# Fractional Conformable Stochastic Integrodifferential Equations: Existence, Uniqueness, and Numerical Simulations Utilizing the Shifted Legendre Spectral Collocation Algorithm

Authors: Haneen Badawi; Nabil Shawagfeh; Omar Abu Arqub
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Theoretical and numerical studies of fractional conformable stochastic integrodifferential equations are introduced in this study. Herein, to emphasize the solution’s existence, we provide proof based on Picard iterations and Arzela−Ascoli’s theorem, whilst the proof of the uniqueness mainly depends on the famous Gronwall’s inequality. Also, we introduce the basic concepts related to shifted Legendre orthogonal polynomials which are utilized to be the basic functions of the spectral collocation algorithm to obtain approximate solutions for the mentioned equations that are not easy to be solved analytically. The substantial idea of the proposed algorithm is to transform such equations into a system containing a finite number of algebraic equations that can be treated using familiar numerical methods. For computational aims, we make a suitable discretization to evaluate the values of the Brownian motion, the noise term considered in our problem, at specific points. In addition, the feasibility and efficiency of the proposed algorithm are proved through convergence analysis and mathematical examples. To exhibit the mathematical simulation, graphs and tables are lucidly shown. Obviously, the physical interpretation of the displayed graphics accurately describes the behavior of the solutions. Despite the simplicity of the presented technique, it produces accurate and reasonable results as notarized in the conclusion section.


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## Body

## 1. Introduction

Recently, greater growth of fractional calculus has been developed by many researchers. Since several real-world phenomena in assorted fields of science are represented successfully via models involving fractional derivatives, mathematicians have focused their attention on doing their best to make deeper studies that improve this branch of calculus and its properties. The mathematicians constructed a variety of definitions of fractional derivative operators with their associated integral inverses together with several important related theories. For example, Hadamard [1] suggested a new approach to fractional derivatives, Khalil et al. [2] constructed the CFD and presented some related theories, Caputo and Fabrizio [3] developed a fractional derivative definition that avoids singularity, and Atangana and Baleanu [4] provided the fractional ABC derivative and discussed its properties. In this regard, authors have employed fractional models for various problems. Ahmed et al. [5] built a fractional-order model to describe cancer with two immune effectors. Rihan [6] was concerned with modeling biological systems in fractional models. Xu [7] constructed a fractional model of the Volterra type of population growth. Debnath [8] sheds light on fractional calculus applications concerning engineering and science with numerical solutions to fractional problems of particular types. The well-posedness and the simulated solutions of the FDEs occupy a great status in applied analysis and engineering applications. For complicated problems that have no exact analytic solutions, various operative numerical techniques have been proposed and employed to provide approximate numerical solutions. Mainly, variational iteration [9], differential transform [10], spectral collocation algorithm [11], finite difference approach [12], and Laplace resolvent kernel scheme [13] are the most popular used methods.More recently, it is noticed that fractional SDEs are more adapted to natural real-world phenomena than deterministic ones. Specifically, the fractional SDEs are FDEs in which one or more of its terms are stochastic processes. These equations have been used to model natural systems in which the past affects the present and future as mechanical systems, population growth, and financial markets. In the last few years, various problems in biology, anthropology, economics, medical models, finance, and engineering have been modeled in terms of fractional SDEs. Dung [14] studied the quasi-linear SDEs driven by fractional SBM and its application to finance. Lima [15] employed a fractional SDE model to utilize COVID-19. Han et al. [16] discussed SDEs of fractional SBM where the stochastic integral is of Stieltjes type. Pei et al. [17] established an averaging principle to deal with SDEs with fractional SBM where the stochastic integral is of Itô’s type. The first issue in solving such equations is to guarantee the presence and assert the exclusivity of their suggested solutions under suitable assumptions. This has been the objective of several studies done by many mathematicians. Guo et al. [18] introduced fractional SDEs in the Caputo sense and provided the existence and uniqueness study of its solution with Caratheodory’s approximation for the considered equation. Zhan et al. [19] studied the existence of mild solution, under suitable conditions, for fractional SDEs where the initial conditions are nonlocal. Moualkia and Xu [20] utilized the Picard iterations’ theory to emphasize the existence and uniqueness of nonlinear fractional k−D of SDEs with Caputo derivative of variable order. Zheng et al. [21] proved the well-posedness of fractional SDEs with stochastic terms of multiplicative white noise. Ahmaadov and Mahmud [22] utilized the Lipschitz condition to establish a sequential proof of the existence of neutral fractional SDEs.Generally, SDEs are difficult to be solved analytically and numerical solutions are employed to afford approximate solutions of good accuracy. In this regard, researchers have constructed effective numerical techniques to obtain highly precise numerical results of different types of these equations; for example, Khodabin et al. [23] used the Taylor scheme to approximate the exact solutions of appointed type of stochastic Volterra integral equations, Kamrani [24] employed the Galerkin−Jacobi method to solve fractional SDEs numerically, Kouhkani et al. [25] developed the operational Tau method to provide numerical solutions for fractional SIDEs, Mohammadi [26] utilized the Chebyshev wavelets to solve a certain class of SIDEs, Cardone et al. [27] constructed the spectral method for solving fractional SDEs, Mirzaee and Hoseini [28] introduced the operational Fibonacci matrices to deal with nonlinear SIDEs, and Asgari et al. [29] proposed the operational matrix algorithm and Bernstein polynomials scheme to handle nonlinear SIDEs.In this study, we are concerned with a specific form of fractional SIDEs of the Volterra type where the derivative is taken to be in the CFD sense. We prove that the existence of a stochastic process satisfies the given equation using the idea of Picard iterations with the use of some topological theorems based on reasonable conditions, and we ensure its uniqueness. Also, a numerical solution based on the SLPs as the basis of the well-known spectral collocation technique is constructed. Indeed, the convergence of the presented algorithm is proved. Anyhow, we study the fractional SIDE of the following form:(1)TϰUT,ω=F1T,UT,ω,ω+λ∫0TF2η,Uη,ω,ωdBη,U0,ω=U0.Herein,T∈Ι=:0,T, ω∈Ω, λ is a real constant, B=BT:T≥0 is the SBM built on a given space Ω,ℱ,P wherein the increasing filtration ℱT0≤T≤T is right continuous. Indeed, Ω is the set of all possible outcomes, and P is a measure defined on ℱ and satisfies PℱT=1. The functions F1 and F2 are given stochastic processes, u is to be determined, and the initial condition u0 is a given random variable presumed to be independent of B. Ultimately, Tϰ denotes the CFD of order ϰ with 0<ϰ≤1.The spectral collocation method is considered a general approximate analytical method used to obtain numerical solutions to various types of differential equations. It is simple, accurate, and does not need a highly qualified programmer. The main principle of this method is to convert extremely difficult differential problems to systems of algebraic equations which are easy to solve. To achieve this goal, the unknown function is assumed to be a linear combination of known base functions with unknown coefficients. Then, the equation is assumed to be satisfied at reasonable collocation points in the given domain. The choice of the base functions and the collocation points affect the efficiency of the method. According to the nice properties of the Legendre polynomials and their orthogonality, they are chosen to be the basis functions of the collocation method in this work. This method was previously used in solving other forms of SDEs as in [30, 31].The remnant sections of our research are outlined next. In Section2, basic definitions and fundamental results of the CFD concept and stochastic processes are given. The Picard iteration idea with the use of some known topological theorems and suitable assumptions is employed to assert the existence of a single solution of (1) in Section 3. The main aim of Section 4 is to present the SL-SCA and employ it for solving (1) numerically followed by a discussion of the convergence study. Section 5 is to provide an algorithm for the scheme presented in Section 4 and uses it to solve some illustrative examples that ensure the accuracy and applicability of the SL-SCA. Finally, in Section 6, the highlight and conclusion are dedicated.


## 2. Preliminaries and Principal Results

In this essential portion, we survey basic definitions, notations, and intrinsic results including CFD, stochastic Itô’s calculus, and some topological spaces which are required for proving our results. Moreover, utilizing the stochastic Fubini theorem, the integral representation of (1) is implemented.


### 2.1. The CFD Requirements

The next mentioned definition is known to be the simplest and the most natural among all fractional derivatives definitions. Although the following definitions and results are considered for0<ϰ≤1, they can be generalized for any ϰ∈ℝ.Definition 1 (see [2]).
The CFD of order0<ϰ<1 for F:0,∞⟶ℝ for all t>0 is(2)TϰFT=limE⟶0FT+ET1−ϰ−FTE.Definition 2 (see [2]).
The conformable integral operator of order0<ϰ≤1 of F:0,∞⟶ℝ is(3)I0ϰFT=∫0TFηηϰ−1dη.Lemma 1 (see [2]).
LetF,G:0,∞⟶ℝ be two functions and t>0. Then, for 0<ϰ≤1 we have the following:(1)
TϰFT=T1−ϰF′T if F is differentiable(2)
I0ϰTϰFT=FT−F0 if TϰFt is continuous(3)
TϰI0ϰFT=FT if F is continuous(4)
TϰaF+bG=aTϰF+bTϰG if F,G are ϰ-differentiable with a,b∈ℝ
Herein,F is called ϰ-differentiable at a fixed point t when TϰFT exists. Indeed, a function could be ϰ-differentiable for some value of ϰ at a point but not differentiable. For example, in FT=2T, T0.5F0=1, but T1F0 does not exist.


### 2.2. Itô’s Calculus

Stochastic calculus is a new subfield of mathematics that is concerned with stochastic processes. The most useful and applicable stochastic process is the SBM. Itô’s integral which requires Itô’s lemma is of big importance and lies in its ability to apply the dominated convergence theorem that is needed for proving the results of equations concerning random terms. Following that, we present the main concepts of stochastic calculus needed in the sequel.Definition 3 (see [30]).
A real-valued stochastic processBT with T∈Ι defines an SBM if the following subsequence properties are satisfied:(1)
For allT∈Ι, the increment BT+ℍ−BT∼N0,ℍ, where Nμ,σ2 denotes the normal distribution with mean μ and variance σ2 wherein ℍ>0(2)
Distinct increments constructed ofBT are independent for 0≤T0≤T1≤⋯≤Tℕ≤T(3)
BT is almost surely a continuous function of t(4)
B0=0 WP1Definition 4 (see [29]).
LetVS,T symbolize the space of all functions Fη,ω:0,∞×Ω⟶ℝ. Then, we can utilize Itô’s integral of F as(4)∫0TFψℕη,ωdBη=limℕ⟶∞∫0Tψℕη,ωdBη,where ψℕℕ=1∞ is the sequence of elementary functions, which satisfies(5)E∫0TF−ψℕ2dη⟶0asℕ⟶∞.Lemma 2 (see [29]).
ForFη,ω∈VS,T, Itô’s Isometry is given by(6)E∫0TFη,ωdBη2=E∫0TF2η,ωdη,T∈I.


### 2.3. Spaces and Notations

LetE denote the expected value of a random variable and GT,ω:Ω,ℱ,P⟶ℝ be a random function; then, the norm of GT,ω is defined as GT,ωE=EGT,ω20.5. Let L2Ω,ℝ refers to the space of each random function GT,ω wherein E∫0TGT,ω2dT<∞. Hereafter, the space of each ℱt-measurable, bounded, and continuous functions realized on I to L2Ω,ℝ is denoted by ∁b=∁I,L2Ω,ℝ.Clearly, the metric space∁b,d with d=maxT∈IUT−WT and U,W∈∁b is separable and complete. Also, the notation M2Ω,∁b,D refers to the space of ∁b-valued random variables equipped with a metric D. Finally, the functions F1T,U,ω,F2T,U,ω:I×∁b×Ω⟶ℝ are assumed to be L2Ω,ℝ jointly measurable and continuous for all T∈I,U∈∁b and almost everywhere ω∈Ω, and U0E<∞. This information can be found in [20].


### 2.4. Integral Representation and Basic Lemmas

This section is to establish an integral form of (1) and state Gronwall’s inequality which is used in completing the desired proofs.Lemma 3.
IfTU is a solution of (1), then its integral form is(7)UT,ω=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.Proof.
Applying (3) for both sides of (1) and using the stochastic Fubini theorem [32], one gained(8)UT,ω=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λ∫0T∫0τF2η,Uη,ω,ωdBητϰ−1dτ=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λ∫0T∫τTηϰ−1dηF2τ,Uτ,ω,ωdBτ=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.Lemma 4 (see [33]).
LetβT and μT be two non-negative continuous functions on Ι and assume that a≥0. If μT≤a+∫0Tβημηdη with t∈Ι, then Gronwall’s inequality is(9)μT≤ae∫0Tβηdη.


## 2.1. The CFD Requirements

The next mentioned definition is known to be the simplest and the most natural among all fractional derivatives definitions. Although the following definitions and results are considered for0<ϰ≤1, they can be generalized for any ϰ∈ℝ.Definition 1 (see [2]).
The CFD of order0<ϰ<1 for F:0,∞⟶ℝ for all t>0 is(2)TϰFT=limE⟶0FT+ET1−ϰ−FTE.Definition 2 (see [2]).
The conformable integral operator of order0<ϰ≤1 of F:0,∞⟶ℝ is(3)I0ϰFT=∫0TFηηϰ−1dη.Lemma 1 (see [2]).
LetF,G:0,∞⟶ℝ be two functions and t>0. Then, for 0<ϰ≤1 we have the following:(1)
TϰFT=T1−ϰF′T if F is differentiable(2)
I0ϰTϰFT=FT−F0 if TϰFt is continuous(3)
TϰI0ϰFT=FT if F is continuous(4)
TϰaF+bG=aTϰF+bTϰG if F,G are ϰ-differentiable with a,b∈ℝ
Herein,F is called ϰ-differentiable at a fixed point t when TϰFT exists. Indeed, a function could be ϰ-differentiable for some value of ϰ at a point but not differentiable. For example, in FT=2T, T0.5F0=1, but T1F0 does not exist.


## 2.2. Itô’s Calculus

Stochastic calculus is a new subfield of mathematics that is concerned with stochastic processes. The most useful and applicable stochastic process is the SBM. Itô’s integral which requires Itô’s lemma is of big importance and lies in its ability to apply the dominated convergence theorem that is needed for proving the results of equations concerning random terms. Following that, we present the main concepts of stochastic calculus needed in the sequel.Definition 3 (see [30]).
A real-valued stochastic processBT with T∈Ι defines an SBM if the following subsequence properties are satisfied:(1)
For allT∈Ι, the increment BT+ℍ−BT∼N0,ℍ, where Nμ,σ2 denotes the normal distribution with mean μ and variance σ2 wherein ℍ>0(2)
Distinct increments constructed ofBT are independent for 0≤T0≤T1≤⋯≤Tℕ≤T(3)
BT is almost surely a continuous function of t(4)
B0=0 WP1Definition 4 (see [29]).
LetVS,T symbolize the space of all functions Fη,ω:0,∞×Ω⟶ℝ. Then, we can utilize Itô’s integral of F as(4)∫0TFψℕη,ωdBη=limℕ⟶∞∫0Tψℕη,ωdBη,where ψℕℕ=1∞ is the sequence of elementary functions, which satisfies(5)E∫0TF−ψℕ2dη⟶0asℕ⟶∞.Lemma 2 (see [29]).
ForFη,ω∈VS,T, Itô’s Isometry is given by(6)E∫0TFη,ωdBη2=E∫0TF2η,ωdη,T∈I.


## 2.3. Spaces and Notations

LetE denote the expected value of a random variable and GT,ω:Ω,ℱ,P⟶ℝ be a random function; then, the norm of GT,ω is defined as GT,ωE=EGT,ω20.5. Let L2Ω,ℝ refers to the space of each random function GT,ω wherein E∫0TGT,ω2dT<∞. Hereafter, the space of each ℱt-measurable, bounded, and continuous functions realized on I to L2Ω,ℝ is denoted by ∁b=∁I,L2Ω,ℝ.Clearly, the metric space∁b,d with d=maxT∈IUT−WT and U,W∈∁b is separable and complete. Also, the notation M2Ω,∁b,D refers to the space of ∁b-valued random variables equipped with a metric D. Finally, the functions F1T,U,ω,F2T,U,ω:I×∁b×Ω⟶ℝ are assumed to be L2Ω,ℝ jointly measurable and continuous for all T∈I,U∈∁b and almost everywhere ω∈Ω, and U0E<∞. This information can be found in [20].


## 2.4. Integral Representation and Basic Lemmas

This section is to establish an integral form of (1) and state Gronwall’s inequality which is used in completing the desired proofs.Lemma 3.
IfTU is a solution of (1), then its integral form is(7)UT,ω=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.Proof.
Applying (3) for both sides of (1) and using the stochastic Fubini theorem [32], one gained(8)UT,ω=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λ∫0T∫0τF2η,Uη,ω,ωdBητϰ−1dτ=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λ∫0T∫τTηϰ−1dηF2τ,Uτ,ω,ωdBτ=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.Lemma 4 (see [33]).
LetβT and μT be two non-negative continuous functions on Ι and assume that a≥0. If μT≤a+∫0Tβημηdη with t∈Ι, then Gronwall’s inequality is(9)μT≤ae∫0Tβηdη.


## 3. Existence and Uniqueness Results

The main target of this section is to establish reasonable proof of under-growth and Lipschitz assumptions asserting the existence of a single solution of (1). Our proofs are performed using the Picard iteration scheme which goes back to Emile Picard 1856–1941. Some topological theorems and lemmas are also used. Herein, ϰ is presumed to be >0.5.Theorem 1.
Suppose that the growth conditionEFiη,Uη,ω,ω2≤Ni1+EUη,ω2 with Ni>0 and i=1,2 holds; then, (1) has a solution in ∁b.Proof.
Using the integral form (7) and defining the Picard sequence Unn=0∞ on I with U0T,ω=U0 as(10)UℕT,ω=U0+∫0Tηϰ−1F1η,Uℕ−1η,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uℕ−1η,ω,ωdBη,ℕ=1,2,…,and let maxi=1,2Ni≤N∗ for some N∗>0.Step 1:
we show thatUℕℕ=0∞ is well-defined and measurable for T∈Ι. First, we denote(11)ℐ1=∫0Tηϰ−1F1η,Uℕ−1η,ω,ωdη,ℐ2=λϰ∫0TTϰ−ηϰF2η,Uℕ−1η,ω,ωdBη.
SinceF1 and F2 are assumed to be in L2Ω,ℝ, and the kernels ηϰ−1 and Tϰ−ηϰ are bounded. Then, the growth condition asserts that(12)∫0TEηϰ−1F1η,Uℕ−1η,ω,ω2dη<∞,∫oTETϰ−ηϰF2η,Uℕ−1η,ω,ω2dη<∞,which indicates that Lebesgue’s integral ℐ1 and Itô’s integral ℐ2 are well-defined. Consequently, the sequence Uℕℕ=0∞ is well-defined.
It remains to prove the measurability of the considered sequence. This can be explained by the measurability ofF1 and F2, and the product of measurable continuous functions is again measurable. So, the maps(13)ω⟶ℐ1,ω⟶ℐ2,are measurable. Thus, we deduce the requirement.Step 2:
we show thatUℕℕ=0∞ is bounded. With the use of CSI, Itô’s isometry, and the growth constraint, (10) gives(14)EUℕT2=EU0+∫0Tηϰ−1F1η,Uℕ−1η,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uℕ−1η,ω,ωdBη2≤3EU02+3E∫0Tηϰ−1F1η,Un−1η,ω,ωdη2+3Eλϰ∫0TTϰ−ηϰF2η,Uℕ−1η,ω,ωdBη2≤3EU02+3T∫0Tη2ϰ−2EF1η,Uℕ−1η,ω,ω2dη+3λ2ϰ2∫0TTϰ−ηϰ2EF2η,Uℕ−1η,ω,ω2dη≤3EU02+3TN∗∫0Tη2ϰ−2EUℕ−1η,ω2+1dη+3N∗λ2ϰ2∫0TTϰ−ηϰ2EUℕ−1η,ω2+1dη≤3EU02+3N∗TT2ϰ−12ϰ−1+3N∗T∫0Tη2ϰ−2EUℕ−1η,ω2dη+3N∗λ2ϰ2T2ϰ+12ϰ+1+3N∗λ2ϰ2∫0TTϰ−ηϰ2EUℕ−1η,ω2dη=3Eu02+3N∗TT2ϰ−12ϰ−1+3N∗λ2ϰ2T2ϰ+12ϰ+1+3N∗∫0tTη2ϰ−2+λtϰ−ηϰϰ2Eun−1η,ω2dη=ℍ1+ℍ2∫0TTη2ϰ−2+λTϰ−ηϰϰ2EUℕ−1η,ω2dη,where ℍ1=3EU02+3N∗TT2ϰ−1/2ϰ−1+3N∗λ2/ϰ2T2ϰ+1/2ϰ+1 and ℍ2=3N∗. Also, for arbitrary p≥1, obviously,(15)max1≤ℕ≤ℙEUℕ−1T,ω2≤max1≤ℕ≤ℙEUℕT,ω2.
Therefore, taking the maximum of both sides of (14) one has(16)max1≤ℕ≤ℙEUℕT,ω2≤ℍ1+ℍ2∫0TTη2ϰ−2+λTϰ−ηϰϰ2max1≤ℕ≤ℙEUℕT,ω2dη.
By Gronwall’s inequality, we have(17)max1≤ℕ≤ℙEUℕT,ω2≤ℍ1eℍ2∫0TTη2ϰ−2+λTϰ−ηϰ/ϰ2dη≤ℍ1eℍ2T2ϰ1/2ϰ−1+Tλ2/2ϰ+1ϰ2<∞.
Since℘ was arbitrary, we have EUℕT,ω2<∞ which guarantees the boundedness of Uℕℕ=0∞. Moreover, Uℕℕ=0∞ is uniformly bounded.Step 3:
we prove the equicontinuity ofUℕℕ=0∞. Let 0<T1<T2<T, T2−T1<δ∗, and max0<T<T1+EUℕ−1T,ω2≤ϑ,ϑ>1. Let ϵ>0 be given and choose δ>0 such that 0<δ∗2ϰ1/2ϰ−1+2λ2δ∗/ϰ22ϰ+1+2Tλ2/ϰ2≤δ and(18)δ≤ϵ2ϑN∗.
Employing CSI, Itô’s isometry, and (10), one obtains(19)UℕT2,ω−UℕT1,ω2≤2∫0T2ηϰ−1F1η,Uℕ−1η,ωωdη−∫0T1ηϰ−1F1η,Uℕ−1η,ω,ωdη2+2λ2ϰ2∫0T2T2ϰ−ηϰF2η,Uℕ−1η,ω,ωdBη−∫0T1T1ϰ−ηϰF2η,Uℕ−1η,ω,ωdBη2≤2∫T1T2ηϰ−1F1η,Uℕ−1η,ω,ωdη2+4λ2ϰ2∫T1T2T2ϰ−ηϰF2η,Uℕ−1η,ω,ωdBη2+4λ2ϰ2∫0T1T2ϰ−T1ϰF2η,Uℕ−1η,ω,ωdBη2.
Taking the expectation of both sides, one has(20)Eunt2,ω−unt1,ω2EUℕT2,ω−UℕT1,ω2≤2∫T1T2η2ϰ−2dη∫T1T2EF1η,Un−1η,ω,ω2dη+4λ2ϰ2∫T1T2T2ϰ−ηϰ2EF2η,Uℕ−1η,ω,ω2dη+4λ2ϰ2T2−T12ϰ∫0T1EF2η,Uℕ−1η,ω,ω2dη≤22ϰ−1T22ϰ−1−T12ϰ−1∫T1T2EF1η,Uℕ−1η,ω,ω2dη+4λ2ϰ2∫T1T2T2ϰ−ηϰ2EF2η,Uℕ−1η,ω,ω2dη+4λ2ϰ2T2−T12ϰ∫0T1EF2η,Uℕ−1η,ω,ω2dη≤22ϰ−1T2−T12ϰ−1N∗∫T1T21+EUℕ−1η,ω2dη+4N∗λ2ϰ2∫T1T2T2−η2ϰ1+EUℕ−1η,ω2dη+4N∗λ2ϰ2T2−T12ϰ∫0T11+EUℕ−1η,ω2dη≤2N∗ϑ2ϰ−1T2−T12ϰ+4λ2N∗ϑϰ22ϰ+1T2−T12ϰ+1+4Tλ2N∗ϑϰ2T2−T12ϰ≤2N∗ϑδ∗2ϰ12ϰ−1+2λ2δ∗ϰ22ϰ+1+2Tλ2ϰ2≤2N∗ϑδ.
By (18), one gets EUℕT2,ω−UℕT1,ω2≤ϵ which proves the equicontinuity of Uℕℕ=0∞.Step 4:
by the results gained in Steps 2 and 3, Arzela-Ascoli’s theorem assures thatUℕℕ=0∞ is a compact subset of ∁b. In addition, Step 2 asserts the boundedness of the sequence Uℕℕ=0∞ which indicates that Uℕ∈M2Ω,∁b. Applying Prohorov’s theorem, one gets the totally D-boundedness of the sequence Uℕℕ=0∞ in M2Ω,∁b. Thus, Uℕℕ=0∞ has a D-Cauchy subsequence UℕMℕ=0∞ denoted by UMM=1∞ [34]. Then, Skorokhod’s representation lemma [35] assures the existence of a convergence sequence WMM=1∞ in M2Ω,∁b that has the same distribution as UMM=1∞ such that the following subsequence properties are satisfied:(1)
DWM,UM=0,M=1,2,…(2)
WM⟶U WP1 as M⟶∞ for some continuous ℱ∂-measurable random variable U in M2Ω,∁b
The abovementioned properties ensure the boundedness ofWM and U WP1. Now, it remains to prove that U, the limit of WMM=1∞, satisfies (1). For this end, we consider the following:(21)EWMT,ω−UT,ω2=E∫0Tηϰ−1F1η,WMη,ω,ω−F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,WMη,ω,ω−F2η,Uη,ω,ωdBη2≤2E∫0Tηϰ−1F1η,WMη,ω,ω−F1η,Uη,ω,ωdη2+2λ2ϰ2E∫0TTϰ−ηϰF2η,WMη,ω,ω−F2η,Uη,ω,ωdBη2≤2∫0Tη2ϰ−2dη∫0TEF1η,WMη,ω,ω−F1η,Uη,ω,ω2dη+2λ2ϰ2∫0TTϰ−ηϰ2EF2η,WMη,ω,ω−F2η,Uη,ω,ω2dη≤2T2ϰ−12ϰ−1∫0TEF1η,WMη,ω,ω−F1η,Uη,ω,ω2dη+2λ2ϰ2∫0TTϰ−ηϰ2EF2η,WMη,ω,ω−F2η,Uη,ω,ω2dη.
By the continuity ofF1 and F2, it appears that for given ε>0, ∃i≥0 with i integer such that, for M>i,(22)∫0TEF1η,WMη,ω,ω−F1η,Uη,ω,ω2dη≤Tϵ2,∫0TTϰ−ηϰ2EF2η,WMη,ω,ω−F2η,Uη,ω,ω2dη≤T2ϰ+12ϰ+1ϵ2.
Hence, for allT∈Ι, one has(23)∫0Tηϰ−1F1η,WMη,ω,ωdη=⟶∫0Tηϰ−1F1η,Uη,ω,ωdη,(24)∫0TTϰ−ηϰF2η,WMη,ω,ωdBη⟶∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.
With the propertyDWM,UM=0 within M=1,2,… mentioned in Step 4, (10) yields(25)WMT,ω=U0+∫0Tηϰ−1F1η,WM−1η,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,WM−1η,ω,ωdBη.
LettingM⟶∞, relations (23)–(25), and the property WM⟶U WP1 shows that(26)UT,ω=U0+∫0Tηϰ−1F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ωdBη.
Consequently, (26) demonstrates that UT,ω is a random solution of (1) as required.Theorem 2.
Under the Lipschitz assumptionEF2η,Wη,ω,ω−F2η,Uη,ω,ω2≤LEWη,ω−Uη,ω2 with L>0 the solution of (1) is unique.Proof.
LetWT,ω and UT,ω be two solutions to (1). Then, using the integral form (7), one has(27)EWT,ω−UT,ω2=E∫0Tηϰ−1F1η,Wη,ω,ω−F1η,Uη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Wη,ω,ω−F2η,Uη,ω,ωdBη2≤2E∫0Tηϰ−1F1η,Wη,ω,ω−F1η,Uη,ω,ωdη2+2λ2ϰ2E∫0TTϰ−ηϰF2η,Wη,ω,ω−F2η,Uη,ω,ωdBη2≤2∫0Tη2ϰ−2dη∫0TEF1η,Wη,ω,ω−F1η,Uη,ω,ω2dη+2λ2ϰ2∫0TTϰ−ηϰ2EF2η,Wη,ω,ω−F2η,Uη,ω,ω2dη≤2T2ϰ−12ϰ−1∫0TEF1η,Wη,ω,ω−F1η,Uη,ω,ω2dη+2λ2ϰ2∫0TTϰ−ηϰ2EF2η,Wη,ω,ω−F2η,Uη,ω,ω2dη.
Again, by the Lipschitz condition, one has(28)EWT,ω−UT,ω2≤2T2ϰ−12ϰ−1L∫0TEWη,ω−Uη,ω2dη+2λ2ϰ2L∫0TTϰ−ηϰ2EWη,ω−Uη,ω2dη=2L∫0TT2ϰ−12ϰ−1+λTϰ−ηϰϰ2EWη,ω−Uη,ω2dη.
Applying Gronwall’s inequality on (28), one yields(29)EWT,ω−UT,ω2≤0.
Hence,WT,ω=UT,ω for all T∈Ι. So, the uniqueness has been proved.


## 4. Numerical Solution and Convergence Analysis

Unlike the deterministic FDEs, which have deterministic solutions, fractional SIDEs solutions are continuously stochastic processes. Aside from differences, solving fractional SIDEs numerically is based on similar techniques that are used for deterministic ones. Herein, the SL-SCA is discussed in detail and employed to grant the numerical solutions of (1). It is indispensable to focus on the fact that each trajectory provided by the presented method is just one realization of the exact stochastic process satisfying (1). To confirm the convergence of the presented algorithm when solving (1), we construct a theorem and apply Gronwall’s inequality in proving it.


### 4.1. Adaptive SL-SCA for Fractional SIDEs

The spectral collocation technique is considered one of the most applicable numerical methods that provide high-precision numerical solutions for various types of FDEs. The main idea of its algorithm is to derive a system of a finite number of algebraic equations by assuming the unknown function to be a truncated series of some suitable basis functions with unknown coefficients. Because of the simplicity, smoothness, and orthogonality properties of the Legendre polynomials, they are selected to be the base functions of the algorithm discussed in the current section.To fulfill the goal of approximating the unknown functionUT,ω in (1), we first present a review of the SLPs and state a theorem that asserts the ability to approximate unknown functions by SLPs expansions. In this sense, the unknown function UT,ω in (1) is replaced by a finite series of SLPs resulting in a residual error function ResT which is assumed to be zero at a suitable choice of collocation points. In addition, the finite series satisfies the initial conditions. Accordingly, (1) is converted to a facilely solvable system of algebraic equations. For simplicity of notation, the variable ω is dropped and we write UT, F1T,UT, and F2T,UT instead of UT,ω, F1T,UT,ω,ω, and F2T,UT,ω,ω, respectively.Legendre polynomials are known to be built on−1,1. Utilizing appropriate change of variables, Legendre polynomials can be restricted to 0,1 and then called the SLPs. They satisfy the orthogonality property concerning the weight function wT=1. SLPs can be expressed in a recursive formula [36] but the most efficient formula is the following analytical series [37]:(30)KℙT=∑ℚ=0ℙ−1ℙ+ℚ1ℙ−ℚ!ℚ!2ℙ+ℚ!Tℚ,ℙ=0,1,….The orthogonality property of SLPs over0,1 is(31)∫01KiTKℙTdT=2i+1−1,i=ℙ,0,i≠ℙ.Definition 5 (see [38]).
For allU,W∈L20,1 the functional of L20,1 is(32)U|W=∫01UTWTdT,UL2=U|U.Theorem 3 (see [36]).
LetPN=Pi,i≤N, where Pi is a polynomial of degree i. Then, for W∈L20,1, there is a unique polynomial qN∗∈PN such that(33)W−qNL2∗=infqN∈PNW−qNL2,where qN∗T=∑i=0NγiφiT with γi=W|φi/φiL2 and the sequence φii=1N is an orthogonal basis for PN in the L2 space.
Since they satisfy the orthogonality property, the SLPs form an orthogonal basis forPN in L2 space. Thus, the previous theorem asserts that UT∈L20,1 can be uniquely approximated by a truncated series of the SLPs as follows:(34)UT≅UNT=∑ℙ=0N−1λℙKℙT,where λℙ=2ℙ+1UT|KℙT with ℙ=0,1,…,N−1.
Using the linearity property of the CFD, one can easily verify that the approximation of the derivative ofUNT in the conformable sense is(35)TϰUNT=∑ℙ=0N−1λℙTϰKℙT,(36)TϰKℙT=∑ℚ=1ℙ−1ℙ+ℚℚℙ+ℚ!ℙ−ℚ!ℚ!2Tℚ−ϰ,ℙ=1,2,…,0,ℙ=0.
Also, using the property∫01ℍηdBS=ℍTBT−∫0TBηdℍη Itô’s integral in (1) can be written as(37)∫0TF2η,UηdBS=F2T,UTBT−∫0T∂F2η,Uη∂ηBηdη.
Inserting relations (34), (35), and (37) into (1), it leads to(38)ResNT=∑ℙ=0N−1λTABCD0αKℙT−FT,∑ℙ=0N−1λℙKℙT−GT,∑ℙ=0N−1λℙKℙTBT+∫0T∂F2η,∑ℙ=0N−1λℙKℙη∂ηBηdη,where ResNT denotes the residual error function that is produced when substituting UNT in (1).
Now, fori=1,2,…,N−1, we substitute each collocation point Ti, arranged as Ti<Ti+1, into (38) and get(39)ResNTi=∑ℙ=0N−1λℙD0αKℙTi−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+∫0Ti∂F2η,∑ℙ=0N−1λℙKℙη∂ηBηdη.
To approximate the integral term in (39), one can use the composite trapezoidal rule as follows:(40)∫0TiℍηBηdη≅Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTiMBℙTℙiM,where M≥2 is an integer and ℍη=∂F2η,UNη/∂η.
Anyhow, the substituting of (40) into (39), leads to(41)ResNTi+εMTi=∑ℙ=0N−1λℙTϰKℙT−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTℙiMBℙTℙiM,where εMTi is the trapezoidal rule error related to the integral approximating using (39). It is known that εMTi⟶0 as M⟶∞.
Now, fori=1,2,…,N−1 one has(42)0=∑ℙ=0N−1λℙTϰKℙTi−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTℙiMBℙTℙiM.
The propertyKℙ0=−1ℙ together with the initial condition gives(43)∑ℙ=0N−1λℙ−1ℙ=U0.
Hence, (42) and (43) form a system containing N algebraic equations. This system has solved the unknowns λℙ wherein ℙ=0,1,…,N−1 with the aid of a suitable method. Consequently, the approximate solution of (1), when substituting the values of λℙ in (34), is obtained.
It remains to treat a crucial issue of simulating the SBM sample paths over an appropriate discretization0<T1<…<Tℙ<…<Tτ of I, for a positive integer τ. The discretization is considered to be equally spaced with Tℙ−Tℙ−1=T/τ=ΔT wherein ℙ=1,…,τ and then BTℙ−BTℙ−1=dBTℙ∼Ν0,ΔT. To obtain an approximate value for BT, a linear spline interpolation [30] is then applied using the points Tℙ,BTℙ wherein ℙ=0,1,…,τ.


### 4.2. Convergence of SL-SCA

This part is to emphasize the convergence of the proposed algorithm. For this aim, it is initially proven that the finite Legendre series expansion of a function inL20,1 converges and has finite modulus coefficients. Then, proof of convergence of SL-SCA considering (1) is constructed.Theorem 4.
The Legendre expansion∑ℙ=0NλℙKℙT of UT∈L20,1 converges to UT as N⟶∞.Proof.
LetUMT=∑ℙ=0MλℙKℙT. Then, for N>M, we have(44)UNx−UMxL22=∑ℙ=M+1NλℙKℙTL22=∑ℙ=M+1NλℙKℙT|∑i=M+1NλiKiT=∑ℙ=M+1N∑i=M+1Nλℙλi¯KℙT|KiT=∑ℙ=M+1N∑ℙ=M+1Nλℙλi¯∫01KℙTKiTdT=∑ℙ=M+1Nλℙ212ℙ+1<∑ℙ=M+1Nλℙ2.
Bessel’s inequality indicates that∑ℙ=M+1Nλℙ2≤∑ℙ=M+1∞λℙ2≤UL22<∞. So, UNT−UMTL22⟶0 as N,M⟶∞. Thus, the sequence UNT is Cauchy in the complete space L20,1 and converges to some WT∈L20,1. Now, to show WT=UT, we consider(45)WT−UT|KℙT=WT|KℙT−WT|KℙT=limN⟶∞UNT,KℙT−λℙ=λℙ−λℙ,=0.
Hence,WT=WT which completes the proof.Theorem 5.
WeconsiderUT,ω and UNT,ω concerning (1) and suppose that the Lipschitz condition EF2η,Uη,ω,ω−F2η,UNη,ω,ω2≤LEUη,ω−UNη,ω2,L>0, is satisfied. Then,(46)UT,ω−UNT,ω2⟶0asN⟶∞.Proof.
LeteNT,ω=UT,ω−UNT,ω be the error function results when the exact solution UT,ω of (1) is approximated by UNT that is given in (34). So, using (7), we have(47)eNT,ω=UT,ω−UNT,ω=∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη+λ2ϰ2∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη+I0ϰεM+I0ϰResNT,
The triangular inequality, CSI, and Itô’s lemma give(48)EUT,ω−UNT,ω2=E∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη+I0ϰεM+I0ϰResNT2≤4E∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη2+4λ2ϰ2E∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη2+4EI0ϰεM2+4EI0ϰResNT2≤4∫0Tη2ϰ−2dη∫0TEF1η,Uη,ω,ω−F1η,UNη,ω,ω2dη+4λ2ϰ2∫0TTϰ−ηϰ2EF2η,Uη,ω,ω−F2η,UNη,ω,ω2dη+4EI0ϰεM2+4EI0ϰResNT2≤4T2ϰ−12ϰ−1∫0TEF1η,Uη,ω,ω−F1η,UNη,ω,ω2dη+4λ2ϰ2∫0TTϰ−ηϰ2EF2η,Uη,ω,ω−F2η,UNη,ω,ω2dη+4EI0ϰεM2+4EI0ϰResNT2.
Now, using the Lipschitz condition and simple calculations, we otain(49)eNTE2≤4T2ϰ−12ϰ−1L∫0TEUη,ω−UNη,ω2dη+4λ2ϰ2L∫0TTϰ−ηϰ2EUη,ω−UNη,ω2dη+4I0ϰεE2+4I0ϰResNTE2=2L∫0TT2ϰ−12ϰ−1+λTϰ−ηϰϰ2eNTE2dη+4I0ϰεE2+4I0ϰResNTE2.
LetϱN,M=4I0ϰεME2+4I0ϰResNTE2 and apply Gronwall’s inequality on (49), which yields(50)eNTE2≤ϱN,Me∫0T2LT2ϰ−1/2ϰ−1+λTϰ−ηϰ/ϰ2dη.
SinceϱN,M⟶0 as N,M⟶∞ and for T∈Ι, ∫0T2LT2ϰ−1/2ϰ−1+λTϰ−ηϰ/ϰ2dη<∞. Then, UT−UNT2⟶0 as N,m⟶∞ as required.


## 4.1. Adaptive SL-SCA for Fractional SIDEs

The spectral collocation technique is considered one of the most applicable numerical methods that provide high-precision numerical solutions for various types of FDEs. The main idea of its algorithm is to derive a system of a finite number of algebraic equations by assuming the unknown function to be a truncated series of some suitable basis functions with unknown coefficients. Because of the simplicity, smoothness, and orthogonality properties of the Legendre polynomials, they are selected to be the base functions of the algorithm discussed in the current section.To fulfill the goal of approximating the unknown functionUT,ω in (1), we first present a review of the SLPs and state a theorem that asserts the ability to approximate unknown functions by SLPs expansions. In this sense, the unknown function UT,ω in (1) is replaced by a finite series of SLPs resulting in a residual error function ResT which is assumed to be zero at a suitable choice of collocation points. In addition, the finite series satisfies the initial conditions. Accordingly, (1) is converted to a facilely solvable system of algebraic equations. For simplicity of notation, the variable ω is dropped and we write UT, F1T,UT, and F2T,UT instead of UT,ω, F1T,UT,ω,ω, and F2T,UT,ω,ω, respectively.Legendre polynomials are known to be built on−1,1. Utilizing appropriate change of variables, Legendre polynomials can be restricted to 0,1 and then called the SLPs. They satisfy the orthogonality property concerning the weight function wT=1. SLPs can be expressed in a recursive formula [36] but the most efficient formula is the following analytical series [37]:(30)KℙT=∑ℚ=0ℙ−1ℙ+ℚ1ℙ−ℚ!ℚ!2ℙ+ℚ!Tℚ,ℙ=0,1,….The orthogonality property of SLPs over0,1 is(31)∫01KiTKℙTdT=2i+1−1,i=ℙ,0,i≠ℙ.Definition 5 (see [38]).
For allU,W∈L20,1 the functional of L20,1 is(32)U|W=∫01UTWTdT,UL2=U|U.Theorem 3 (see [36]).
LetPN=Pi,i≤N, where Pi is a polynomial of degree i. Then, for W∈L20,1, there is a unique polynomial qN∗∈PN such that(33)W−qNL2∗=infqN∈PNW−qNL2,where qN∗T=∑i=0NγiφiT with γi=W|φi/φiL2 and the sequence φii=1N is an orthogonal basis for PN in the L2 space.
Since they satisfy the orthogonality property, the SLPs form an orthogonal basis forPN in L2 space. Thus, the previous theorem asserts that UT∈L20,1 can be uniquely approximated by a truncated series of the SLPs as follows:(34)UT≅UNT=∑ℙ=0N−1λℙKℙT,where λℙ=2ℙ+1UT|KℙT with ℙ=0,1,…,N−1.
Using the linearity property of the CFD, one can easily verify that the approximation of the derivative ofUNT in the conformable sense is(35)TϰUNT=∑ℙ=0N−1λℙTϰKℙT,(36)TϰKℙT=∑ℚ=1ℙ−1ℙ+ℚℚℙ+ℚ!ℙ−ℚ!ℚ!2Tℚ−ϰ,ℙ=1,2,…,0,ℙ=0.
Also, using the property∫01ℍηdBS=ℍTBT−∫0TBηdℍη Itô’s integral in (1) can be written as(37)∫0TF2η,UηdBS=F2T,UTBT−∫0T∂F2η,Uη∂ηBηdη.
Inserting relations (34), (35), and (37) into (1), it leads to(38)ResNT=∑ℙ=0N−1λTABCD0αKℙT−FT,∑ℙ=0N−1λℙKℙT−GT,∑ℙ=0N−1λℙKℙTBT+∫0T∂F2η,∑ℙ=0N−1λℙKℙη∂ηBηdη,where ResNT denotes the residual error function that is produced when substituting UNT in (1).
Now, fori=1,2,…,N−1, we substitute each collocation point Ti, arranged as Ti<Ti+1, into (38) and get(39)ResNTi=∑ℙ=0N−1λℙD0αKℙTi−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+∫0Ti∂F2η,∑ℙ=0N−1λℙKℙη∂ηBηdη.
To approximate the integral term in (39), one can use the composite trapezoidal rule as follows:(40)∫0TiℍηBηdη≅Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTiMBℙTℙiM,where M≥2 is an integer and ℍη=∂F2η,UNη/∂η.
Anyhow, the substituting of (40) into (39), leads to(41)ResNTi+εMTi=∑ℙ=0N−1λℙTϰKℙT−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTℙiMBℙTℙiM,where εMTi is the trapezoidal rule error related to the integral approximating using (39). It is known that εMTi⟶0 as M⟶∞.
Now, fori=1,2,…,N−1 one has(42)0=∑ℙ=0N−1λℙTϰKℙTi−F1Ti,∑ℙ=0N−1λℙKℙTi−F2Ti,∑ℙ=0N−1λℙKℙTiBTi+Ti2iMℍTiBTi+2∑ℙ=1iM−1ℍℙTℙiMBℙTℙiM.
The propertyKℙ0=−1ℙ together with the initial condition gives(43)∑ℙ=0N−1λℙ−1ℙ=U0.
Hence, (42) and (43) form a system containing N algebraic equations. This system has solved the unknowns λℙ wherein ℙ=0,1,…,N−1 with the aid of a suitable method. Consequently, the approximate solution of (1), when substituting the values of λℙ in (34), is obtained.
It remains to treat a crucial issue of simulating the SBM sample paths over an appropriate discretization0<T1<…<Tℙ<…<Tτ of I, for a positive integer τ. The discretization is considered to be equally spaced with Tℙ−Tℙ−1=T/τ=ΔT wherein ℙ=1,…,τ and then BTℙ−BTℙ−1=dBTℙ∼Ν0,ΔT. To obtain an approximate value for BT, a linear spline interpolation [30] is then applied using the points Tℙ,BTℙ wherein ℙ=0,1,…,τ.


## 4.2. Convergence of SL-SCA

This part is to emphasize the convergence of the proposed algorithm. For this aim, it is initially proven that the finite Legendre series expansion of a function inL20,1 converges and has finite modulus coefficients. Then, proof of convergence of SL-SCA considering (1) is constructed.Theorem 4.
The Legendre expansion∑ℙ=0NλℙKℙT of UT∈L20,1 converges to UT as N⟶∞.Proof.
LetUMT=∑ℙ=0MλℙKℙT. Then, for N>M, we have(44)UNx−UMxL22=∑ℙ=M+1NλℙKℙTL22=∑ℙ=M+1NλℙKℙT|∑i=M+1NλiKiT=∑ℙ=M+1N∑i=M+1Nλℙλi¯KℙT|KiT=∑ℙ=M+1N∑ℙ=M+1Nλℙλi¯∫01KℙTKiTdT=∑ℙ=M+1Nλℙ212ℙ+1<∑ℙ=M+1Nλℙ2.
Bessel’s inequality indicates that∑ℙ=M+1Nλℙ2≤∑ℙ=M+1∞λℙ2≤UL22<∞. So, UNT−UMTL22⟶0 as N,M⟶∞. Thus, the sequence UNT is Cauchy in the complete space L20,1 and converges to some WT∈L20,1. Now, to show WT=UT, we consider(45)WT−UT|KℙT=WT|KℙT−WT|KℙT=limN⟶∞UNT,KℙT−λℙ=λℙ−λℙ,=0.
Hence,WT=WT which completes the proof.Theorem 5.
WeconsiderUT,ω and UNT,ω concerning (1) and suppose that the Lipschitz condition EF2η,Uη,ω,ω−F2η,UNη,ω,ω2≤LEUη,ω−UNη,ω2,L>0, is satisfied. Then,(46)UT,ω−UNT,ω2⟶0asN⟶∞.Proof.
LeteNT,ω=UT,ω−UNT,ω be the error function results when the exact solution UT,ω of (1) is approximated by UNT that is given in (34). So, using (7), we have(47)eNT,ω=UT,ω−UNT,ω=∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη+λ2ϰ2∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη+I0ϰεM+I0ϰResNT,
The triangular inequality, CSI, and Itô’s lemma give(48)EUT,ω−UNT,ω2=E∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη+λϰ∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη+I0ϰεM+I0ϰResNT2≤4E∫0Tηϰ−1F1η,Uη,ω,ω−F1η,UNη,ω,ωdη2+4λ2ϰ2E∫0TTϰ−ηϰF2η,Uη,ω,ω−F2η,UNη,ω,ωdBη2+4EI0ϰεM2+4EI0ϰResNT2≤4∫0Tη2ϰ−2dη∫0TEF1η,Uη,ω,ω−F1η,UNη,ω,ω2dη+4λ2ϰ2∫0TTϰ−ηϰ2EF2η,Uη,ω,ω−F2η,UNη,ω,ω2dη+4EI0ϰεM2+4EI0ϰResNT2≤4T2ϰ−12ϰ−1∫0TEF1η,Uη,ω,ω−F1η,UNη,ω,ω2dη+4λ2ϰ2∫0TTϰ−ηϰ2EF2η,Uη,ω,ω−F2η,UNη,ω,ω2dη+4EI0ϰεM2+4EI0ϰResNT2.
Now, using the Lipschitz condition and simple calculations, we otain(49)eNTE2≤4T2ϰ−12ϰ−1L∫0TEUη,ω−UNη,ω2dη+4λ2ϰ2L∫0TTϰ−ηϰ2EUη,ω−UNη,ω2dη+4I0ϰεE2+4I0ϰResNTE2=2L∫0TT2ϰ−12ϰ−1+λTϰ−ηϰϰ2eNTE2dη+4I0ϰεE2+4I0ϰResNTE2.
LetϱN,M=4I0ϰεME2+4I0ϰResNTE2 and apply Gronwall’s inequality on (49), which yields(50)eNTE2≤ϱN,Me∫0T2LT2ϰ−1/2ϰ−1+λTϰ−ηϰ/ϰ2dη.
SinceϱN,M⟶0 as N,M⟶∞ and for T∈Ι, ∫0T2LT2ϰ−1/2ϰ−1+λTϰ−ηϰ/ϰ2dη<∞. Then, UT−UNT2⟶0 as N,m⟶∞ as required.


## 5. Numerical Results and Examples

To emphasize the results gained in previous sections, we provide an algorithm consisting of clear steps based on the SL-SCA described in Section4 and utilize it to solve examples of linear and nonlinear forms of (1) numerically. Hither, the values of the SBM are calculated for τ=128, and for the collocation points, the subsequence formula is used [30].(51)Ti=12−12cos2i+1π2N+2,i=1,2,…,N−1.Generally, the exact solution of (1) does not exist. So, we consider eNT=UNT−UT to be the error function which is calculated only for examples that have no stochastic terms (see Example 1 for λ=0).Example 1.
Consider the following linear fractional SIDE:(52)TϰUT=UT+T0.25+2T1.25−T2−T+λ∫0TηdBη,U0=0,wherein T∈Ι=0,1 and 0.5<ϰ<1. Hither, for ϰ=0.75 and λ=0, the exact solution is UT=T2+T.Example 2.
Linear fractional SIDE is as follows:(53)TϰUT=UT+T0.5−T2+∫0TUηdBη,U0=0,wherein T∈Ι=0,1 and 0.5<ϰ<1.Example 3.
Linear fractional SIDE is as follows:(54)TϰUT=T2sinT+∫0TUηdBη,U0=0,wherein T∈Ι=0,1 and 0.5<ϰ<1.Example 4.
Nonlinear fractional SIDE is as follows:(55)TϰUT=U2T+UT−T+∫0T0.5dBη,U0=0,wherein t∈Ι=0,1 and 0.5<ϰ<1.Example 5.
Nonlinear fractional SIDE is as follows:(56)Tϰtt=ttt+∫01ηdBη,t0=0,wherein T∈Ι=0,1 and 0.5<ϰ<1.
It is important to shed light on the fact that different SBM paths produce different approximated realizations. Fixedω∈Ω, a path of an SBM BT,ω=BT, is a continuous nowhere differentiable starting from zero WP1 and has independent normally distributed discrete increments. As it is presented in Section 4, BT can be simulated at fixed values of T. The paths that appear in Figure 1 are performed using Mathematica 11 program.
For Example1, the subsequence results have been obtained by employing Algorithm 1. Firstly, in Table 1, data outcomes for UNT and UT at N∈4,6,9, ϰ=0.75, and λ=0 are tabulated. Secondly, in Table 2, data outcomes for UNT at ϰ∈0.8,0.85,0.9,0.95, N=7, and λ=1 are tabulated. Thirdly, in Figure 2 graphical outcomes for UNT, UT, and eNT at λ=0,ϰ=0.75, and N=4 are sketched. Fourthly, in Figure 3 graphical outcomes for UNT, UU, and eNT at λ=0,ϰ=0.75, and N=9 are sketched. Fifthly, in Figure 4 graphical outcomes for UNT at ϰ∈0.80,0.85,0.90, N∈3,4,6, and λ=1 are sketched.
The following algorithm steps are executed using MATHEMATICA 11 software:
Algorithm 1: Executions of the SL-SCA in handling fractional SIDEs (1).
Input: Initial data:ϰ, M, T, N, U0,λ. Functions: F1, and F2. SBM: B.Execution A: Compute SLPsKiT for i=0,1,⋯,N−1 from (30).Execution B: Compute forTϰKiTi=1,2,⋯,N−1 from (36).Execution C: ComputeKi0 for i=1,2,⋯,N−1.Execution D: Compute the collocation pointsTi for i=1,2,⋯,N−1 using (51).Execution E: Approximate∫0Ti∂F2η,∑ℙ=0N−1λℙKℙη/∂ηBηdη using (40).Execution G: Solve the system of N algebraic equations of (42) and (43).Output: The approximationUNT=∑i=0N−1λiKiT.
For Example2, the subsequence results have been obtained by employing Algorithm 1. Firstly, in Table 3, data outcomes for UNT at ϰ∈0.65,0.7,0.75,0.8 and N=3 are tabulated. Secondly, in Figure 5, graphical outcomes for UNT at ϰ∈0.65,0.7,0.75 and N∈4,5,6 are sketched.
For Example3, the subsequence results have been obtained by employing Algorithm 1. Firstly, in Table 4, data outcomes for UNT at ϰ∈0.70,0.75,0.8,0.85 and N=5 are tabulated. Secondly, in Figure 6, graphical outcomes for UNT at ϰ∈0.75,0.80,0.85 and N∈4,5,6 are sketched.
For Example4, the subsequence results have been obtained by employing Algorithm 1. Firstly, in Table 5, data outcomes for UNT at ϰ∈0.65,0.7,0.75,0.8 and N=6 are tabulated. Secondly, in Figure 7,graphical outcomes for UNT at ϰ∈0.65,0.75,0.85 and N∈4,5,6 are sketched.
For Example5, the subsequence results have been obtained by employing Algorithm 1. Firstly, in Table 6, data outcomes for UNT at ϰ∈0.8,0.85,0.9,0.95 and N=5 are tabulated. Secondly, in Figure 8, graphical outcomes for UNT at ϰ∈0.7,0.8,0.9 and N∈3,4,5 are sketched.
Figures1 and 2 indicate the accuracy and capacity of the presented algorithm. It should be noted that a few terms of SLPs produce approximate solutions with few errors. Other figures show that the approximate paths of the exact stochastic process paths for example with stochastic terms are smooth. This could be explained by the algorithm of approximation used to obtain these approximations which are mentioned in Section 4.Figure 1
Path for the values of the SBMBT as (a) the results shown in Examples 1–3 and (b) the results shown in Examples 4-5.
(a)(b)Table 1
Comparison results usingUTi and UNTi at N∈4,6,9, ϰ=0.75, and λ=0 in Example 1.
tiutiu4tiu6tiu9ti0.10.110.10999999999999970.11000000000000080.11000000000000230.20.240.23999999999999980.240000000000000770.24000000000000290.30.390.38999999999999970.39000000000000090.39000000000000310.40.560.55999999999999970.56000000000000120.56000000000000360.50.750.74999999999999970.75000000000000120.75000000000000410.60.960.95999999999999960.96000000000000150.96000000000000450.71.191.18999999999999951.19000000000000151.19000000000000480.81.441.43999999999999951.44000000000000201.44000000000000550.91.711.70999999999999931.71000000000000221.7100000000000062121.99999999999999912.00000000000000272.0000000000000058Table 2
Comparison of results usingUNTi at ϰ∈0.8.85,0.9,0.95, N=7, and λ=1 in Example 1.
Ti,ϰ0.80.850.90.950.10.18254996927835840.16149137209319480.14329185506125910.12747650434039530.20.31200615273262180.27533539170035650.24377978123504750.21646470647456200.30.45337400206592980.40173178928353750.35728789351527160.31879418631169850.40.62426996549680270.55697644733532380.49894922547827880.44857189922344690.50.81815627463828470.73477323280685700.66273605598804840.60005903748527410.61.02190515617788740.92250679270181820.83653667269825060.76163785315696230.71.22769246835834431.11247631283636481.01277857235117090.92586844240220940.81.43922076225917021.30809023976505051.19459809787351301.09563549124734380.91.67227176787902471.52502196587266781.39755651226904831.286384982779143411.94958830501889221.78732647763208871.64690250930816971.5244508657817661Figure 2
Schematic effects in Example1 within ϰ=0.75, λ=0, and N=4 as (a) UNT (blue) and UT (red) and (b) eNT.
(a)(b)Figure 3
Schematic effects in Example1 within ϰ=0.75, λ=0, and N=9 as (a) UNT (blue) and UT (red) and (b) eNT.
(a)(b)Figure 4
Schematic effects in Example1 as (a) UNT at ϰ∈0.8,0.85,0.9, N=7, and λ=1 as red: ϰ=0.9, blue: ϰ=0.85, and green: ϰ=0.8 and (b) UNT at N∈3,4,6, ϰ=0.8, and λ=1 as red: N=6, blue: N=4, and green: N=3.
(a)(b)Table 3
Comparison of results usingUNTi at ϰ∈0.65,0.7,0.75,0.8 and N=3 in Example 2.
ti,ϰ0.650.70.750.80.10.14798266695086540.12931426146366590.11367707139282840.10041247123521070.20.29354313245671240.25722063412988630.22679098976584160.20097325957554560.30.43668139651754090.38371911799866100.33934175511903960.30168236502100460.40.57739745913335130.50880971306999050.45132936745242210.40253978757158780.50.71569132030414310.63249241934387410.56275382676598940.50354552722729510.60.85156298002991660.75476723682031230.67361513305974130.60469958398812650.70.98501243831067150.87563416549930520.78391328633367790.70600195785408200.81.11603969514640870.99509320538085260.89364828658779900.80745264882516190.91.24464475053712681.11314435646495411.00282013382210520.909051656901365711.37082760448282701.22978761875161041.11142882803659561.0107989820826937Figure 5
Schematic effects in Example2 as (a) UNT at ϰ∈0.65,0.7,0.75 and N=3 as red: ϰ=0.75, blue: ϰ=0.7, and green: ϰ=0.65 and (b) UNT at N∈4,5,6 and ϰ=0.75 as red: N=6, blue: N=5, and green: N=4.
(a)(b)Table 4
Comparison of results usingUNTi at ϰ∈0.7,0.75,0.8,0.85 and N=5 in Example 3.
ti,ϰϰ=0.7ϰ=0.75ϰ=0.8ϰ=0.850.1−0.0004127422014683−0.0002893298202559−0.0001833975053145−0.00009413910900360.20.00005884104531590.00015236972237390.00023140910419490.00029616164034820.30.00265909072300180.00256436101968010.00246989059509570.00237465480120710.40.00885407115702290.00842380069793360.00801332279259250.00762034292265180.50.02033157001559660.01944573961788540.01859878658052650.01778747854968770.60.03900109830972460.03758312287476710.03621916790137610.03490556422324690.70.06699389039319270.06502678979829040.06312315775625640.06127935248018870.80.10666290396257070.10420547395264740.10181525220491940.09948884585329910.90.16058282005721260.15778580313651030.15505575236575400.152389296871290710.23155004305925650.22867229938303190.22586076441578590.2231112080588032Figure 6
Schematic effects in Example3 as (a) UNT at ϰ∈0.75,0.80,0.85 and N=5 as red: ϰ=0.85, blue: ϰ=0.8, and green: ϰ=0.7 and (b) UNT at N∈4,5,6 and ϰ=0.7 as red: N=6, blue: N=5, and green: N=4.
(a)(b)Table 5
Comparison of results usingUNTi at ϰ∈0.65,0.7,0.75,0.8 and N=6 in Example 4.
Ti,ϰ0.650.70.750.80.10.39131201263388040.42068868482224440.45131060614391010.48314873668536070.20.53830375604256220.57078575415456840.60434604902395690.63896263170938830.30.66373903450862910.69536356647013790.72772676534060520.76081440588673650.40.84941768539806000.88331431724368950.91778956667945040.95282798783744840.51.10948064266289941.14810597627314161.18720140548672331.22674363039813360.61.46371500034391571.50853822500118721.55357314104478301.59878302703313930.72.01085907607327562.06949839383690402.12807330544761802.18651442824572230.83.00190747457721603.10271739947735273.20404186957635403.30571775798921760.94.91341615117870105.12752568222920505.34560400907477605.567249730078245018.52080747530008508.99160914333029409.47628387032472809.9739089645998360Figure 7
Schematic effects in Example4: (a)UNT at ϰ∈0.65,0.75,0.85 and N=6 as red: ϰ=0.85, blue: ϰ=0.875, and green: ϰ=0.65 and (b) UNT at N∈4,5,6 and ϰ=0.85 as red: N=6, blue: N=5, and green: N=4.
(a)(b)Table 6
Comparison of results usinguNti at ϰ∈0.8,0.85,0.9,0.95 and N=5 in Example 5.
Ti,ϰ0.80.850.90.950.10.03283501153866410.03197550797742590.03116413530159080.03368024061890570.20.04684905157820340.04516943108939530.04360476313637990.04487706987550130.30.06105669350639540.05823840526929080.05561951459828130.05368716506667980.40.08946383655721170.08497639464471500.08078786278561390.07498492373982330.50.14106770581081860.13431469153749000.12797112280110170.11842246369280270.60.21985685219357670.21032191646365790.20131245175187400.18842962297397850.70.32481115247804080.31220401813348110.30023684874946430.28421395988220000.80.44990180928296040.43430427345144130.41945115490981210.39976075296680550.90.58409135107327930.56610328751624000.54894405335326150.523833001027622610.71133363216013530.69221899362079850.67398606920456100.6399714231149685Figure 8
Graphical results in Example5: (a) UNT at ϰ∈0.7,0.8,0.9 and N=5 as red: ϰ=0.9, blue: ϰ=0.8, and green: ϰ=0.7 and (b) UNT at N∈3,4,5 and ϰ=0.95 as red: N=5, blue: N=4, and green: N=3.
(a)(b)


## 6. Conclusions and Future Works

This work is mainly proposed to prove and confirm the uniqueness and existence of the suggested solution of a particular form of fractional SIDEs and construct a straightforward, simple, and effective numerical algorithm that provides approximate solutions to complicated problems modeling real-world phenomena. For the theoretical part, the Picard iteration criterion with some topological theorems has been applied to the integral form of (1). For achieving the numerical aim, we have introduced the SLPs and employed them as base functions for a spectral collocation method to have the so-called SL-SCA. The Itô’s integral in (1) has been approximated using a well-known numerical integration method and the values of the SBM have been approximated via an effective method. Also, the convergence analysis of the proposed algorithm has been discussed. Moreover, using the algorithm given in Section 5, three linear and two nonlinear examples of fractional SIDEs have been successfully solved. The outcomes assert that the SL-SCA is a powerful and convenient method. Accurate approximations can be achieved via a few terms of SLPs. For upcoming research, this algorithm is suggested to be employed for solving fractional SIDEs with fractional stochastic SBM of order H∈0,1.



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