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.

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 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:

Herein, , , is a real constant, is the SBM built on a given space wherein the increasing filtration is right continuous. Indeed, is the set of all possible outcomes, and is a measure defined on and satisfies . The functions and are given stochastic processes, is to be determined, and the initial condition is a given random variable presumed to be independent of . Ultimately, denotes the CFD of order with .

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 Section 2, 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 for , they can be generalized for any .

Definition 1 (see [2]). The CFD of order for for all is

Definition 2 (see [2]). The conformable integral operator of order of is

Lemma 1 (see [2]). Let be two functions and . Then, for we have the following:(1) if is differentiable(2) if is continuous(3) if is continuous(4) if are -differentiable with Herein, is called -differentiable at a fixed point when exists. Indeed, a function could be -differentiable for some value of at a point but not differentiable. For example, in , , but 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 process with defines an SBM if the following subsequence properties are satisfied:(1)For all the increment , where denotes the normal distribution with mean and variance wherein (2)Distinct increments constructed of are independent for (3) is almost surely a continuous function of (4) WP1

Definition 4 (see [29]). Let symbolize the space of all functions . Then, we can utilize Itô’s integral of aswhere is the sequence of elementary functions, which satisfies

Lemma 2 (see [29]). For , Itô’s Isometry is given by

2.3. Spaces and Notations

Let denote the expected value of a random variable and be a random function; then, the norm of is defined as . Let refers to the space of each random function wherein . Hereafter, the space of each -measurable, bounded, and continuous functions realized on to is denoted by .

Clearly, the metric space with and is separable and complete. Also, the notation refers to the space of -valued random variables equipped with a metric . Finally, the functions are assumed to be jointly measurable and continuous for all and almost everywhere , and . 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. If is a solution of (1), then its integral form is

Proof. Applying (3) for both sides of (1) and using the stochastic Fubini theorem [32], one gained

Lemma 4 (see [33]). Let and be two non-negative continuous functions on and assume that . If with , then Gronwall’s inequality is

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, is presumed to be .

Theorem 1. Suppose that the growth condition with and holds; then, (1) has a solution in .

Proof. Using the integral form (7) and defining the Picard sequence on I with asand let for some .Step 1:we show that is well-defined and measurable for . First, we denoteSince and are assumed to be in , and the kernels and are bounded. Then, the growth condition asserts thatwhich indicates that Lebesgue’s integral and Itô’s integral are well-defined. Consequently, the sequence is well-defined.
It remains to prove the measurability of the considered sequence. This can be explained by the measurability of and , and the product of measurable continuous functions is again measurable. So, the mapsare measurable. Thus, we deduce the requirement.Step 2:we show that is bounded. With the use of CSI, Itô’s isometry, and the growth constraint, (10) giveswhere and . Also, for arbitrary , obviously,Therefore, taking the maximum of both sides of (14) one hasBy Gronwall’s inequality, we haveSince was arbitrary, we have which guarantees the boundedness of . Moreover, is uniformly bounded.Step 3:we prove the equicontinuity of . Let , , and Let be given and choose such that andEmploying CSI, Itô’s isometry, and (10), one obtainsTaking the expectation of both sides, one hasBy (18), one gets which proves the equicontinuity of Step 4:by the results gained in Steps 2 and 3, Arzela-Ascoli’s theorem assures that is a compact subset of . In addition, Step 2 asserts the boundedness of the sequence which indicates that . Applying Prohorov’s theorem, one gets the totally -boundedness of the sequence in . Thus, has a -Cauchy subsequence denoted by [34]. Then, Skorokhod’s representation lemma [35] assures the existence of a convergence sequence in that has the same distribution as such that the following subsequence properties are satisfied:(1)(2) WP1 as for some continuous -measurable random variable in The abovementioned properties ensure the boundedness of and WP1. Now, it remains to prove that , the limit of , satisfies (1). For this end, we consider the following:By the continuity of and , it appears that for given , with integer such that, for ,Hence, for all , one hasWith the property within mentioned in Step 4, (10) yieldsLetting relations (23)–(25), and the property WP1 shows thatConsequently, (26) demonstrates that is a random solution of (1) as required.

Theorem 2. Under the Lipschitz assumption with the solution of (1) is unique.

Proof. Let and be two solutions to (1). Then, using the integral form (7), one hasAgain, by the Lipschitz condition, one hasApplying Gronwall’s inequality on (28), one yieldsHence, for all . 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 function 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 in (1) is replaced by a finite series of SLPs resulting in a residual error function 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 is dropped and we write , , and instead of , , and , respectively.

Legendre polynomials are known to be built on . Utilizing appropriate change of variables, Legendre polynomials can be restricted to and then called the SLPs. They satisfy the orthogonality property concerning the weight function . SLPs can be expressed in a recursive formula [36] but the most efficient formula is the following analytical series [37]:

The orthogonality property of SLPs over is

Definition 5 (see [38]). For all the functional of is

Theorem 3 (see [36]). Let , where is a polynomial of degree . Then, for there is a unique polynomial such thatwhere with and the sequence is an orthogonal basis for in the space.
Since they satisfy the orthogonality property, the SLPs form an orthogonal basis for in space. Thus, the previous theorem asserts that can be uniquely approximated by a truncated series of the SLPs as follows:where with .
Using the linearity property of the CFD, one can easily verify that the approximation of the derivative of in the conformable sense isAlso, using the property Itô’s integral in (1) can be written asInserting relations (34), (35), and (37) into (1), it leads towhere denotes the residual error function that is produced when substituting in (1).
Now, for , we substitute each collocation point , arranged as into (38) and getTo approximate the integral term in (39), one can use the composite trapezoidal rule as follows:where is an integer and .
Anyhow, the substituting of (40) into (39), leads towhere is the trapezoidal rule error related to the integral approximating using (39). It is known that as