Research paper
An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay

https://doi.org/10.1016/j.cnsns.2020.105475Get rights and content

Highlights

  • Nonlinear fractional stochastic differential equations with constant time delay is under consideration.

  • A new computational scheme based on a piecewise integro quadratic spline interpolation is proposed.

  • The convergence properties of the scheme are investigated.

  • The accuracy of the proposed scheme is analyzed in the perspective of the expected mean absolute norm error and experimental convergence order.

  • 1he statistical indicators are analyzed for assessing the performance of the proposed scheme.

Abstract

This paper proposes an accurate and computationally efficient technique for the approximate solution of a rich class of fractional stochastic differential equations with constant delay driven by Brownian motion. In this regard, a piecewise integro quadratic spline interpolation approach is adopted for approximating the fractional-order integral. The performance of the computational scheme is evaluated by statistical indicators of the exact solutions. Moreover, the computational convergence is also analysed. Three families of models with stochastic excitations illustrate the accuracy of the new approach as compared with the M-scheme.

Introduction

Stochastic differential equations (SDEs) have been increasingly applied for mathematical modelling [28]. Indeed, SDEs are effectively used to solve important real-world problems where the influence of randomness can not be neglected [22]. This characteristic and the elegant formulation attracted not only the attention of mathematicians, but also of engineers, physicians, economists among others [7], [15]. Following these ideas, in the present paper, a special and highly important class of problems is considered which contains fractional SDEs with constant time delay.

Fractional calculus is the theory of derivatives and integrals of any order that constitutes a valuable tool within the scientific community. In practice, fractional calculus has been applied for the study of a variety of problems and areas where non-locality plays a key role. For more details, interested readers can see [13], [17], [26], [27], [29], [40], [46]. However, in several applied problems, it is anticipated that it is not enough that the models have a precise formulation, but also that they behave more closely to the real processes. In particular, many of the problems include after-effect phenomena in their inner dynamics, and the introduction of state delayed equations is necessary. The applications of such equations can be found in engineering, biology, mechanics, economics, and population dynamics, just to mention a few. Key studies can be found, for instance, in the papers [5], [6], [8], [18], [19], [20], [24], [34], [41], [42] and the monographs [11], [23], [36].

In recent years, significant research has been conducted in the development of efficient numerical schemes and approximations with high order of accuracy for solving several fractional SDEs [21], [35].

In this paper, we consider the following fractional stochastic differential equations with time delays (FSDDE),{CD0,tβy(t)=K(t,y(t),y(tτ))+Q(t,y(t))dϖ(t)dt,t(0,T]y(t)=Φ(t),t[τ,0],where 12<β<1. Furthermore, Ψ=[0,T], K:Ψ×R×RR and Q:Ψ×RR are measurable functions, τ indicates the delay time, Φ(t) is the history function defined on the interval t[τ,0], and ϖ(t), t ≥ 0, represents a given standard Wiener process.

We use the Caputo [2], [3] fractional derivative which is formulated, see [44], as follows,CD0,tβy(t)=1Γ(pβ)0ty(p)(ς)(tς)β+1pdς,0p1<βpN,where y(t) is the unknown function that is (p1)- times continuously differentiable.

A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present states) depends only upon the present state, not on the sequence of events that preceded it [45]. Therefore, the solutions of the FSDDE (see, Eq. (1)) do not follow the Markov property and it is a challenging problem to derive their representations, let alone to obtain the exact analytical forms. In this regard, a numerical technique for FSDDE needs to be formulated to solve the stochastic dynamic problems following Eq. (1). For more details, readers can refer to [39], [45]. Recently, a numerical solution was proposed [30] based on a linear B-spline interpolation. The scheme determines the statistical indicators (SIs) of the stochastic responses for the fractional Langevin [14], [49] and Mackey-Glass [10] models with stochastic excitations. Following the ideas discussed in [35], we consider integro quadratic spline interpolation [32], [47] for the FSDDE. We determine the SIs of the stochastic responses for the fractional stochastic human postural sway, population and Nicholsons blowflies models with stochastic excitations. The accuracy and superiority of the proposed algorithm are compared with the forward finite-difference approximation [31], known as M-scheme. Furthermore, we must mention for consistency that the existence and uniqueness of the solutions of the FSDDE were studied in [33].

The remaining of the paper is structured as follows. Section 2, presents an explicit approximation procedure facilitating the integro quadratic spline interpolation for discretizing the FSDDE. Section 3, studies the convergence of the proposed approach. Section 4, investigates the accuracy of the new algorithm considering the fractional stochastic postural sway, population and Nicholson blowflies models. Finally, Section 5, presents the concluding remarks.

Section snippets

Theoretical Results

The aim of this section is twofold. First, obtains an approximation of the fractional-order integral operators and, second, develops an accurate and computationally efficient scheme for the FSDDE solution (see, Eq. (1)). In what follows, some necessary concepts and results and introduced. Thereafter, we consider tn=nh, where n={j,j+1,,1,0,1,,m}, h=Tm for the uniform step size, j=τh and j,mN.

Let L2(Ω,Ft,P) be a fixed probability space with a normal filtration (Ft)t0, mean square

Convergence Analysis

In this section, we investigate the convergence of the IQS-scheme (see, Eq. (21)) for approximating the solution of the FSDDE (see, Eq. (1)).

From Eq. (17), and for the time instant tm ∈ [0, T], we obtainy(tm)=y0+J0,tmβK(t,y(t),y(tτ))+Jtmβ(Q(t,u(t))dϖ(t)dt).Let E(tm)=y(tm)ym and E0=E(t0)=0. By means of the elementary inequality,n=1mxn2mn=1mxn2,subtracting Eq. (24) from Eq. (21), and the Ho¨lder inequality, we have

E(tm)ms2=E[1Γ(β)0tm(tmς)β1K(ς,y(ς),y(ςτ))dς+1Γ(β)0tm(tmς)β1Q(ς,y(

Illustrative examples

In this section, the accuracy and computational efficiency of the IQS-scheme are assessed using three illustrative examples. For evaluating the computational performance of the IQS-scheme, the expected mean absolute error (EMAE) (E¯mms) and the experimental convergence order (ECOms) are consideredE¯mms=1mn=1mEAEn,andECOms=log2(E¯2mmsE¯mms),where EAEn=(E[ynmy2n2m2])12, ynm and y2n2m are approximate values of y(tn), and m represents the number of interior mesh points. All the

Conclusion

In this paper, an explicit numerical algorithm was designed for calculating the approximate solutions of an important class of fractional stochastic equations with constant time delay driven by Brownian motion. The numerical algorithm is based on the integro quadratic spline interpolation for approximating adequately the fractional-order integral operators. Three illustrative examples with stochastic excitations illustrated the accuracy of compared the results obtained by means of the M- and

CRediT authorship contribution statement

B.P. Moghaddam: Conceptualization, Methodology, Software. Z.S. Mostaghim: Data curation, Writing - original draft. A.A. Pantelous: Writing - review & editing. J.A. Tenreiro Machado: Writing - review & editing.

Declaration of Competing 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 paper.

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