0.06ϱδ.Case 2.
The Caputo-Fabrizio fractional derivative operator and the Yang decomposition approach are now used to solve the (6).
Assuming (65) and implementing the Yang transform’s differentiation criteria, we obtain
(73)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing (7), we obtain
(74)YUy1,ϱ=φ.maxy1−25exp−0.06,0+1+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Employing the inverse Yang transform gives(75)Uy1,ϱ=Y−1φ.maxy1−25exp−0.06,0+Y−11+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Employing the Yang decomposition approach, we obtain(76)Q0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0.
We predict the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(77)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12∑p=0∞Uy1,ϱy1y1−0.06y1∑p=0∞Uy1,ϱy1+0.06∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(78)U1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−−0.000216y1+0.00216maxy1−25exp−0.06,0A3δ×1−δ3+3ϱ2δ21−δϱ22+3ϱδ1−δ2+ϱ3δ33,⋮
For Example2, the series form solution is developed as follows:
(79)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0−0.06y1−0.06maxy1−25exp−0.06,01−δϱ−1−0.0036y1−0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22−.⋯
Considering the Taylor series expansion and assigningδ=1, the exact findings of Example 2 can be determined as
(80)Uy1,ϱ=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,01−exp−0.06ϱ.


### 5.2. Yang Iterative Transform Method

Example 3 (see [29]).
Surmise the fractional-order BSM (4) supplemented with the (5).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
(81)φ−δYUy1,ϱ=∑p=0n−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
It follows that(82)YUy1,ϱ=φ.maxexpy1−1,0+φδϱY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
In view of the proposed algorithm in Section3.2, we find
(83)U0y1,ϱ=Y−1φδmaxexpy1−1,0=maxexpy1−1,0,U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1ϕsφδ+1s=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−1φδYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−1φδYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
The result in series representation is(84)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Eventually, we have(85)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.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
(86)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0p−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
It follows that(87)YUy1,ϱ=φ.maxexpy1−1,0+1+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
In view of the proposed algorithm in Section3.2, we find
(88)U0y1,ϱ=Y−1φ.maxexpy1−1,0=maxexpy1−1,0,U1y1,ϱ=Y−11+δφ−1YU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1YU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1YU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0×1−δ3+3ϱδ1−δ2+3ϱ22δ21−δ+ϱ3δ33.⋮
The result in series representation is(89)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Eventually, we have(90)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζ1+δϱ−1−ζ21−δ2+2ϱδ1−δ+ϱ2δ22−⋯+maxexpy1,0−ζ1+δϱ−1−ζ21−δ2+2ϱδ1−δ+ϱ2δ22−⋯.Example 4 (see [30]).
Surmise the fractional-order BSM (6) supplemented with the (7).Case 1.
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
(91)φ−δYUy1,ϱ=∑p=0p−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
It follows that(92)YUy1,ϱ=φ.maxy1−25exp−0.06,0+φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
In view of the proposed algorithm in Section3.2, we have
(93)U0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0,U1y1,ϱ=Y−1φδY−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,0Y−1φδ+1s=−0.06y1+0.06maxy1−25exp−0.06,0ϱδΓδ+1,U2y1,ϱ=Y−1φδY−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−1φδY−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−0.000216y1+0.00216maxy1−25exp−0.06,0ϱ3δΓ3δ+1,⋮
The result in series representation is(94)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Consequently, we have(95)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯,=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06×1−1−0.06ϱδΓδ+1−0.0036ϱ2δΓ2δ+1−0.000216ϱ3δΓ3δ+1+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.061−Eδ0.06ϱδ.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
(96)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
It follows that(97)YUy1,ϱ=φ.maxy1−25exp−0.06,0+1+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
In view of the proposed algorithm in Section3.2, we have
(98)U0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0,U1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−−0.000216y1+0.00216maxy1−25exp−0.06,0×1−δ3+3ϱδ1−δ2+3δ21−δϱ2δ22+ϱ3δ33,⋮
The result in series representation is(99)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Consequently, we have(100)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0−0.06y1−0.06maxy1−25exp−0.06,01+δϱ−1−0.0036y1−0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22−⋯.


### 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 differenty1 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 1
The actual,YDMCFD and YDMCF results of Example 1 for multiple fractional-orders with changing terms of y1 and ϱ.
y1ϱδ=0.7δ=0.8δ=0.9δ=1YDMCFDδ=1YDMCFExact0.10.60.47051923880.47566901380.47411904790.47051923880.47041622590.47051681330.70.61422597720.59116764800.56971057600.55020606710.55020606710.55019404690.80.48440253260.49042748560.49199509430.49051185800.49026767920.49050424020.90.69566928980.69566928980.65765492720.63806909430.63806909430.63788193741.00.72816829040.71214116190.69543142600.67778238270.67778238270.67733155110.30.60.57866897130.58098344550.57909031280.57469349620.57456767590.57469053350.70.58550311780.59030215970.59022163760.58700000280.58680020490.58699453150.80.80436926850.77826944820.75250691600.72750648170.72750648170.72743316620.90.48896493110.49711964260.50043834090.50027281180.49992514300.50026064761.00.49312685490.50342526070.50858814200.50988067670.50940376490.50986219470.50.60.70678787750.70961478280.70730250530.70193222130.70177854400.70192860270.70.71513512300.72099668600.72099668600.72089833610.71696342250.71695673970.80.72264366190.73163183540.73397043290.73175770440.73139343240.73174634000.90.72944996070.74161536060.74656627670.74631933620.74580067550.74630118951.00.73565882130.75102223740.75872435120.76065258650.75994111770.76062501460.70.60.86327266290.86672545280.86390123080.85734195110.85715424930.85733753140.70.87346801170.88062734090.88050721600.87570110170.87540303810.87569293940.80.88263896170.89361714170.89647351110.89377087840.89332595560.89375699790.90.89095219370.90581104680.91185810950.91155649570.91092300190.91153433121.00.89853571330.91730063210.92670801520.92906316710.92819417710.92902949060.90.61.0544036121.0586208591.0551713461.0471598241.0469305641.0471544260.71.0668562391.0756006631.0754539421.0695837411.0692196851.0695737710.81.0780576621.0914664421.0949552191.0916542161.0911107861.0916372620.91.0882114671.1063601111.1137460101.1133776181.1126038671.1133505461.01.0974739981.1203935221.1318837261.1347603151.1336989281.134719182Table 2
For estimated outcomes ofUy1,ϱ at δ=1 considering multiple choices of y1 and ϱ, examine HPM [35], YDMCFD, and YDMCF of Example 1.
y1ϱExact−HPMExact−YDMCFDExact−YDMCF0.10.67.900000×10−32.425509×10−61.005999×10−40.71.099099×10−34.479656×10−61.591007×10−40.84.244800×10−27.617867×10−62.365698×10−40.99.107607×10−21.216420×10−63.355046×10−41.05.478550×10−21.848200×10−64.584298×10−40.30.69.980000×10−42.962700×10−61.228576×10−40.75.400763×10−35.471300×10−61.943266×10−40.87.800998×10−29.304400×10−62.889363×10−40.93.900567×10−31.485740×10−64.097862×10−41.05.100562×10−22.257400×10−65.599274×10−40.50.63.009801×10−33.618600×10−61.500587×10−40.76.400789×10−36.682800×10−62.373511×10−40.89.660009×10−31.136440×10−53.529076×10−40.92.934890×10−21.814670×10−55.005140×10−41.06.000989×10−32.757190×10−56.838969×10−40.70.62.560000×10−34.419700×10−61.832821×10−40.72.200000×10−28.162300×10−62.899013×10−40.81.056900×10−31.388050×10−54.310423×10−40.934.008890×10−32.216450×10−56.113293×10−41.08.000956×10−33.367650×10−58.353135×10−40.90.67.789435×10−25.398000×10−62.238620×10−60.73.000897×10−29.970000×10−63.540860×10−40.82.788609×10−21.695400×10−55.264760×10−40.96.560000×10−32.707200×10−57.466790×10−41.02.000043×10−24.113300×10−51.020254×10−3Table3 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.Table 3
The actual,YDMCFD and YDMCF results of Example 2 for multiple fractional-orders with changing terms of y1 and ϱ.
y1ϱδ=0.7δ=0.8δ=0.9δ=1JDMCFDδ=1JDMABCExact0.10.60.00261213750.00202939220.00156398150.00119640720.00119640720.0011964070.70.47936940850.48329853170.48323260580.48059495430.48043137350.48059047470.80.00557972080.00484499710.00417325290.00356779440.00356779440.0035677930.90.00679611940.00607549490.00538806750.00474286080.00474286080.00474285801.00.00791454290.00723655650.00656454690.00591090000.00591090000.00591089330.30.60.00522427500.00405878450.00312796300.00239281440.00239281440.00239281430.70.00844087060.00703451540.00581517600.00477131520.00477131520.00477131480.80.0111594410.00968999420.00834650590.00713558880.00713558880.00713558700.90.0135922380.0121509890.0107761350.00948572160.00948572160.00948571601.00.0158290850.0144731130.0131290930.0118218000.0118218000.01182178660.50.60.00783641250.00608817670.00469194450.00358922160.00358922160.00358922150.70.0126613050.0105517730.00872276410.00715697280.00715697280.00715697220.80.0167391620.0145349910.0125197580.0107033830.0107033830.01070338050.90.0203883580.0182264840.0161642020.0142285820.0142285820.01422857411.00.0237436280.0217096690.0196936400.0177327000.0177327000.01773267990.70.60.0104485500.00811756900.00625592610.00478562880.00478562880.00478562870.70.0168817410.0140690300.0116303520.00954263040.00954263040.00954262960.80.0223188830.0193799880.0166930110.0166930110.0166930110.01427117400.90.0271844770.0243019790.0215522700.0189714430.0189714430.01897143211.00.0316581710.0289462260.0262581870.0236436000.0236436000.02364357320.90.60.0130606870.0101469610.00781990760.00598203600.00598203600.0059820350.70.0211021760.0175862880.0145379400.0119282880.0119282880.0119282870.80.0278986040.0242249850.0208662640.0178389720.0178389720.0178389670.90.0339805970.0303774740.0269403370.0237143040.0237143040.0237142901.00.0395727140.0361827820.0328227340.0295545000.0295545000.029554466Table 4
For estimated outcomes ofUy1,ϱ at δ=1 considering multiple choices of y1 and ϱ, examine HPM [35], YDMCFD,and YDMCF of Example 2.
y1ϱExact−HPMExact−YDMCFDExact−YDMCF0.10.63.00450×10−102.00000×10−112.00000×10−110.72.00000×10−94.4796×10−61.80000×10−100.89.50000×10−98.80000×10−108.80000×10−100.93.89000×10−82.76000×10−92.76000×10−91.07.70000×10−86.70000×10−96.70000×10−90.30.65.780000×10−104.00000×10−114.00000×10−110.74.80000×10−93.60000×10−103.60000×10−100.82.98000×10−81.76000×10−91.76000×10−90.96.45000×10−85.52000×10−95.52000×10−91.02.47000×10−71.34000×10−81.34000×10−80.50.67.96600×10−106.00000×10−116.00000×10−110.76.785000×10−95.40000×10−105.40000×10−100.83.98000×10−82.64000×10−92.64000×10−90.99.31000×10−88.28000×10−98.28000×10−91.03.003000×10−72.01000×10−82.01000×10−80.70.69.890000×10−108.00000×10−118.00000×10−110.79.80000×10−97.20000×10−107.20000×10−100.84.94000×10−83.52000×10−93.52000×10−90.92.89000×10−71.10400×10−81.10400×10−81.03.60089×10−72.68000×10−82.68000×10−80.90.62.9900×10−91.00000×10−101.00000×10−100.711.00011×10−99.00000×10−109.00000×10−100.86.40000×10−84.40000×10−94.40000×10−90.92.87000×10−71.38000×10−81.38000×10−81.04.89000×10−73.35000×10−83.35000×10−8For Example1, Figure 1 displays the evolution of the Yang decomposition technique’s data from Uy1,ϱ. Figures 1(a) and 1(b) exhibit the performance of precise and approximation BSe option pricing findings using the CFD operator, whilst Figures 2(a) and 2(b) presents the profile for different Brownian motion δ=0.9 and δ=0.8, respectively. Figures 3(a) and 3(b) indicate the absolute errors conducted and fractional-order fluctuation of Uy1,ϱ. At δ=0.7,0.8,0.9,1.0. The multiple fractional orders act similarly.Figure 1
Three-dimensional illustration via CFD of Example1 when δ=1. (a) Exact solution. (b) Approximate solution.
(a)(b)Figure 2
Three-dimensional illustration of the approximate solution via CFD of Example1 when (a) δ=0.9 and δ=0.8.
(a)(b)Figure 3
Three-dimensional illustration via the CFD of Example1. (a) Absolute error. (b) Multiple fractional-order.
(a)(b)Trying to continue in the analogous trend, Figures4(a) and 4(b) visually depict the precise-approximate repercussions Uy1,ϱ for (6) using the Yang decomposition approach on the contents of the choices, whilst Figures 5(a) and 5(b) present the profile for Brownian motion δ=0.9 and δ=0.8, respectively. The absolute error and sensitivity of gathered information for (6) involving different numerical and Brownian movements of δ=0.7,0.8,0.9 and 1 are illustrated in Figures 6(a) and 6(a). Furthermore, Figure 6(b) refers to the dynamic of the two-dimensional alternatives of the analysis values Uy1,ϱ 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.Figure 4
Three-dimensional illustration via CFD of Example1 when δ=1. (a) Exact solution. (b) Approximate solution.
(a)(b)Figure 5
Three-dimensional illustration of the approximate solution via CFD of Example1 when (a) δ=0.9 and δ=0.8.
(a)(b)Figure 6
Three-dimensional illustration via the CFD of Example1. (a) Absolute error. (b) Multiple fractional-order.
(a)(b)


## 5.1. Yang Decomposition Method

Example 1 (see [29]).
Surmise the fractional-order BSM (4) supplemented with the (5).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
(50)YDϱδUy1,ϱ=Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing the Yang transform’s differentiation criteria gives(51)φ−δϱYUy1,ϱ=ϕϱ∑p=0p−1φ1−δ−pUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing (5), we find
(52)YUy1,ϱ=φmaxexpy1−1,0+φδY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Applying the inverse Yang transform produces(53)Uy1,ϱ=Y−1φ.maxexpy1−1,0+Y−1φδY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
To determine this, apply the Yang decomposition approach as follows:(54)U0y1,ϱ=Y−1φ.maxexpy1−1,0=maxexpy1−1,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(55)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−1φδY∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(56)U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1φδ+1=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−11φδϱYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−11φδϱYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
For Example1, the series form solution is developed as follows:
(57)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.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
(58)11+δ1−φYUy1,ϱ=11+δ1−φ∑p=0p−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Utilizing (5), we obtain
(59)YUy1,ϱ=φmaxexpy1−1,0+1+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ,Uy1,ϱ=Y−1φ.maxexpy1−1,0+Y−11+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
Employing the Yang decomposition approach produces(60)Q0y1,ϱ=Y−1φmaxexpy1−1,0=maxexpy1−1,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(61)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−11+δφ−1Y∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(62)U1y1,ϱ=Y−11+δφ−1YU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,01+δϱ−1,U2y1,ϱ=Y−11+δφ−1YU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,01−δ2+2ϱδ1−δ+ρ2δ22,U3y1,ϱ=Y−11+δφ−1YU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0A3δ×1−δ3+3ϱ2δ21−δρ22+3ϱδ1−δ2+ϱ3δ33,⋮
For Example1, the series form solution is developed as follows:
(63)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯,=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.
Considering the Taylor series expansion and assigningδ=1, the exact findings of Example 1 can be determined as
(64)Uy1,ϱ=maxexpy1−1,0exp−ζϱ+maxexpy1,01−exp−ζϱ.Example 2 (see [30]).
Surmise the fractional-order BSM (6) supplemented with the (7).Case 1.
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
(65)YDϱδUy1,ϱ=Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing the Yang transform’s differentiation criteria, gives(66)φ−δYUy1,ϱ=∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ρ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing (7), we find
(67)YUy1,ϱ=φ.maxy1−25exp−0.06,0+φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Applying the inverse Yang transform produces(68)Uy1,ϱ=Y−1φ.maxy1−25exp−0.06,0+Y−1φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
To determine this, apply the Yang decomposition approach as follows:(69)Q0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0.
We predict that the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(70)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−1φδY∑p=0∞Uy1,ϱy1y1+ζ−1∑p=0∞Uy1,ϱy1−ζ∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(71)U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1φδ+1=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−11φδϱYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−11φδϱYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
For Example2, the series form solution is developed as follows:
(72)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,0×1−1−0.06ϱδΓδ+1−0.0036ϱ2δΓ2δ+1−0.000216ϱ3δΓ3δ+1+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,01−Eδ0.06ϱδ.Case 2.
The Caputo-Fabrizio fractional derivative operator and the Yang decomposition approach are now used to solve the (6).
Assuming (65) and implementing the Yang transform’s differentiation criteria, we obtain
(73)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Utilizing (7), we obtain
(74)YUy1,ϱ=φ.maxy1−25exp−0.06,0+1+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Employing the inverse Yang transform gives(75)Uy1,ϱ=Y−1φ.maxy1−25exp−0.06,0+Y−11+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
Employing the Yang decomposition approach, we obtain(76)Q0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0.
We predict the unidentified mappingUy1,ϱ may be expressed as an infinite series of the pattern
(77)Uy1,ϱ=∑p=0∞Upy1,ϱ,∑p=0∞Uq+1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12∑p=0∞Uy1,ϱy1y1−0.06y1∑p=0∞Uy1,ϱy1+0.06∑p=0∞Uy1,ϱ,p=0,1,2,⋯,(78)U1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−−0.000216y1+0.00216maxy1−25exp−0.06,0A3δ×1−δ3+3ϱ2δ21−δϱ22+3ϱδ1−δ2+ϱ3δ33,⋮
For Example2, the series form solution is developed as follows:
(79)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0−0.06y1−0.06maxy1−25exp−0.06,01−δϱ−1−0.0036y1−0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22−.⋯
Considering the Taylor series expansion and assigningδ=1, the exact findings of Example 2 can be determined as
(80)Uy1,ϱ=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06,01−exp−0.06ϱ.


## 5.2. Yang Iterative Transform Method

Example 3 (see [29]).
Surmise the fractional-order BSM (4) supplemented with the (5).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
(81)φ−δYUy1,ϱ=∑p=0n−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
It follows that(82)YUy1,ϱ=φ.maxexpy1−1,0+φδϱY∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
In view of the proposed algorithm in Section3.2, we find
(83)U0y1,ϱ=Y−1φδmaxexpy1−1,0=maxexpy1−1,0,U1y1,ϱ=Y−1φδYU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,0Y−1ϕsφδ+1s=ζmaxexpy1,0−ζmaxexpy1−1,0ϱδΓδ+1,U2y1,ϱ=Y−1φδYU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−1φδYU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0ϱ3δΓ3δ+1.⋮
The result in series representation is(84)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Eventually, we have(85)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζϱδΓδ+1+ζ2ϱ2δΓ2δ+1−ζ3ϱ3δΓ3δ+1+⋯+maxexpy1,01−1+ζϱδΓδ+1−ζ2ϱ2δΓ2δ+1+ζ3ϱ3δΓ3δ+1+⋯=maxexpy1−1,0Eδ−ζϱδ+maxexpy1,0Eδ1−ζϱδ.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
(86)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0p−1φ1−δ−pϱUp0+Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
It follows that(87)YUy1,ϱ=φ.maxexpy1−1,0+1+δφ−1Y∂2Uy1,ϱ∂y12+ζ−1∂Uy1,ϱ∂y1−ζUy1,ϱ.
In view of the proposed algorithm in Section3.2, we find
(88)U0y1,ϱ=Y−1φ.maxexpy1−1,0=maxexpy1−1,0,U1y1,ϱ=Y−11+δφ−1YU0y1,ϱy1y1+ζ−1U0y1,ϱy1+ζU0=ζmaxexpy1,0−ζmaxexpy1−1,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1YU1y1,ϱy1y1+ζ−1U1y1,ϱy1+ζU1=−ζ2maxexpy1,0+ζ2maxexpy1−1,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1YU2y1,ϱy1y1+ζ−1U2y1,ϱy1+ζU2=−ζ3maxexpy1,0+ζ3maxexpy1−1,0×1−δ3+3ϱδ1−δ2+3ϱ22δ21−δ+ϱ3δ33.⋮
The result in series representation is(89)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Eventually, we have(90)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxexpy1−1,01−ζ1+δϱ−1−ζ21−δ2+2ϱδ1−δ+ϱ2δ22−⋯+maxexpy1,0−ζ1+δϱ−1−ζ21−δ2+2ϱδ1−δ+ϱ2δ22−⋯.Example 4 (see [30]).
Surmise the fractional-order BSM (6) supplemented with the (7).Case 1.
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
(91)φ−δYUy1,ϱ=∑p=0p−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
It follows that(92)YUy1,ϱ=φ.maxy1−25exp−0.06,0+φδY−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
In view of the proposed algorithm in Section3.2, we have
(93)U0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0,U1y1,ϱ=Y−1φδY−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,0Y−1φδ+1s=−0.06y1+0.06maxy1−25exp−0.06,0ϱδΓδ+1,U2y1,ϱ=Y−1φδY−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,0ϱ2δΓ2δ+1,U3y1,ϱ=Y−1φδY−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−0.000216y1+0.00216maxy1−25exp−0.06,0ϱ3δΓ3δ+1,⋮
The result in series representation is(94)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Consequently, we have(95)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯,=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.06×1−1−0.06ϱδΓδ+1−0.0036ϱ2δΓ2δ+1−0.000216ϱ3δΓ3δ+1+⋯=maxy1−25exp−0.06,0+y1−maxy1−25exp−0.061−Eδ0.06ϱδ.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
(96)11+δφ−1YUy1,ϱ=11+δφ−1∑p=0n−1φ1−δ−pϱUp0+Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
It follows that(97)YUy1,ϱ=φ.maxy1−25exp−0.06,0+1+δφ−1Y−0.082+siny12y12∂2Uy1,ϱ∂y12−0.06y1∂Uy1,ϱ∂y1+0.06Uy1,ϱ.
In view of the proposed algorithm in Section3.2, we have
(98)U0y1,ϱ=Y−1φ.maxy1−25exp−0.06,0=maxy1−25exp−0.06,0,U1y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U0y1,ϱy1y1−0.06y1U0y1,ϱy1−0.06U0=−0.06y1+0.06maxy1−25exp−0.06,01−δϱ−1,U2y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U1y1,ϱy1y1−0.06y1U1y1,ϱy1−0.06U1=−0.0036y1+0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22,U3y1,ϱ=Y−11+δφ−1Y−0.082+siny12y12U2y1,ϱy1y1−0.06y1U2y1,ϱy1−0.06U2=−−0.000216y1+0.00216maxy1−25exp−0.06,0×1−δ3+3ϱδ1−δ2+3δ21−δϱ2δ22+ϱ3δ33,⋮
The result in series representation is(99)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+.⋯
Consequently, we have(100)Uy1,ϱ=U0y1,ϱ+U1y1,ϱ+U2y1,ϱ+U3y1,ϱ+⋯=maxy1−25exp−0.06,0−0.06y1−0.06maxy1−25exp−0.06,01+δϱ−1−0.0036y1−0.0036maxy1−25exp−0.06,01−δ2+2ϱδ1−δ+ϱ2δ22−⋯.


## 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 differenty1 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 1
The actual,YDMCFD and YDMCF results of Example 1 for multiple fractional-orders with changing terms of y1 and ϱ.
y1ϱδ=0.7δ=0.8δ=0.9δ=1YDMCFDδ=1YDMCFExact0.10.60.47051923880.47566901380.47411904790.47051923880.47041622590.47051681330.70.61422597720.59116764800.56971057600.55020606710.55020606710.55019404690.80.48440253260.49042748560.49199509430.49051185800.49026767920.49050424020.90.69566928980.69566928980.65765492720.63806909430.63806909430.63788193741.00.72816829040.71214116190.69543142600.67778238270.67778238270.67733155110.30.60.57866897130.58098344550.57909031280.57469349620.57456767590.57469053350.70.58550311780.59030215970.59022163760.58700000280.58680020490.58699453150.80.80436926850.77826944820.75250691600.72750648170.72750648170.72743316620.90.48896493110.49711964260.50043834090.50027281180.49992514300.50026064761.00.49312685490.50342526070.50858814200.50988067670.50940376490.50986219470.50.60.70678787750.70961478280.70730250530.70193222130.70177854400.70192860270.70.71513512300.72099668600.72099668600.72089833610.71696342250.71695673970.80.72264366190.73163183540.73397043290.73175770440.73139343240.73174634000.90.72944996070.74161536060.74656627670.74631933620.74580067550.74630118951.00.73565882130.75102223740.75872435120.76065258650.75994111770.76062501460.70.60.86327266290.86672545280.86390123080.85734195110.85715424930.85733753140.70.87346801170.88062734090.88050721600.87570110170.87540303810.87569293940.80.88263896170.89361714170.89647351110.89377087840.89332595560.89375699790.90.89095219370.90581104680.91185810950.91155649570.91092300190.91153433121.00.89853571330.91730063210.92670801520.92906316710.92819417710.92902949060.90.61.0544036121.0586208591.0551713461.0471598241.0469305641.0471544260.71.0668562391.0756006631.0754539421.0695837411.0692196851.0695737710.81.0780576621.0914664421.0949552191.0916542161.0911107861.0916372620.91.0882114671.1063601111.1137460101.1133776181.1126038671.1133505461.01.0974739981.1203935221.1318837261.1347603151.1336989281.134719182Table 2
For estimated outcomes ofUy1,ϱ at δ=1 considering multiple choices of y1 and ϱ, examine HPM [35], YDMCFD, and YDMCF of Example 1.
y1ϱExact−HPMExact−YDMCFDExact−YDMCF0.10.67.900000×10−32.425509×10−61.005999×10−40.71.099099×10−34.479656×10−61.591007×10−40.84.244800×10−27.617867×10−62.365698×10−40.99.107607×10−21.216420×10−63.355046×10−41.05.478550×10−21.848200×10−64.584298×10−40.30.69.980000×10−42.962700×10−61.228576×10−40.75.400763×10−35.471300×10−61.943266×10−40.87.800998×10−29.304400×10−62.889363×10−40.93.900567×10−31.485740×10−64.097862×10−41.05.100562×10−22.257400×10−65.599274×10−40.50.63.009801×10−33.618600×10−61.500587×10−40.76.400789×10−36.682800×10−62.373511×10−40.89.660009×10−31.136440×10−53.529076×10−40.92.934890×10−21.814670×10−55.005140×10−41.06.000989×10−32.757190×10−56.838969×10−40.70.62.560000×10−34.419700×10−61.832821×10−40.72.200000×10−28.162300×10−62.899013×10−40.81.056900×10−31.388050×10−54.310423×10−40.934.008890×10−32.216450×10−56.113293×10−41.08.000956×10−33.367650×10−58.353135×10−40.90.67.789435×10−25.398000×10−62.238620×10−60.73.000897×10−29.970000×10−63.540860×10−40.82.788609×10−21.695400×10−55.264760×10−40.96.560000×10−32.707200×10−57.466790×10−41.02.000043×10−24.113300×10−51.020254×10−3Table3 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.Table 3
The actual,YDMCFD and YDMCF results of Example 2 for multiple fractional-orders with changing terms of y1 and ϱ.
y1ϱδ=0.7δ=0.8δ=0.9δ=1JDMCFDδ=1JDMABCExact0.10.60.00261213750.00202939220.00156398150.00119640720.00119640720.0011964070.70.47936940850.48329853170.48323260580.48059495430.48043137350.48059047470.80.00557972080.00484499710.00417325290.00356779440.00356779440.0035677930.90.00679611940.00607549490.00538806750.00474286080.00474286080.00474285801.00.00791454290.00723655650.00656454690.00591090000.00591090000.00591089330.30.60.00522427500.00405878450.00312796300.00239281440.00239281440.00239281430.70.00844087060.00703451540.00581517600.00477131520.00477131520.00477131480.80.0111594410.00968999420.00834650590.00713558880.00713558880.00713558700.90.0135922380.0121509890.0107761350.00948572160.00948572160.00948571601.00.0158290850.0144731130.0131290930.0118218000.0118218000.01182178660.50.60.00783641250.00608817670.00469194450.00358922160.00358922160.00358922150.70.0126613050.0105517730.00872276410.00715697280.00715697280.00715697220.80.0167391620.0145349910.0125197580.0107033830.0107033830.01070338050.90.0203883580.0182264840.0161642020.0142285820.0142285820.01422857411.00.0237436280.0217096690.0196936400.0177327000.0177327000.01773267990.70.60.0104485500.00811756900.00625592610.00478562880.00478562880.00478562870.70.0168817410.0140690300.0116303520.00954263040.00954263040.00954262960.80.0223188830.0193799880.0166930110.0166930110.0166930110.01427117400.90.0271844770.0243019790.0215522700.0189714430.0189714430.01897143211.00.0316581710.0289462260.0262581870.0236436000.0236436000.02364357320.90.60.0130606870.0101469610.00781990760.00598203600.00598203600.0059820350.70.0211021760.0175862880.0145379400.0119282880.0119282880.0119282870.80.0278986040.0242249850.0208662640.0178389720.0178389720.0178389670.90.0339805970.0303774740.0269403370.0237143040.0237143040.0237142901.00.0395727140.0361827820.0328227340.0295545000.0295545000.029554466Table 4
For estimated outcomes ofUy1,ϱ at δ=1 considering multiple choices of y1 and ϱ, examine HPM [35], YDMCFD,and YDMCF of Example 2.
y1ϱExact−HPMExact−YDMCFDExact−YDMCF0.10.63.00450×10−102.00000×10−112.00000×10−110.72.00000×10−94.4796×10−61.80000×10−100.89.50000×10−98.80000×10−108.80000×10−100.93.89000×10−82.76000×10−92.76000×10−91.07.70000×10−86.70000×10−96.70000×10−90.30.65.780000×10−104.00000×10−114.00000×10−110.74.80000×10−93.60000×10−103.60000×10−100.82.98000×10−81.76000×10−91.76000×10−90.96.45000×10−85.52000×10−95.52000×10−91.02.47000×10−71.34000×10−81.34000×10−80.50.67.96600×10−106.00000×10−116.00000×10−110.76.785000×10−95.40000×10−105.40000×10−100.83.98000×10−82.64000×10−92.64000×10−90.99.31000×10−88.28000×10−98.28000×10−91.03.003000×10−72.01000×10−82.01000×10−80.70.69.890000×10−108.00000×10−118.00000×10−110.79.80000×10−97.20000×10−107.20000×10−100.84.94000×10−83.52000×10−93.52000×10−90.92.89000×10−71.10400×10−81.10400×10−81.03.60089×10−72.68000×10−82.68000×10−80.90.62.9900×10−91.00000×10−101.00000×10−100.711.00011×10−99.00000×10−109.00000×10−100.86.40000×10−84.40000×10−94.40000×10−90.92.87000×10−71.38000×10−81.38000×10−81.04.89000×10−73.35000×10−83.35000×10−8For Example1, Figure 1 displays the evolution of the Yang decomposition technique’s data from Uy1,ϱ. Figures 1(a) and 1(b) exhibit the performance of precise and approximation BSe option pricing findings using the CFD operator, whilst Figures 2(a) and 2(b) presents the profile for different Brownian motion δ=0.9 and δ=0.8, respectively. Figures 3(a) and 3(b) indicate the absolute errors conducted and fractional-order fluctuation of Uy1,ϱ. At δ=0.7,0.8,0.9,1.0. The multiple fractional orders act similarly.Figure 1
Three-dimensional illustration via CFD of Example1 when δ=1. (a) Exact solution. (b) Approximate solution.
(a)(b)Figure 2
Three-dimensional illustration of the approximate solution via CFD of Example1 when (a) δ=0.9 and δ=0.8.
(a)(b)Figure 3
Three-dimensional illustration via the CFD of Example1. (a) Absolute error. (b) Multiple fractional-order.
(a)(b)Trying to continue in the analogous trend, Figures4(a) and 4(b) visually depict the precise-approximate repercussions Uy1,ϱ for (6) using the Yang decomposition approach on the contents of the choices, whilst Figures 5(a) and 5(b) present the profile for Brownian motion δ=0.9 and δ=0.8, respectively. The absolute error and sensitivity of gathered information for (6) involving different numerical and Brownian movements of δ=0.7,0.8,0.9 and 1 are illustrated in Figures 6(a) and 6(a). Furthermore, Figure 6(b) refers to the dynamic of the two-dimensional alternatives of the analysis values Uy1,ϱ 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.Figure 4
Three-dimensional illustration via CFD of Example1 when δ=1. (a) Exact solution. (b) Approximate solution.
(a)(b)Figure 5
Three-dimensional illustration of the approximate solution via CFD of Example1 when (a) δ=0.9 and δ=0.8.
(a)(b)Figure 6
Three-dimensional illustration via the CFD of Example1. (a) Absolute error. (b) Multiple fractional-order.
(a)(b)


## 6. 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 operators 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.



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