# Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Authors: Muhammad Imran Liaqat; Eric Okyere
Journal: Journal of Mathematics
 (2023)

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## Abstract

The residual power series method is effective for obtaining solutions to fractional-order differential equations. However, the procedure needs then−1ϖ derivative of the residual function. We are all aware of the difficulty of computing the fractional derivative of a function. In this article, we considered the simple and efficient method known as the Laplace residual power series method (LRPSM) to find the analytical approximate and exact solutions of the time-fractional Black–Scholes option pricing equations (BSOPE) in the sense of the Caputo derivative. This approach combines the Laplace transform and the residual power series method. The suggested method just needs the idea of an infinite limit, so the computations required to determine the coefficients are minimal. The obtained results are compared in the sense of absolute errors against those of other approaches, such as the homotopy perturbation method, the Aboodh transform decomposition method, and the projected differential transform method. The results obtained using the provided method show strong agreement with different series solution methods, demonstrating that the suggested method is a suitable alternative tool to the methods based on He’s or Adomian polynomials.


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## Body

## 1. Introduction

Fractional calculus (FC) deals with fractional derivatives and integrations. The pioneers of FC were two mathematicians, Leibniz and L’Hospital, and the date September 30, 1695, is regarded as its exact birthday. Many scientists and researchers have been drawn to FC in recent decades because it is commonly used in scientific contexts such as engineering, image processing, physics, biochemistry, biology, fluid mechanics, and entropy theory [1–5]. Although fractional derivatives can be defined in a variety of ways, not all of them are generally used. The Atangana–Baleanu, Riemann–Liouville (R-L), Caputo–Fabrizio, Caputo, and conformable operators are the most frequently used [6–12]. In some cases, fractional derivatives are preferable to integer-order derivatives for modeling because they can simulate and analyze complicated systems having complicated nonlinear processes and higher-order behaviors. There are two main causes of this. First, we can select any order for the derivative operator, rather than being restricted to an integer order. Second, depending on both the past and current circumstances, noninteger order derivatives are advantageous when the system has long-term memory. The primary components of FC, which is the generalized version of classical calculus and has piqued the interest of numerous academics and scientists due to its wide range of applications, are fractional order differential equations (FODEs). The FODEs are frequently used for their logical support in the mathematical framework of physical problems, including technology, healthcare, monetary markets, and decision theory. As a result, the solutions provided by the FODEs are significant and useful. Applications regularly face FODEs that are too complex for close-form solutions. Under the specified initial and boundary conditions, numerical methods present a potent alternative tool for solving FODEs. Several numerical techniques, including the Shehu decomposition approach [13], the differential transform method [14], the variational iteration method [15], the operational matrix approach [16], the homotopy analysis technique [17], the Aboodh transform decomposition method [18], the finite difference method [19], the fractional power series method [20], the Chebyshev polynomials method [21], the residual power series method [22], and the natural transform homotopy perturbation method [23], have been developed in recent years for solving FODEs.It is efficient to obtain approximate analytical solutions to FODEs using the residual power series method (RPSM). However, the algorithm requires then−1ϖ derivative of the residual function. We are all aware of how difficult it is to calculate a function’s fractional derivative. As a result, the use of conventional RPSM is somewhat constrained. To overcome the limitations of the RPSM, Eriqat et al. introduced a new technique called LRPSM [24]. Several FODEs have been solved through the recommended method [24–29]. The provided equation is transformed into the Laplace transform (LT) space in accordance with the set of rules for this novel approach; the solution to the new form of the problem is established; the solution to the original equation is then achieved by applying the inverse LT.The pricing of financial derivatives is a topic that has generated a great deal of interest and literature. A financial derivative is an asset whose price is based on the value of another asset. Frequently, a stock or bond serves as the underlying asset. Financial derivatives are not a completely novel concept. It is generally agreed that Charles Castelly’s work, which was published in 1877, was the first attempt at contemporary derivative pricing, despite some historical controversy regarding the exact year of the birth of financial derivatives [30]. Although it lacked mathematical rigor, Castell’s book served as a general introduction to ideas such as hedging and speculative trading.The first widely used mathematical method to calculate the theoretical value of an option contract using prevailing equity markets’ predicted dividends, the option’s strike price, projected interest rates, time till the cessation, and expected unpredictability was the Black–Scholes (BS) model, developed in 1997 by Fischer Black, Robert Merton, and Myron Scholes. The Pricing of Options and Corporate Liabilities by Black and Scholes, published in the Journal of Political Economy in 1973, provided the initial formulation of the equation. Robert C. Merton contributed to the editing of that document. Later that year, he wrote his own work, “Theory of Rational Option Pricing,” expanding the model’s mathematical capabilities and applications while also coining the term “BS theory of options pricing” [31]. One of the most significant mathematical representations of a financial market is the BS equation. The value of financial derivatives is controlled by a second-order parabolic partial differential equation. Many different commodities and payout structures have been used in the BS model for pricing stock options. The following equation describes the BS model for the value of an option [32]:(1)∂ϖwϰ,ζ∂ζϖ+σ2ϰ22∂2wϰ,ζ∂ϰ2+ℑζϰ∂wϰ,ζ∂ϰ−ℑζwϰ,ζ=0,where ϖ is the order of the Caputo derivative (CD), wϰ,ζ is the European offer’s value at the fundamental market cap of ϰ and time ζ, the fluctuation component of the company’s shares, often referred to as σ, quantifies the variance of the stock’s return, Y is the expiration date, and ℑζ is the risk-free interest rate. wcϰ,ζ and wpϰ,ζ, correspondingly, stand for the call and put values on European options. Then, the payoff functions are given by wcϰ,ζ=maxϰ−U,0 and wpϰ,ζ=maxU−ϰ,0, where U indicates the termination price for the option and the function maxϰ,0 gives the larger value between ϰ and 0. The important financial view of the BS equation is that it minimizes risk by allowing for the prudent selection of purchasing and selling the stock under scrutiny. It is a sign that, according to the BS financial model, there is only one correct price for the option. The important financial view of the BS equation is that it minimizes risk by allowing for the prudent selection of purchasing and selling the stock under scrutiny. It is a sign that, according to the BS financial model, there is only one correct price for the option. In this article, a fractional model that may be used to model the pricing of various financial derivatives is presented.Numerous techniques have been used to examine the time-fractional BSOPE [33–40]. Each of these techniques has specific restrictions and flaws. These techniques involve a lot of computing work and a long running time. In this article, we considered LRPSM, which is a simple and efficient technique to solve BSOPE. The advantage of the recommended method over the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM) is its strength in handling problems without the use of He’s and Adomian polynomials. The advantage of this approach is that the problem does not involve any physical parameter assumptions, no matter how big or small. For a series such as the RPSM, the coefficients must be determined each time using the fractional derivative. Since LRPSM just needs the idea of an infinite limit, the computations required to determine the coefficients are minimal. The LRPSM results are also compared to those of other approaches, including the projected differential transform method (PDTM) [34], the Aboodh transform decomposition method (ATMD) [35], and the homotopy perturbation method (HPM) [36]. The results obtained using the suggested method show strong agreement with numerous methodologies, proving that the LRPSM is a useful substitute for approaches using He’s or Adomian polynomials. Additionally, error functions are used to compare the exact and approximative solutions graphically and numerically. The higher degrees of accuracy and convergence rates were confirmed by the error analysis, demonstrating the suggested method’s efficacy and reliability. The process is rapid, exact, and simple to use, and it produces excellent results.The article’s structure is as follows: In Section2, we use a number of important definitions and conclusions from the theory of FC. The main concept of LRPSM is examined in Section 3 in order to establish and determine the solution of the time-fractional BSOPE. Section 4 investigates the potential, capability, and simplicity of the suggested approach using three numerical models. In Section 5, graphics and tables are used to investigate the numerical outcomes and discussion. Section 6, towards the end, has the conclusion.


## 2. Preliminaries

In this section, we examine several common definitions, properties, and some useful consequences that we used in this article.Definition 1 (see [24]).
We assume that the functionwϰ,ζ is of exponential order δ and piecewise continuous. Then, the LT of wϰ,ζ for ζ≥0 is formulated as(2)ℒwϰ,ζ=Wϰ,s=∫0∞wϰ,ζe−sζdζ,s>δ,and the inverse LT is defined by(3)ℒ−1Wϰ,s=wϰ,ζ=∫h−ι∞h+ι∞esζWϰ,sds,h=Res>h0,where h0 lies in the right half plane of the absolute convergence of the Laplace integral.Lemma 1 (see [24–29]).
Letw1ϰ,ζ and w2ϰ,ζ be piecewise continuous on 0,∞ and be of exponential order. We assume that ℒw1ϰ,ζ=W1ϰ,s, ℒw2ϰ,ζ=W2ϰ,s, and c1, c2 are constants. Then, the properties mentioned as follows are valid:(i)
ℒc1w1ϰ,ζ+c2w2ϰ,ζ=c1W1ϰ,s+c2W2ϰ,s(ii)
ℒ−1c1W1ϰ,s+c2W2ϰ,s=c1w1ϰ,ζ+c2w2ϰ,ζ(iii)
ϕ0ϰ=lims⟶∞sWϰ,s=wϰ,0(iv)
ℒDζϖwϰ,ζ=sϖWϰ,s−∑j=0n−1wjϰ,0/sj−ϖ+1,n−1<ϖ≤n,n∈ℕ(v)
ℒDζnϖwϰ,ζ=snϖWϰ,s−∑j=0n−1sϖn−j−1Dζjϖwϰ,0,0<ϖ≤1Definition 2 (see [41]).
The fractional derivative ofwϰ,ζ I order ϖ in the CD sense is defined as follows:(4)Dζϖwϰ,ζ=Jζn−ϖwnϰ,ζ,ζ≥0,n−1<ϖ≤n,where Jζn−ϖ is the R-L integral of wϰ,ζ.Theorem 1 (see [24]).
We assume that the multiple fractional power series (MFPS) representation for the functionℒwϰ,ζ=Wϰ,s is given by(5)Wϰ,s=∑n=0∞ϕnϰsnϖ+1,s>0,then we have(6)ϕnϰ=Dζnϖwϰ,0,where Dζnϖ=Dζϖ.Dζϖ…Dζϖn−times.The conditions for the convergence of the MFPS are determined in the following theorem.Theorem 2 (see [24]).
Letℒwϰ,ζ=Wϰ,s can be denoted as the new form of MFPS explained in Theorem 1. If sℒDζj+1ϖwϰ,ζ≤Z, on 0<s≤v with 0<ϖ≤1, then the remainder Rjϰ,s of the new form of MFPS satisfies the following inequality:(7)Rjϰ,s≤Zsj+1ϖ+1,0<s≤v.


## 3. Methodology for the LRPSM for the Time-Fractional BSOPE

This section examines the steps for using the suggested method to find solutions to BSOPEs. Running the LT on the BSOPE and then considering MFPS as the BSOPE’s new space solution constitute the main idea of LRPSM. The way in which the coefficients of this series utilize the limit idea is the main difference between the LRPSM and the RPSM. The generated consequents are then transformed into real space using the inverse LT. The guidelines for using the LRPSM to find solutions are as follows:Step 1: Equation (1) should be changed as follows:(8)Dζϖwϰ,ζ−fwϰ,ζ,σ2ϰ2∂2wϰ,ζ2∂ϰ2,ℑζϰ∂wϰ,ζ∂ϰ,ℑζwϰ,ζ=0.Step 2: by considering LT on both sides of equation (8), we obtain the following:(9)Wϰ,s−wϰ,0s−1sϖFϰ,s=0,where(10)Wϰ,s=ℒwϰ,ζ, and(11)Fϰ,s=ℒfσ2ϰ2∂2wϰ,ζ2∂ϰ2,ℑζwϰ,ζ,wϰ,ζ,ℑζϰ∂wϰ,ζ∂ϰ.Step 3: we assume that the solution of equation (9) is the following series:(12)Wϰ,s=∑n=0∞ϕnϰsnϰ+1,s>0.Step 4: we obtained the following as a result of using Lemma1(iii):(13)ϕ0ϰ=lims⟶∞sWϰ,s=wϰ,0=0.Step 5: we define thekth-truncated expansion of Wϰ,s as(14)Wkϰ,s=ϕ0ϰs+∑n=1kϕnϰsnϖ+1.Step 6: we introduce Laplace residual function (LRF) of equation (9) and the kth-LRF, respectively, as follows:(15)ℒResϰ,s=Wϰ,s−ϕ0ϰs−1sϖFs,(16)ℒReskϰ,s=Wkϰ,s−ϕ0ϰs−1sϖFks.Step 7: we use the expansion form ofWkϰ,s into ℒReskϰ,s.Step 8: we multiply both sides ofℒReskϰ,s with skϖ+1.Step 9: by utilizing the fact in equation (16), we solve the following sequence of algebraic equations for ϕnϰ, where n=1,2,3,…,k, step by step:(17)lims⟶∞sϖ+1ℒReskϰ,s=0,k=1,2,3,…Step 10: we use the attained values ofϕnϰ into the kth-truncated expansion of Wϰ,s for each n=1,2,3,…,k to attain the kth-approximate solution of the algebraic equation in equation (9).Step 11: we apply the inverse LT on the final form ofWkϰ,s to attain kth-approximate solution, wkϰ,ζ of the suggested problem.


## 4. Some Illustrated Problems

In this section, three time-fractional BSOPE in the CD sense are solved in order to assess the effectiveness and suitability of the suggested approach.Problem 1.
consider the time-fractional BSOPE that follows [34]:(18)∂ϖwϰ,ζ∂ζϖ+ϰ2∂2wϰ,ζ∂ϰ2+0.5ϰ∂wϰ,ζ∂ϰ−wϰ,ζ=0,0<ϖ≤1,under the following initial conditions:(19)wϰ,0=maxϰ3,0=ϰ3forϰ>0,0forϰ≤0.
We will examine the case whenϰ>0. Applying LT on both sides of equation (18), we have(20)ℒ∂ϖwϰ,ζ∂ζϖ+ϰ2∂2wϰ,ζ∂ϰ2+0.5ϰ∂wϰ,ζ∂ϰ−wϰ,ζ=0.
Making use of the process outlined in Section3, we get the findings from equation (20) as follows:(21)Wϰ,s=ϰ3s−1sϖϰ2Dϰϰℒwϰ,ζ−1sϖ0.5ϰDϰℒwϰ,ζ+1sϖℒwϰ,ζ.
We suppose that the expansion ofWϰ,s is as follows:(22)Wϰ,s=∑n=0∞ϕnϰsnϖ+1.
Thekth-truncated expansion is given as follows:(23)Wkϰ,s=∑n=0kϕnϰsnϖ+1.
We obtained the following as a result of using Lemma1(iii):(24)ϕ0ϰ=lims⟶∞sWs=wϰ,0=ϰ3.
As a result, thekth-truncate expansion of equation (21) is as follows:(25)Wkϰ,s=ϰ3s+∑n=1kϕnϰsnϖ+1.
The LRF of equation (21) is as follows:(26)ℒResϰ,s=Wϰ,s−ϰ3s+1sϖϰ2DϰϰWϰ,s+1sϖ0.5ϰDϰWϰ,s−1sϖWϰ,s.
Thekth-LRF of equation (21) is as follows:(27)ℒReskϰ,s=Wkϰ,s−ϰ3s+1sϖϰ2DϰϰWkϰ,s+1sϖ0.5ϰDϰWkϰ,s−1sϖWkϰ,s.
We expand the characteristics of the RPSM to highlight the following details [42, 43]:(i)
ℒResϰ,s=0 and limk⟶∞ℒReskϰ,s(ii)
lims⟶∞sℒResϰ,s=0⇒lims⟶∞sℒReskϰ,s=0(iii)
lims⟶∞skϖ+1ℒResϰ,s=lims⟶∞skϖ+1ℒReskϰ,s=0, where k=1,2,3,…. To determine the first unknown co-efficientϕ1ϰ in (25), we have to use the 1st truncated series W1ϰ,s=ϰ3/s+ϕ1ϰ/sϖ+1 into the 1st-LRF, ℒRes1ϰ,s, to obtain(28)ℒRes1ϰ,s=ϰ3s+ϕ1ϰsϖ+1−ϰ3s+1sϖϰ2Dϰϰϰ3s+ϕ1ϰsϖ+1+1sϖ0.5ϰDϰϰ3s+ϕ1ϰsϖ+11sϖϰ3s+ϕ1ϰsϖ+1.sϖ+1 is used on both sides of equation (28):(29)sϖ+1ℒRes1ϰ,s=ϕ1ϰ+ϰ2Dϰϰϰ3+ϕ1ϰsϖ+0.5ϰDϰϰ3+ϕ1ϰsϖ−ϰ3+ϕ1ϰsϖ.
We use the fact that(30)lims⟶∞skϖ+1ℒReskϰ,s=0,fork=1.
As a result, we obtained as follows:(31)ϕ1ϰ=−6.5ϰ3.
Similarly, to find out values of the second undefined coefficientϕ2ϰ, we have to use the 2nd-truncates series W2ϰ,s=ϰ3/s+ϕ1ϰ/sϖ+1+ϕ2ϰ/s2ϖ+1 into the 2nd-LRF to obtain(32)ℒRes2ϰ,s=ϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1−ϰ3s+1sϖϰ2Dϰϰϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1+1sϖ0.5ϰDϰϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1−1sϖϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1.
Usings2ϖ+1 on both sides of equation (32), we get the following equation:(33)s2ϖ+1ℒRes2ϰ,s=sϖϕ1ϰ+ϕ2ϰ+sϖ+1ϰ2Dϰϰϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1+sϖ+10.5Dϰϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1−sϖ+1ϰ3s+ϕ1ϰsϖ+1+ϕ2ϰs2ϖ+1.
Again, we use the fact that(34)lims⟶∞skϖ+1ℒReskϰ,s=0,fork=2.
As a result, we obtained the2nd coefficient ϕ1ϰ in the following form:(35)ϕ2ϰ=6.52ϰ3.
Therefore, the2nd-approximate LRPS solution of equation (21) is(36)W2ϰ,s=1sϰ3−6.5sϖ+1ϰ3+6.52s2ϖ+1ϰ3.
Typically, to find the coefficientsϕkϰ, first we use the kth-truncated series in equation (25), then we utilize it in the kth-LRF, equation (27), we multiply ℒReskϰ,s by skϖ+1, and then we solve the algebraic equation as follows:(37)lims⟶∞skϖ+1ℒReskϰ,s=0,forϕkϰ.
We get the following results by utilizing the previous procedure:(38)ϕ3ϰ=−6.53ϰ3,ϕ4ϰ=6.54ϰ3,ϕ5ϰ=−6.55ϰ3.
The approximate solution of equation (21) is obtained by five iterations as follows:(39)W5ϰ,s=ϰ3s−6.5ϰ3sϖ+1+6.52ϰ3s2ϖ+1−6.53ϰ3s3ϖ+1+6.54ϰ3s4ϖ+1−6.55ϰ3s5ϖ+1.
By applying the inverse LT to equation (39), we are able to approximate the fifth step solution in the original feature space:(40)w5ϰ,ζ=ϰ3−6.5ϰ3ζϖΓϖ+1+6.52ϰ3ζ2ϖΓ2ϖ+1−6.53ϰ3ζ3ϖΓ3ϖ+1+6.54ϰ3ζ4ϖΓ4ϖ+1−6.55ϰ3ζ5ϖΓ5ϖ+1.
When we useϖ=1 in equation (40), we get the following form:(41)w5ϰ,ζ=ϰ31+−6.5ζ1!+−6.5ζ22!+−6.5ζ33!+−6.5ζ44!+−6.5ζ55!,which are the first six terms of the expansion ϰ3e−6.5ζ and, thus, is the exact solution of equations (18) and (19) at ϖ=1.Problem 2.
consider the following time-fractional BSOPE [35]:(42)∂ϖwϰ,ζ∂ζϖ+0.082+sinϰ2ϰ2∂2wϰ,ζ∂ϰ2+0.06ϰ∂wϰ,ζ∂ϰ=0.06wϰ,ζ,0<ϖ≤1,subject to the following initial conditions:(43)wϰ,0=maxϰ−25e−0.06,0.
First, we perform LT on both sides of equation (42), we use the initial condition from equation (43), and then we format the resulting equation as follows:(44)Wϰ,s=1smaxϰ−25e−0.06,0−1sϖ0.082+sinϰ2ϰ2DϰϰWϰ,s−0.06ϰsϖDϰWϰ,s+0.06sϖWϰ,s.
We describe the expansion solution of the algebraic equation (44). So, we suppose that the series of Wϰ,s is as follows:(45)Wϰ,s=∑n=0∞ϕnϰsnϖ+1.
Thekth-truncated series of the expansion of Wϰ,s is as follows:(46)Wkϰ,s=∑n=0kϕnϰsnϖ+1.
We obtained the following as a result of using Lemma1(iii):(47)ϕ0ϰ=lims⟶∞sWϰ,s=wϰ,0=maxϰ−25e−0.06,0.
So, thekth-truncated expansion becomes as follows:(48)Wkϰ,s=1smaxϰ−25e−0.06,0+∑n=1kϕnϰsnϖ+1.
The LRF of (44) is as follows:(49)ℒResϰ,s=Wϰ,s−1smaxϰ−25e−0.06,0+1sϖ0.082+sinϰ2ϰ2DϰϰWϰ,s+0.06ϰsϖDϰWϰ,s−0.06sϖWϰ,s.
Thekth-LRF of equation (44) is designed as follows:(50)ℒReskϰ,s=Wkϰ,s−1smaxϰ−25e−0.06,0+1sϖ0.082+sinϰ2ϰ2DϰϰWkϰ,s+0.06ϰsϖDϰWkϰ,s−0.06sϖWkϰ,s.
To determine the first unknown coefficientϕ1ϰ in equation (46), we have to use 1st-truncated expansion W1ϰ,s=1/smaxϰ−25e−0.06,0+ϕ1ϰ/sϖ+1 into the 1st-LRF ℒRes1ϰ,s, then we multiply by sϖ+1 on both sides, and then we use the following fact lims⟶∞sϖ+1ℒRes1ϰ,s=0 to obtain(51)ϕ1ϰ=−0.06ϰ−maxϰ−25e−0.06,0.
Similarly, to establish the value of the second undefined coefficientϕ2ϰ, we have to utilize the 2nd truncated expansion W2ϰ,s=1/smaxϰ−25e−0.06,0+ϕ1ϰ/sϖ+1+ϕ2ϰ/s2ϖ+1 into the 2nd-LRF and use the following fact lims⟶∞s2ϖ+1ℒRes2ϰ,s=0, we have(52)ϕ2ϰ=−0.062ϰ−maxϰ−25e−0.06,0.
Therefore, the approximate solution derived from the2nd iteration of equation (44) is as follows:(53)W2ϰ,s=1smaxϰ−25e−0.06,0−0.06sϖ+1ϰ−maxϰ−25e−0.06,0−0.062s2ϖ+1ϰ−maxϰ−25e−0.06,0.
To determine the3rd, 4th, and 5th unknown coefficients, we repeat the same process. We get as follows:(54)ϕ3ϰ=−0.063ϰ−maxϰ−25e−0.06,0,ϕ4ϰ=−0.064ϰ−maxϰ−25e−0.06,0,ϕ5ϰ=−0.065ϰ−maxϰ−25e−0.06,0.
Therefore, the approximate solution derived from the5th iteration of equation (44) is as follows:(55)W5ϰ,s=1smaxϰ−25e−0.06,0−0.06sϖ+1+0.062s2ϖ+1+0.063s3ϖ+1+0.064s4ϖ+1+0.065s5ϖ+1ϰ−maxϰ−25e−0.06,0.
By utilizing the inverse LT on both sides of equation (55), the approximate solution derived from the 5th iteration by LRPSM of equations (42) and (43) is as follows:(56)w5ϰ,ζ=maxϰ−25e−0.06,0−0.06ζϖΓϖ+1+0.06ζϖ2Γ2ϖ+1+0.06ζϖ3Γ3ϖ+1+0.06ζϖ4Γ4ϖ+1+0.06ζϖ5Γ5ϖ+1ϰ−maxϰ−25e−0.06,0.
Whenϖ=1 is used in equation (56), we get as follows:(57)w5ϰ,ζ=maxϰ−25e−0.06,0−0.06ζ+0.06ζ22!+0.06ζ33!+0.06ζ44!+0.06ζ55!ϰ−maxϰ−25e−0.06,0.
As a result, the following is the exact solution of equations (42) and (43) for ϖ=1:(58)wϰ,ζ=maxϰ−25e−0.06,0+1−e0.06ζϰ−maxϰ−25e−0.06,0.Problem 3.
consider the following time-fractional BSOPE [36]:(59)∂ϖwϰ,ζ∂ζϖ=∂2wϰ,ζ∂ϰ2+λ−1∂wϰ,ζ∂ϰ−λwϰ,ζ,0<ϖ≤1,subject to the initial condition:(60)wϰ,0=maxeϰ−1,0.
First, we perform LT on both sides of equation (59), using the initial condition from equation (60), and then format the resulting equation as follows:(61)Wϰ,s=1smaxeϰ−1,0+1sϖDϰϰWϰ,s+λ−1sϖDϰWϰ,s−λsϖWϰ,s.
We describe the expansion solution of the algebraic equation (61). Therefore, we assume that the expansion of Wϰ,s is as follows:(62)Wϰ,s=∑n=0∞ϕnϰsnϖ+1.
Thekth-truncated expansion of equation (61) is as follows:(63)Wkϰ,s=1smaxeϰ−1,0+∑n=1kϕnϰsnϖ+1.
The LRF of equation (61) takes the following form:(64)ℒResϰ,s=Wϰ,s−1smaxeϰ−1,0−1sϖDϰϰWϰ,s−λ−1sϖDϰWϰ,s+λsϖWϰ,s.
Accordingly, thekth-LRF takes the following form:(65)ℒReskϰ,s=Wkϰ,s−1smaxeϰ−1,0−1sϖDϰϰWkϰ,s−λ−1sϖDϰWkϰ,s+λsϖWkϰ,s.
We substitute thekth-truncated series equations (63) into (64), multiply the resulting equation by skϖ+1, and then solve the equation lims⟶∞skϖ+1ℒReskϰ,s=0, where k=1,2,3,4,5, for ϕkϰ gives(66)ϕ0ϰ,s=maxeϰ−1,0,ϕ1ϰ,s=λmaxeϰ,0−maxeϰ−1,0,ϕ2ϰ,s=−λ2maxeϰ,0−maxeϰ−1,0,ϕ3ϰ,s=λ3maxeϰ,0−maxeϰ−1,0,ϕ4ϰ,s=−λ4maxeϰ,0−maxeϰ−1,0,ϕ5ϰ,s=λ5maxeϰ,0−maxeϰ−1,0.
Therefore, the approximate solution derived from the5th iteration of equation (61) is as follows:(67)W5ϰ,s=1smaxeϰ−1,0+λsϖ+1maxeϰ,0−maxeϰ−1,0−λ2s2ϖ+1maxeϰ,0−maxeϰ−1,0+λ3s3ϖ+1maxeϰ,0−maxeϰ−1,0−λ4s4ϖ+1maxeϰ,0−maxeϰ−1,0+λ5s5ϖ+1maxeϰ,0−maxeϰ−1,0.
By utilizing the inverse LT on both sides of equation (67), the approximate solution derived from the 5th iteration by LRPSM of equations (59) and (60) is as follows:(68)w5ϰ,ζ=maxeϰ−1,0+λζϖΓϖ+1maxeϰ,0−maxeϰ−1,0−λ2ζ2ϖΓ2ϖ+1maxeϰ,0−maxeϰ−1,0+λ3ζ3ϖΓ3ϖ+1maxeϰ,0−maxeϰ−1,0−λ4ζ4ϖΓ4ϖ+1maxeϰ,0−maxeϰ−1,0+λ5ζ5ϖΓ5ϖ+1maxeϰ,0−maxeϰ−1,0.
Whenϖ=1 is used in equation (68), we get as follows:(69)w5ϰ,ζ=maxeϰ−1,0+λζ1!−λ2ζ22!+λ3ζ33!−λ4ζ44!+λ5ζ55!maxeϰ,0−maxeϰ−1,0.
As a result, the following is the exact solution of equations (59) and (60) for ϖ=1:(70)wϰ,ζ=maxeϰ−1,0e−λζ+maxeϰ,01−e−λζ.


## 5. Numerical Simulation and Discussion

The findings of the results of the models presented in Problems1–3 are evaluated graphically and numerically in this section. Error functions can be used to evaluate the numerical method’s correctness and competency. It is important to provide the errors of the approximation analytical solution that the LRPSM offers in terms of an infinite fractional power series. We used the absolute and recurrence error functions to demonstrate the accuracy and strength of LRPSM.The 2D graphs of the comparative analysis of the exact and approximative solutions derived by the suggested method in Problems 1–3 are shown in Figures1(a)–1(c). These figures display the 2D plots of the precise solution and the approximate solution attained from the fifth iteration attained by LRPSM for Examples 1–3 when ϖ=0.6,0.7,0.8,0.9, and 1.0 in the range of ζ∈0,0.5. These graphs indicate that when ϖ⟶1.0 is applied, the approximative solution converges to the precise solution. The precise result and the approximation overlap at ϖ=1.0, demonstrating the efficacy and accuracy of the recommended method.Figure 1
The approximate result of five iterations, as well as the exact result ofwϰ,ζ for various values of ϖ in the range ζ∈0,0.5 for (a) Problem 1, when ϰ=0.002, (b) Problem 2, when ϰ=1.0, and (c) Problem 3, when ϰ=1.0 and λ=2.0.
(a)(b)(c)Figures2(a)–2(c) display the 2D curve of calculating the similarity with the help of absolute error of the approximation formed in the fifth step and the precise solution found by the suggested technique for Examples 1–3, respectively, for ϖ=1.0 in the range of ζ∈0,0.5. The article has shown that the fifth-step approximation solution of the recommended approach is very close to the exact solution. By showing the absolute error of the precise and approximate outputs on a graph, LRPSM’s precision is proven.Figure 2
In Figure2, the 2D curve of wϰ,ζ−w5ϰ,ζ in the range ζ∈0,0.5, when ϖ=1.0 for (a) Problem 1, when ϰ=0.002, (b) Problem 2, when ϰ=1.0, and (c) Problem 3, when ϰ=1.0 and λ=2.0.
(a)(b)(c)The comparison article using the 3D curve is shown in Figures3(a)–3(c) in the sense of the absolute error of the approximate finding from the fifth iteration and the exact finding found using the suggested method to Examples 1–3, respectively, at ϖ=1.0 in the ranges of ζ∈0,0.2 and ϖ∈0,0.2. The article has revealed that the fifth-step approximation of the recommended approach is very similar to the precise result. The absolute error of the precise and approximation findings on 3D graphs serves as a demonstration of the precision of LRPSM.Figure 3
In Figure3, the 3D curve of wϰ,ζ−w5ϰ,ζ in the ranges ζ∈0,0.2 and ϖ∈0,0.2, when ϖ=1.0 for (a) Problem 1, (b) Problem 2, and (c) Problem 3, when λ=2.0.
(a)(b)(c)In Tables1–6, the numerical convergence of the approximation to the precise solution has been demonstrated by w4ϰ,ζ−w3ϰ,ζ and w5ϰ,ζ−w4ϰ,ζ in the range ζ∈0,0.1. Tables 1–6 shows that w4ϰ,ζ and w5ϰ,ζ obtained by the suggested method quickly approaches the wϰ,ζ when ϖ⟶1.0. We can see from Tables 1, 3, and 5 that all of the test problems for the fourth stage have very low recurrence errors. The recurrence error will further decrease if we take into account the fifth-step approximation shown in Tables 2, 4, and 6. The approximation is rapidly approaching the exact solution as a result of the accuracy of our suggested strategy being demonstrated by the recurring error process. We arrived at the conclusion that the suggested approach is a feasible and efficient technique for solving particular classes of FODEs with fewer calculations and iteration steps.Table 1
w4ϰ,ζ−w3ϰ,ζ of Problem 1 at different values of ϖ at ϰ=0.002 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.017.64160×10−127.32938×10−136.73357×10−145.95021×10−150.025.32192×10−116.73539×10−128.16495×10−139.52033×10−140.031.65624×10−102.46522×10−113.51465×10−124.81967×10−130.043.70640×10−106.18955×10−119.90060×10−121.52325×10−120.056.92309×10−101.26407×10−102.21074×10−113.71888×10−120.061.15347×10−92.26543×10−104.26177×10−117.71147×10−120.071.77606×10−93.71006×10−107.42333×10−111.42865×10−110.082.58129×10−95.68794×10−101.20052×10−102.43721×10−110.093.58974×10−98.29169×10−101.83450×10−103.90393×10−110.14.82152×10−91.84106×10−103.37478×10−115.95021×10−12Table 2
w5ϰ,ζ−w4ϰ,ζ of Problem 1 at different values of ϖ at ϰ=0.002 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.017.98018×10−133.86764×10−141.77337×10−157.73527×10−170.029.02854×10−126.18822×10−134.01268×10−142.47529×10−150.033.73196×10−113.13278×10−122.48797×10−131.87967×10−140.041.02146×10−109.90115×10−129.07967×10−137.92092×10−140.052.23053×10−102.41727×10−112.47836×10−122.41727×10−130.064.22223×10−105.01246×10−115.62964×10−126.01495×10−130.077.24195×10−109.28619×10−111.12653×10−111.30007×10−120.081.15565×10−91.58418×10−102.05449×10−112.53469×10−120.091.74526×10−92.53756×10−103.49053×10−114.56760×10−120.12.52355×10−93.86764×10−105.60790×10−117.73527×10−12Table 3
w4ϰ,ζ−w3ϰ,ζ of Problem 2 at different values of ϖ at ϰ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.016.93499×10−126.65164×10−136.11093×10−145.40000×10−150.024.82981×10−116.11258×10−127.40995×10−138.64000×10−140.031.50309×10−102.23726×10−113.18965×10−124.37400×10−130.043.36367×10−105.61721×10−118.98510×10−121.38240×10−120.056.28292×10−101.14718×10−102.00632×10−113.37500×10−120.061.04681×10−92.05595×10−103.86769×10−116.99840×10−120.071.61183×10−93.36699×10−106.73690×10−111.29654×10−110.082.34260×10−95.16198×10−101.08951×10−102.21184×10−110.093.25780×10−97.52497×10−102.50264×10−93.54294×10−110.14.37568×10−91.05421×10−92.43280×10−105.40000×10−11Table 4
w5ϰ,ζ−w4ϰ,ζ of Problem 2 at different values of ϖ at ϰ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.016.68516×10−153.24000×10−161.48559×10−176.48000×10−190.027.56340×10−145.18400×10−153.36151×10−162.07360×10−170.033.12634×10−132.62440×10−142.08423×10−151.57464×10−160.048.55701×10−138.29440×10−147.60623×10−156.63552×10−160.051.86856×10−122.02500×10−132.07618×10−142.02500×10−150.063.53705×10−124.19904×10−134.71607×10−145.03885×10−150.076.06674×10−127.77924×10−139.43715×10−141.08909×10−140.089.68115×10−121.32710×10−121.72109×10−132.12337×10−140.091.46205×10−112.12576×10−122.92409×10−133.82638×10−140.12.11403×10−113.24000×10−124.69785×10−136.48000×10−14Table 5
w4ϰ,ζ−w3ϰ,ζ of Problem 3 at different values of ϖ at ϰ=0.002 when λ=2.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.011.71234×10−51.64238×10−61.50887×10−71.33333×10−80.021.19254×10−41.5092810×−51.82962×10−62.13333×10−70.033.71134×10−45.52410×10−57.87569×10−61.08000×10−60.048.30537×10−41.38697×10−42.21854×10−53.41333×10−60.051.55134×10−32.83255×10−44.95386×10−58.33333×10−60.062.58472×10−35.07642×10−49.54985×10−51.72800×10−50.073.97984×10−38.31357×10−41.66343×10−43.20133×10−50.085.78419×10−31.27456×10−32.69015×10−45.46133×10−50.098.04396×10−31.85802×10−34.11079×10−48.74800×10−50.11.08042×10−22.60300×10−36.00692×10−41.33333×10−4Table 6
w5ϰ,ζ−w4ϰ,ζ of Problem 3 at different values of ϖ at ϰ=0.002 when λ=2.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζϖ=0.7ϖ=0.8ϖ=0.9ϖ=1.00.012.75110×10−71.33333×10−86.11355×10−102.66667×10−110.023.11251×10−62.13333×10−71.38334×10−88.53333×10−100.031.28656×10−51.08000×10−68.57707×10−86.48000×10−90.043.52140×10−53.41333×10−63.13014×10−72.73067×10−80.057.68955×10−58.33333×10−68.54394×10−78.33333×10−80.061.45558×10−41.72800×10−51.94077×10−62.07360×10−70.072.49660×10−43.20133×10−53.88360×10−64.48187×10−70.083.98401×10−45.46133×10−57.08269×10−68.73813×10−70.096.01665×10−48.74800×10−51.20333×10−51.57464×10−60.18.69973×10−41.33333×10−41.93327×10−52.66667×10−6For appropriately selected points,|wϰ,ζ−w6ϰ,ζ and |wϰ,ζ−w7ϰ,ζ in the range ζ∈0,0.1 obtained by LRPSM at ϖ=1.0 in Examples 1–3 are displayed in Tables 7–12 for comparison article in the sense of the absolute error of the approximate and the exact finding. We can see from Tables 7, 9, and 11 that approximate solutions derived from the 6th iteration for all of the test problems have very low errors. The absolute error will further decrease if we take into account the 7th-step approximation shown in Tables 8, 10, and 12. By quantitatively comparing the absolute inaccuracy of the precise and approximative findings, LRPSM’s precision is shown.Table 7
wϰ,ζ−w6ϰ,ζ of Problem 1 at ϰ=0.003 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw6ϰ,ζw6ϰ,ζ−wϰ,ζ0.012.53008×10−82.53008×10−82.60496×10−200.022.37086×10−82.37086×10−83.30768×10−180.032.22165×10−82.22165×10−85.60646×10−170.042.08184×10−82.08184×10−84.16685×10−160.051.95082×10−81.95082×10−81.97128×10−150.061.82805×10−81.82805×10−87.00821×10−150.071.71301×10−81.71301×10−82.04572×10−140.081.60521×10−81.60521×10−85.16919×10−140.091.50419×10−81.50420×10−81.16989×10−130.11.40952×10−81.40955×10−82.42728×10−13Table 8
wϰ,ζ−w7ϰ,ζ of Problem 1 at ϰ=0.003 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw7ϰ,ζwϰ,ζ−w7ϰ,ζ0.012.53008×10−82.53008×10−82.11758×10−220.022.37086×10−82.37086×10−85.38461×10−200.032.22165×10−82.22165×10−81.37022×10−180.042.08184×10−82.08184×10−81.35902×10−170.051.95082×10−81.95082×10−88.0436×10−170.061.82805×10−81.82805×10−83.43448×10−160.071.71301×10−81.71301×10−81.17061×10−150.081.60521×10−81.60521×10−83.38332×10−150.091.50419×10−81.50418×10−88.62135×10−150.11.40952×10−81.40952×10−81.98913×10−14Table 9
wϰ,ζ−w6ϰ,ζ of Problem 2 at ϰ=1.0 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw6ϰ,ζwϰ,ζ−w6ϰ,ζ0.01−0.00060018−0.000600182.95987×10−170.02−0.00120072−0.001200724.42354×10−170.03−0.00180162−0.001801627.69784×10−170.04−0.00240288−0.002402887.19910×10−170.05−0.00300450−0.003004505.46438×10−170.06−0.00360649−0.003606491.00614×10−160.07−0.00420883−0.004208835.72459×10−170.08−0.00481154−0.004811543.12250×10−170.09−0.00541461−0.005414617.28584×10−170.1−0.00601804−0.006018045.89806×10−17Table 10
wϰ,ζ−w7ϰ,ζ of Problem 2 at ϰ=1.0 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw7ϰ,ζwϰ,ζ−w7ϰ,ζ0.01−0.00060018−0.000600182.94903×10−170.02−0.00120072−0.001200724.40186×10−170.03−0.00180162−0.001801627.67615×10−170.04−0.00240288−0.002402887.15573×10−170.05−0.00300450−0.003004505.50775×10−170.06−0.00360649−0.003606491.00614×10−160.07−0.00420883−0.004208835.63785×10−170.08−0.00481154−0.004811543.12250×10−170.09−0.00541461−0.005414617.28584×10−170.1−0.00601804−0.006018045.89806×10−17Table 11
wϰ,ζ−w6ϰ,ζ of Problem 3 at ϰ=1.0 and λ=2.0 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw6ϰ,ζwϰ,ζ−w6ϰ,ζ0.011.738081.738082.22045×10−160.021.757491.757493.24185×10−140.031.776521.776525.51115×10−130.041.795171.795174.11982×10−120.051.813441.813441.95961×10−110.061.831361.831367.00426×10−110.071.848921.848922.05550×10−100.081.866141.866145.22144×10−100.091.883011.883011.18793×10−90.11.738081.899552.47757×10−9Table 12
wϰ,ζ−w7ϰ,ζ of Problem 3 at ϰ=1.0 and λ=2.0 when ϖ=1.0 determined by LRPSM at plausible locations in the range ζ∈0,0.1.
ζwϰ,ζw7ϰ,ζwϰ,ζ−w7ϰ,ζ0.011.738081.738080.00.021.757491.757490.00.031.776521.776524.21885×10−150.041.795171.795174.10783×10−140.051.813441.813442.45359×10−130.061.831361.831361.05249×10−120.071.848921.848923.60423×10−120.081.866141.866141.04659×10−110.091.883011.883012.67941×10−110.11.738081.899556.21085×10−11Tables13–15 also compares the absolute error of the approximations from the fifth iteration obtained by the LRPSM of Examples 1–3 at plausible short-listed grid points in the range ζ∈0,0.1 with the absolute error of the fifth-step approximations obtained by various methods, including the PDTM [34], ATDM [35], and HPM [36]. Strong agreement between the results produced using the suggested method and various series solution techniques shows that the LRPSM is a useful substitute for approaches based on He’s or Adomian polynomials.Table 13
wϰ,ζ−w5ϰ,ζ in different approaches for Problem 1 at ϰ=0.002 when ϖ=1.0 at plausible locations in the range ζ∈0,0.1.
ζAbs.errorAbs.errorLRPSMPDTM [34]0.018.30269×10−198.30269×10−190.025.26512×10−175.26512×10−170.035.94281×10−165.94281×10−160.043.30894×10−153.30894×10−150.051.25095×10−141.25095×10−140.063.70206×10−143.70206×10−140.079.25270×10−149.25270×10−140.082.04357×10−132.04357×10−130.094.10678×10−134.10678×10−130.17.66068×10−137.66068×10−13Table 14
wϰ,ζ−w5ϰ,ζ in different approaches for Problem 2 at ϰ=1.0 when ϖ=1.0 at plausible locations in the range ζ∈0,0.1.
ζAbs.errorAbs.errorLRPSMATDM [35]0.012.94903×10−172.94903×10−170.024.40186×10−174.40186×10−170.037.67615×10−177.67615×10−170.047.11237×10−177.11237×10−170.055.63785×10−175.63785×10−170.061.03216×10−161.03216×10−160.076.41848×10−176.41848×10−170.081.47451×10−171.47451×10−170.091.07553×10−161.07553×10−160.11.23165×10−161.23165×10−16Table 15
wϰ,ζ−w5ϰ,ζ in different approaches for Problem 3 at ϰ=1.0 and λ=2.0 when ϖ=1.0 at plausible locations in the range ζ∈0,0.1.
ζAbs.errorAbs.errorLRPSMHPM [36]0.018.88178×10−148.88178×10−140.025.65636×10−125.65636×10−120.036.42488×10−116.42488×10−110.043.59969×10−103.59969×10−100.051.36929×10−91.36929×10−90.064.07716×10−94.07716×10−90.071.02521×10−81.02521×10−80.082.27795×10−82.27795×10−80.094.60513×10−84.60513×10−80.18.64113×10−88.64113×10−8


## 6. Conclusion

In this article, we used the LRPSM to solve time-fractional BSOPE in the sense of CD. The effectiveness of the LRPSM has been demonstrated by results in both graphs and numerically. The approximate solutions achieved using LRPSM are in perfect agreement with the corresponding exact solutions, as can be seen from the graphs and tables. The numerical evidence for the convergence of the approximative solution to the exact solution is presented in Tables1–6. The comparison article is established in Tables 7–12 in terms of the absolute error of the approximate and exact solutions. The results of these numerical approaches are also compared with the PDTM, ATDM, and HPM in terms of absolute errors in Tables 13–15.The results obtained using the suggested approach, which demonstrates excellent agreement with PDTM, ATDM, and HPM, show that LRPSM is a suitable replacement tool for He’s or Adomian polynomial-based methods used to solve FODEs. When determining the coefficients for a series such as the RPSM, the fractional derivative needs to be determined each time. However, LRPSM just needs the idea of an infinite limit. As a result, the computations required to determine the coefficients are minimal. The results led us to the conclusion that our technique is easy to use, accurate, flexible, and effective.



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