Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules

Aisha F. Fareed; Mourad S. Semary

doi:10.1515/nleng-2024-0070

Article Open Access

Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules

Aisha F. Fareed and Mourad S. Semary

Published/Copyright: February 10, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

This research introduces a novel and robust numerical approach, the stochastic improved Simpson Method, specifically developed to solve Itô and Stratonovich stochastic nonlinear system of differential equations with fractional order. By extending the classical Simpson’s one-third rule with the explicit product integration rectangle rule, the proposed method efficiently handles fractional derivatives of orders between 0 and 1, based on the Caputo derivative. The novelty of this approach lies in its enhanced accuracy and stability in addressing the unique challenges posed by both Itô and Stratonovich systems, outperforming traditional numerical techniques. Rigorous order analysis, conducted with Mathematica 12 software, demonstrates the robustness and precision of the method. Its effectiveness is further validated through four distinct numerical case studies, highlighting its superior performance in solving stochastic nonlinear system of differential equations with fractional order.

Keywords: stochastic differential equations; white noise; fractional-order systems; Simpson’s rule; Caputo derivative

1 Introduction

Fractional calculus dates back to the origins of calculus, stemming from a discussion between L’Hôpital and Leibniz in 1695, where they explored the idea of raising the differential operator to the power of 1/2 [1,2]. Fractional calculus is the study of non-integer derivatives and integrals, where the order can be real, rational, or complex [1]. Due to their versatility and ability to represent memory-dependent systems, fractional-order systems (FOSs) have garnered significant interest from researchers over the past few decades [3]. A notable property of FOS analysis is that it produces more accurate responses than actual systems while using simpler mathematical equations compared to integer-order counterparts [1]. This advantage is directly related to the increased tunability achieved by incorporating fractional orders as new model parameters. Engineering applications of fractional orders include bioengineering [4,5], control [6,7], filters [8], oscillators [9], energy [10], encryption [11], and chaos theory [12].

Stochastic differential equations (SDEs) represent processes more accurately in practical scenarios by accounting for stochastic factors. SDE theory has been applied to a wide range of study domains, including stochastic control, stochastic neural networks, financial economics, electronic engineering, and population dynamics [13]. Due to their nonlinear nature and complexity, SDEs are notoriously difficult to solve exactly. Consequently, scholars have focused on developing efficient mathematical approaches to simulate the solutions of SDEs. Various aspects of the numerical methods framework for SDEs have been studied in recent decades [14,15,16,17,18,19,20,21,22,23]. Recently, some analytical and numerical methods have been improved to handle fixed and variable order fractional stochastic Volterra integro-differential equations [24,25,26,27]. The Caputo definition is of the classical variety and is given as follows.

Definition I

[22]. The Caputo αth order fractional derivative and the fractional integral for u : [ a , b ] → R , m − 1 < α ≤ m and m ∈ N , are described as

D a C t α u ( t ) = 1 Γ ( n − α ) ∫ a t u ( m ) ( s ) ( t − s ) m − α − 1 d s ,

I t α u ( t ) = 1 Γ ( α ) ∫ a t u ( s ) ( t − s ) α − 1 d s .

To ensure the existence of the Caputo fractional derivative and the fractional integral as defined in Definition 1, certain conditions on the function u : [ a , b ] → R with order α such that m − 1 < α ≤ m and m ∈ N must be met. Specially, the Caputo fractional derivative D a C t α u ( t ) exists if u has a well-defined mth order derivative on [ a , b ] , i.e., u ∈ C m ( [ a , b ] ). This ensures that u ( m ) ( s ) is integrable over [ a , t ] . Additionally, for the fractional integral, I t α u ( t ) must be locally integrable on [ a , b ] , meaning u ∈ L 1 ( [ a , b ] ) . These conditions provide a sufficient foundation for the existence of the Caputo fractional derivative and fractional integral in this context.

Simpson’s one-third rule is a classical numerical integration technique widely used due to its simplicity and effectiveness in approximating definite integrals. However, the classical Simpson’s rule, based on quadratic polynomials, can face challenges when applied to more complex systems, such as those involving fractional-order derivatives or stochastic processes. To address these limitations, a modified version of this rule, known as the improved Simpson’s (IS) one-third rule, has been developed. This approach replaces the quadratic polynomial with a linear trigonometric function, yielding a simpler and more accurate solution compared to the classical method [28]. The improved rule enhances precision, particularly for systems where traditional Simpson’s rule struggles.

While IS’s one-third rule has shown promise in deterministic systems, its application to stochastic systems, particularly those governed by fractional-order dynamics, is still an emerging field. SDEs with fractional order, particularly in the Itô and Stratonovich frameworks, introduce additional complexity due to the interplay between randomness and memory effects inherent in fractional calculus. These systems have been explored through various numerical methods, yet the integration of fractional derivatives within stochastic systems remains challenging. Existing numerical techniques often struggle to balance the precision needed for fractional derivatives with the stochastic nature of these systems.

This study addresses this gap by extending the IS’s one-third rule to solve stochastic nonlinear differential equations with fractional order, focusing on both Itô and Stratonovich formulations. By incorporating the product integration (PI) rectangle rule to handle fractional derivatives between 0 and 1, the proposed method is uniquely suited to tackle the difficulties posed by stochastic fractional systems. The novelty of this approach lies in its ability to enhance the accuracy and stability of solutions for such systems, surpassing existing methods in both performance and applicability. To the best of our knowledge, this is the first time the combination of the IS’s rule with a PI scheme has been applied to stochastic nonlinear FOSs, providing a novel and robust numerical framework for solving these equations.

The principal objective and contribution of this research endeavor lie in addressing the solution of Itô and Stratonovich stochastic fractional nonlinear differential equation systems through the introduction of a novel numerical approach based on the IS’s one-third rule [28]. This newly proposed numerical methodology integrates the IS’s one-third rule with the explicit PI rectangular rule to accommodate fractional derivatives, thus yielding an elegant and efficient numerical framework capable of resolving stochastic fractional nonlinear differential equation systems of the following form:

(1) D t α 0 C u ( t ) = F 1 ( t , u , V ) + ( G 1 ( t , u , V ) + ρ ƒ 1 ( t ) ) ᶇ ( t ) , u ( 0 ) = a , D t α 0 C V ( t ) = F 2 ( t , u , V ) + ( G 2 ( t , u , V ) + ρ ƒ 2 ( t ) ) ᶇ ( t ) , V ( 0 ) = b ,

where the fractional time derivative D t α 0 C is of Caputo type, u ( t ) , V ( t ) are the unidentified functions, t is represents the independent variable, F 1 ( t , u , V ) , F 2 ( t , u , V ) and G 1 ( t , u , V ) , G 2 ( t , u , V ) represent the linear or nonlinear functions where ᶇ ( t ) is Gaussian white noise. The relation between Gaussian white noise ᶇ ( t ) and the Wiener process ɷ ( t ) is defined by

(2) ᶇ ( t ) = ϱ d ɷ ( t ) d t ,

the expectation E [ ᶇ ( t ) ] = 0 , and finite variance Var [ ᶇ ( t ) ] = ϱ 2 . The study is organized as follows: Section 2 presents the proposed numerical approach based on the IS rule. The numerical simulations of Itô and Stratonovich stochastic systems are discussed in Section 3. Finally, the concluding remarks of this study are summarized in Section 4.

2 Stochastic improved Simpson (SIS) rule

Given the next Itô stochastic fractional differential equation system which is characterized by the Caputo derivative as [29] follows:

(3) D t 0 α 0 C u ( t ) = F 1 ( t , u , V ) + ( G 1 ( t , u , V ) + ρ ƒ 1 ( t ) ) ᶇ ( t ) , u ( t 0 ) = u 0 , D t 0 α 0 C V ( t ) = F 2 ( t , u , V ) + ( G 2 ( t , u , V ) + ρ ƒ 2 ( t ) ) ᶇ ( t ) , V ( t 0 ) = V 0 ,

where F 1 ( t , u , V ) , F 2 ( t , u , V ) , G 1 ( t , u , V ) , and G 2 ( t , u , V ) are continuous functions and u 0 , V 0 are the initial values. Extending the Riemann–Liouville fractional integral to each side of Eq. (3) yields [30]:

(4) u ( t ) = T m − 1 [ u ; t 0 ] ( t ) + 1 Γ ( α ) ∫ t 0 t ( t − τ ) α − 1 ( F 1 ( τ , u ( τ ) , V ( τ ) ) + ( G 1 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 1 ( τ ) ) ᶇ ( τ ) ) d τ , V ( t ) = T m − 1 [ V ; t 0 ] ( t ) + 1 Γ ( α ) ∫ t 0 t ( t − τ ) α − 1 ( F 2 ( τ , u ( τ ) , V ( τ ) ) + ( G 2 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 2 ( τ ) ) ᶇ ( τ ) ) d τ ,

where T m − 1 [ u ; t 0 ] ( t ) is the Taylor polynomial of degree m − 1 for the function u ( t ) centered at t 0 , for 0 < α < 1 . Eq. (4) is also referred to as weak singular Volterra integral equation.

Differential equations can be solved in one or more steps using numerical approaches. The solution’s approximation result from the previous step is used in assessing the present step in the former. In the latter case, more than one past step is used to assess the current step. Because fractional differential equations have been proven to be memory dependent, multistep solutions are necessary. Such multi-step approaches can be regarded as a convolutional quadruple, which is defined in general as [30]

(5) u n = φ n + ∑ j = 0 n c n − j ( F 1 ( t j , u j , V j ) + ( G 1 ( t j , u j , V j ) + ρ ƒ 1 ( t j ) ) ᶇ ( τ ) ) , V n = ψ n + ∑ j = 0 n d n − j ( F 2 ( t j , u j , V j ) + ( G 2 ( t j , u j , V j ) + ρ ƒ 2 ( t j ) ) ᶇ ( τ ) ) ,

where φ n , ψ n , and c n are recognized as coefficients that vary depending on the technique used, and t n = t 0 + n h , h > 0 where h represents the equally-spaced grid. Product integration rules and fractional linear multi-step methods are the only two categories of multi-step methods that have been examined [30]. The product-integration rules are the sole thing that this work addresses.

Depending on the PI principals, the finding of Eq. (4) can be expressed as [30] follows:

(6) u ( t n ) = u 0 + 1 Γ ( α ) ∑ j = 0 n − 1 ∫ t j t j + 1 ( t n − τ ) α − 1 ( F 1 ( τ , u ( τ ) , V ( τ ) ) + ( G 1 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 1 ( τ ) ) ᶇ ( τ ) ) d τ , V ( t n ) = v 0 + 1 Γ ( α ) ∑ j = 0 n − 1 ∫ t j t j + 1 ( t n − τ ) α − 1 ( F 2 ( τ , u ( τ ) , V ( τ ) ) + ( G 2 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 2 ( τ ) ) ᶇ ( τ ) ) d τ ,

where 0 < α < 1 . F 1 ( τ , u ( τ ) , V ( τ ) ) , G 1 ( τ , u ( τ ) , V ( τ ) ) , F 2 ( τ , u ( τ ) , V ( τ ) ) , and G 2 ( τ , u ( τ ) , V ( τ ) ) are divided into sub-intervals [ t j , t j + 1 ] and the integral is approximated using polynomials. Different polynomial choices result in different PI rules. Based on how the integral approximation is performed, implicit or explicit rules are implemented. Implicit methods entail solving nonlinear equations where u n is dependent on F 1 ( t n , u n ) , G 1 ( t n , u n ) . However, in explicit methods, u n is dependent on F 1 ( t n − 1 , u n − 1 , V n − 1 ) , G 1 ( t n − 1 , u n − 1 , V n − 1 ) , and previous approximates of F 1 , G 1 , and u rather than F 1 ( t n , u n , V n ) , G 1 ( t n , u n , V n ) . V n is dependent on F 2 ( t n − 1 , u n − 1 , V n − 1 ) , G 2 ( t n − 1 , u n − 1 , V n − 1 ) , and previous approximates of F 2 , G 2 , and V rather than F 2 ( t n , u n , V n ) , G 2 ( t n , u n , V n ) . As a result, explicit methods are more suitable for hardware implementations.

The IS’s rule [28] is given by

(7) ∫ t j t j + 2 F ( τ ) d τ = 2 h s ∑ j = 0 n − 1 F t j + 2 + t j 2 .

To handle the integrations

(8) ∫ t j t j + 2 ( t n − τ ) α − 1 ( F 1 ( τ , u ( τ ) , V ( τ ) ) + ( G 1 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 1 ( τ ) ) ᶇ ( τ ) ) d τ , ∫ t j t j + 2 ( t n − τ ) α − 1 ( F 2 ( τ , u ( τ ) , V ( τ ) ) + ( G 2 ( τ , u ( τ ) , V ( τ ) ) + ρ ƒ 2 ( τ ) ) ᶇ ( τ ) ) d τ .

In (6), the IS’s rule (7) and relation (2) were used by setting

h s = T N , t n = n h s , τ = ( n − 1 ) 2 + n 2 h s , d ɷ ( t j + 1 ) d t = ɷ j + 1 − ɷ j h s , n = 0 , 1 , 2 , … , N ∈ Z + .

Now, Eq. (6) can be written as

(9) u ( t n ) = u 0 + 1 Γ ( α ) h s α 2 α − 2 ∑ j = 0 n − 1 F 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + G 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + ρ ƒ 1 t j + 1 + t j 2 ɷ j + 1 − ɷ j h s , V ( t n ) = V 0 + 1 Γ ( α ) h s α 2 α − 2 ∑ j = 0 n − 1 F 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + G 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + ρ ƒ 2 t j + 1 + t j 2 ɷ j + 1 − ɷ j h s .

By approximating the functions

F 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 , F 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 ,

G 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 , and G 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 ,

in the resulting formula (10) using the following explicit PI rectangular rule [30]:

(10) u ̌ ( t n ) = u 0 + h α ∑ j = 0 n − 1 P n − j − 1 ( α ) f ( t j , u j , V j ) , n = 1 , 2 , 3 , … ,

where

(11) P n ( α ) = ( n + 1 ) α − n α Γ ( α ) .

We obtain the SIS rule in conjunction with PI (SIS-PI) rectangular formula for n = 1 , 2 , 3 , … in the following form:

(12) u ̌ ( t n ) = u 0 + h α ∑ j = 0 n − 1 P n − j − 1 ( α ) F 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + G 1 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + ρ ƒ 1 t j + 1 + t j 2 ɷ j + 1 − ɷ j h s , V ̌ ( t n ) = V 0 + h α ∑ j = 0 n − 1 P n − j − 1 ( α ) F 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + G 2 t j + 1 + t j 2 , u t j + 1 + t j 2 , V t j + 1 + t j 2 + ρ ƒ 2 t j + 1 + t j 2 ɷ j + 1 − ɷ j h s .

To calculate the mean value and variance of the solution, take the mean and variance of u ̌ ( t n ) and V ̌ ( t n ) . This smart combination of the SIS’s rule and the PI rectangle formula. The SIS-PI rectangle formula creates a rapid and effective technique for dealing with fractional stochastic nonlinear differential equation systems, which is one of its countless advantages. In contrast to existing approaches such as HAM, WHEP, and even TAM, which are unable to execute many iterations due to the complexity of mean value and variance calculations, we simply compute multiple iterations without time expenditure.

3 Numerical assessments

In this section, we demonstrate the efficacy of the SIS rule by solving some classical physics problems intuitively. We apply the proposed method to two types of stochastic systems.

3.1 Itô stochastic system

Without missing generality, we consider the overall autonomous Itô stochastic fractional systems in this section.

Example 1

The Brusselator system is suggested in an Ito stochastic version [19].

(13) D t 0 α 0 C u ( t ) = ( χ − 1 ) u ( t ) + χ u 2 ( t ) + ( 1 + u ( t ) ) 2 V ( t ) + γ u ( t ) ( 1 + u ( t ) ) d ω ( t ) d t , u ( t 0 ) = − 0.1 D t 0 α 0 C V ( t ) = − χ u ( t ) − χ u 2 ( t ) − ( 1 + u ( t ) ) 2 V ( t ) − γ u ( t ) ( 1 + u ( t ) ) d ω ( t ) d t , V ( t 0 ) = 0 .

Unforced periodic oscillations in these chemical reactions constitute this nonlinear system. When χ = 1.9 and γ = 0.1 , using the proposed numerical scheme (12) with F 1 ( t , u , V ) = ( χ − 1 ) u ( t ) + χ u 2 ( t ) + ( 1 + u ( t ) ) 2 V ( t ) , G 1 ( t , u , V ) = γ u ( t ) ( 1 + u ( t ) ) ,

F 2 ( t , u , V ) = − χ u ( t ) − χ u 2 ( t ) − ( 1 + u ( t ) ) 2 V ( t ) and G 2 ( t , u , V ) = − γ u ( t ) ( 1 + u ( t ) ) ,

we simulate this system, and the results are shown in Figure 1.

Figure 1 illustrates the proximity of the approximate paths generated by the proposed scheme (13) to the origin, resembling the behavior patterns of the accurate solution. The figure also presents the expectation and variance of the suggested SIS scheme, alongside the solution acquired through the stochastic Runge–Kutta technique (SRK) integrated into Mathematica software. These findings validate that the numerical scheme solutions (12) align effectively with the SRK method.

Figure 1

The Itô SIS-PI rectangular (a) mean value, (b) variance, and (c) solution for the Brusselator system.

Example (2)

Recognize the Davis-Skodje system [31]

(14) D t 0 α 0 C u ( t ) = − u ( t ) + μ u ( t ) d ω 1 ( t ) d t u ( t 0 ) = 6 , D t 0 α 0 C V ( t ) = − γ V ( t ) + γ u ( t ) 1 + u ( t ) − u ( t ) ( 1 + u ( t ) ) 2 + σ γ V ( t ) d ω 2 ( t ) d t V ( t 0 ) = 0.85 .

The chemical reaction system (14) as outlined in the study by Nouri et al. [31] provides insight into the system’s spectral gap or stiffness when γ > 1 . Utilizing the parameters μ = σ = 0.01 and γ = 1,000 allows for the visualization of this system, employing the suggested algorithm (12) with appropriate substitutions

F 1 ( t , u , V ) = − u ( t ) , G 1 ( t , u , V ) = μ u ( t ) ,

F 2 ( t , u , V ) = − γ V ( t ) + γ u ( t ) 1 + u ( t ) − u ( t ) ( 1 + u ( t ) ) 2 and G 2 ( t , u , V ) = σ γ V ( t ) .

we simulate this system, and the results are shown in Figure 2.

The research by Nouri et al. [31] proves that the steady equilibrium position for the nonlinear system (14) is (0, 0). Figure 2 depicts the suggested scheme for simulating the nonlinear system (14). We observe that the current method approaches the asymptotic solution (0, 0).

Figure 2

The Itô SIS-PI rectangular solution for (a) mean value, (b) variance, and (c) solution for Davis-Skodje system.

3.2 Stratonovich stochastic system

Stratonovich SDEs [32,33] are similar to Itô differential equations in that they involve stochastic integrals in the Stratonovich sense instead of Itô integrals. The Stratonovich integral is indicated by a small circle before the Brownian differential to distinguish Itô and Stratonovich SDEs. The general autonomous Stratonovich stochastic fractional nonlinear differential equation system is considered.

(15) D t α 0 C u ( t ) = F 1 ( t , u , V ) + ρ ƒ 1 ( t , u , V ) ᴏ ᶇ ( t ) , u ( 0 ) = a D t α 0 C V ( t ) = F 2 ( t , u , V ) + ρ ƒ 2 ( t , u , V ) ᴏ ᶇ ( t ) , V ( 0 ) = b .

A Stratonovich SDE can be converted into an equivalent Itô equation by using simple transformation formulas [32].

Eq. (15) is equivalent to the following SDE in the Itô sense:

(16) D t α 0 C u ( t ) = F ∼ 1 ( t , u , V ) + ρ ƒ 1 ( t , u , V ) ᶇ ( t ) , u ( 0 ) = a D t α 0 C V ( t ) = F ∼ 2 ( t , u , V ) + ρ ƒ 2 ( t , u , V ) ᶇ ( t ) , V ( 0 ) = b ,

where

(17) F i ( t , u , V ) = F ∼ i ( t , u , V ) + ρ 2 ∑ j , l ∂ ƒ ij ( t , u , V ) ∂ u l ƒ lj ( t , u , V ) .

Now, we are ready to use the proposed scheme (12) after converting the Stratonovich SDE into an equivalent Itô equation.

Example (3)

Consider the stochastic Kubo oscillator [29].

(18) D t α 0 C u ( t ) = − c 1 V ( t ) − c 2 V ( t ) ᴏ ᶇ ( t ) , u ( 0 ) = 1 D t α 0 C V ( t ) = c 1 u ( t ) + c 2 u ( t ) ᴏ ᶇ ( t ) , V ( 0 ) = 1 ,

where the values of the parameters are c 1 = 1 and c 2 = 0.1 .

Furthermore, the exact solution to (18) is provided by Figure 3.

(19) u ( t ) = u ( 0 ) cos ( c 1 t + c 2 ɷ ( t ) ) − V ( 0 ) sin ( c 1 t + c 2 ɷ ( t ) ) , V ( t ) = u ( 0 ) sin ( c 1 t + c 2 ɷ ( t ) ) + V ( 0 ) cos ( c 1 t + c 2 ɷ ( t ) ) ,

we simulate this system, and the results are shown in Figure 3.

Figure 3

The Stratonovich SIS-PI rectangular solution for Kubo oscillator system for (a) mean value, (b) variance, (c) solution for the stochastic Kubo oscillator and (d) comparison with the exact solution.

Example (4)

Take a look at the stochastic mathematical pendulum system [29].

(20) D t α 0 C u ( t ) = − c 1 sin V ( t ) − c 2 sin V ( t ) ᴏ ᶇ ( t ) , u ( 0 ) = 0 , D t α 0 C V ( t ) = c 1 u ( t ) + c 2 u ( t ) ᴏ ᶇ ( t ) , V ( 0 ) = π 2 .

Let c 1 = 1 and c 2 = 0.1 be the parameters values.

For α = 1, Eq. (20) represents the 2D stochastic Hamiltonian system with non-quadratic Hamiltonian function.

(21) I ( u , V ) = 1 2 p 2 − cos ( V ) .

The solution of the stochastic mathematical pendulum system is shown in Figure 4 by the proposed method.

Figure 4

The Stratonovich SIS-PI rectangular solution for mathematical pendulum system.

3.3 Discussion and analysis

Several studies have addressed the development of numerical methods for solving complex systems of differential equations, particularly those involving nonlinear and linear fractional-order dynamics. In the work of Saini et al. [34], the authors tackle a coupled system of reaction-diffusion problems with mixed boundary conditions. This study focuses on computational cost reduction while ensuring the accuracy of solutions in the presence of boundary layers. Similarly, Shakti et al. [35] proposed an optimal and accurate computational method using a moving mesh refinement for a system of reaction-diffusion problems, demonstrating convergence and accuracy even with small diffusion terms.

For nonlinear problems, Das [36] presented a rigorous a posteriori convergence analysis for nonlinear singularly perturbed systems, highlighting the importance of adaptive meshing techniques to achieve optimal accuracy. Additionally, Das and Natesan [37] introduced an optimal error estimation method for reaction-diffusion boundary-value problems using mesh equidistribution techniques, further enhancing the precision of numerical solutions. For linear problems, Das [38] compared a priori and a posteriori meshes for solving singularly perturbed nonlinear parameterized problems, providing a detailed analysis of the benefits of adaptive meshes for achieving higher accuracy.

More recent studies, such as Choudhary et al. [39], have explored stable numerical approximations for time-fractional nonlinear equations using quintic B-spline methods, further expanding the capabilities of numerical techniques in handling FOSs. Das et al. [40,41] also explored the application of the homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations, contributing to the growing body of methods designed for FOSs.

While the present system of differential equations is simpler compared to the coupled systems studied in these works, our contribution lies in developing the SIS method for solving stochastic nonlinear differential systems with fractional order. Unlike traditional methods, our approach integrates the explicit PI rectangle rule with Simpson’s one-third rule to effectively handle fractional derivatives in stochastic systems, addressing both Itô and Stratonovich formulations. Although this study does not include a formal convergence analysis, we demonstrate the method’s precision and robustness through numerical simulations and order analysis. Future work will extend the study by conducting a comprehensive convergence analysis and comparing the performance of our method with the advanced techniques presented in the aforementioned studies.

Regarding the well-posedness of fractional differential equations, we acknowledge the importance of establishing the necessary conditions under which numerical solutions can be obtained. Ensuring the well-posedness of fractional differential equations is crucial, as it guarantees the existence, uniqueness, and stability of solutions – key aspects for the reliability of any numerical method. Several recent studies have extensively explored this area. For instance, Srivastava et al. [42] investigated the well-posedness of fractional-order p-Laplacian nonlinear models, demonstrating Ulam–Hyers stability and providing conditions essential for stable solutions. Similarly, Santra et al. [43] performed error analysis on fractional-order integro-parabolic partial differential equations using adaptive mesh techniques, further underscoring the significance of well-posedness conditions for ensuring solution accuracy.

Additionally, Das et al. [40] introduced a homotopy perturbation method for solving fractional-order Volterra-Fredholm integro-differential equations, alongside a detailed convergence analysis that highlights the robustness of their approach when well-posedness conditions are satisfied. In another study, Kumar et al. [44] developed a graded mesh refinement method for singularly perturbed time-delayed parabolic convection-diffusion problems, reinforcing the need for proper numerical frameworks that account for well-posedness conditions to obtain reliable solutions.

In recent studies, higher-order convergence analysis for reaction-diffusion systems has been explored, such as in Zarred et al. [45], where the convergence behavior of boundary layers in a semi-linear system networked by a k-star graph is analyzed. Additionally, Kumar et al. [46] provided adaptive mesh-based approximations for Darcy scale precipitation-dissolution models in porous media. Das et al. [47] examined higher-order accurate approximations for boundary layer-originated systems on equidistributed meshes, while Kumar et al. [48] focused on error estimations for nonlinear singularly perturbed problems.

First, we derive the error truncation of the proposed method. The global truncation error of the IS’s rule (7). The rule in the first subinterval is

(22) ∫ t 0 t 2 F ( τ ) d τ = 2 h s F ( t 1 ) .

The local truncation error for the rule (22) is

(23) LTE = ∫ t 0 t 2 F ( τ ) d τ − 2 h s F ( t 1 ) .

By using Taylor series, the functions F ( t 1 ) and F ( τ ) are given by

(24) F ( t 1 ) = F 0 + h s F 0 ′ + h s 2 2 ! F 0 ″ + h s 3 3 ! F 0 ‴ + … ,

(25) F ( τ ) = F 0 + ( τ − t 0 ) F 0 ′ + ( τ − t 0 ) 2 2 ! F 0 ″ + ( τ − t 0 ) 3 3 ! F 0 ‴ + ⋯ ,

Now, we substitute Eqs. (24) and (25) in (23), then

(26) LTE = h s 3 3 F 0 ″ ( ζ ) ∀ ζ ∈ ( t 0 , t 2 ) .

From (26), the global truncation error of the IS’s rule is

(27) GTE = h s 3 3 [ F 0 ″ ( ζ ) + F 2 ″ ( ζ ) + ⋯ + F 2 n − 2 ″ ( ζ ) ] + o ( h s 4 ) .

In the solution interval [ a , b ] , at least one value of ξ stratifies

(28) F ″ ( ξ ) = [ F 0 ″ ( ζ ) + F 2 ″ ( ζ ) + ⋯ + F 2 n − 2 ″ ( ζ ) ] n , b − a = 2 n h s .

We substitute Eq. (28) in (27), then the global truncation error of (7) is

(29) GTE = h s 2 6 F ″ ( ξ ) .

The IS’s rule is used to handle the integrations in the Eqs. (9) and (12); therefore, we show that the error truncation of the proposed method is o ( h 1 + α ) .

The second target, we show the convergence of the IS’s rule to exact integral. Assume F ( τ ) = e τ , the exact solution of ∫ 0 t F ( τ ) d τ is equal e t − 1 . By using the IS’s rule for ∫ 0 t e τ d τ ,

(30) ∫ 0 t e τ d τ = 2 h s [ e h s + e 3 h s + ⋯ + e ( 2 n − 1 ) h s ] ,

when h → 0 , then Eq. (30) becomes

(31) lim h → 0 2 h s e ( 2 n − 1 ) h s − e h s e 2 h s − 1 = e t − 1 .

Solution (31) means the rule used is converge to the exact solution.

Finally, the SIS’s rule is designed to address system (1) featuring random function excitation and a derivative of order α = 1 . To extend its applicability to calculate fractional derivatives for α ∈ ( 0 , 1 ) , we incorporate the explicit PI rectangle rule, accommodating fractional derivatives of D α u ( t ) , D α V ( t ) . The proposed SIS method offers several benefits and some limitations that need to be clearly outlined.

For complex systems, the suggested SIS method significantly saves computational time. Unlike the classical Simpson’s rule, which requires three interval points for each calculation, the IS’s rule is computed simply at the interval midpoint. This simplification reduces the computational burden, making the SIS method more efficient for complex and high-dimensional problems. Moreover, the incorporation of the PI rectangle rule enhances the method’s capability to handle fractional derivatives for α ∈ (0, 1), where α denotes the order of the fractional derivatives of the differential systems. This combination, referred to as the SIS-PI method, provides an innovative stochastic approach for approximating a wide range of fractional stochastic systems, thereby broadening the applicability of the method to various practical scenarios.

Despite its advantages, the SIS method encounters inherent limitations, particularly related to the stiffness associated with integrating random functions. This stiffness imposes constraints on the number of estimable iterations, potentially limiting the method’s effectiveness in certain scenarios. To address this limitation, we propose the combined utilization of the SIS’s rule and the PI formula (SIS-PI) method for resolving fractional stochastic nonlinear systems. This combined approach leverages the strengths of both methods, offering a robust solution to the challenges posed by stiffness and enabling more accurate and efficient approximations of fractional stochastic systems.

4 Conclusion

The study presents the results of fractional systems employing two types of stochastic terms. A comparative analysis is conducted between the innovative algorithms proposed and the SRK approach to establish their efficacy and motivation. Four distinct systems are investigated to showcase the efficiency and versatility of the proposed strategy. These systems encompass the stochastic Davis-Skodje and Brusselator systems of the Itô stochastic type, as well as the stochastic Kubo oscillators and mathematical pendulum systems of the Stratonovich stochastic type. Through numerical examples, the study underscores the practicality and superiority of the proposed approach in terms of computational efficiency. The results affirm the accuracy and effectiveness of the newly introduced method, which operates without necessitating restrictive assumptions for nonlinear terms. Future research will introduce additional theoretical investigations on the proposed method to further enhance its robustness and efficiency. Moreover, the approach will be expanded to solve stochastic fractional order partial differential systems and stochastic delayed differential equations with various fractional orders. Additionally, we aim to apply the method to stochastic fuzzy differential equations, thereby broadening its applicability. Practical applications in the fields of medical and engineering sciences will also be explored, such as modeling the stochastic behavior of the coronavirus and addressing various engineering problems. These efforts will contribute to the broader field of fractional calculus and SDEs, providing innovative solutions to complex real-world issues.

Acknowledgment

This study is supported via funding from Prince sattam bin Abdulaziz University project number (PSAU/2024/R/1446).

Funding information: Authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. A.F. Fareed wrote the main manuscript text and M.S. Semary prepared the results. All authors reviewed the manuscript.
Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
Ethical approval: Not applicable.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-03-30

Revised: 2024-11-23

Accepted: 2024-11-29

Published Online: 2025-02-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0070

Keywords for this article

stochastic differential equations; white noise; fractional-order systems; Simpson’s rule; Caputo derivative

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