Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

El-Beltagy, Mohamed A.; Etman, A. A. M.; Maged, Sroor

doi:10.1016/j.chaos.2022.111847

Cited by 3 publications

(6 citation statements)

References 24 publications

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“…We shall follow [2] and extend the development of FWHE basis functionals and use their properties to derive the solution statistics. The Hermite functions will be selected to represent the basis of FWHE due to their properties that are suitable for the Gaussian processes such as fBM.…”

Section: Construction Of Fwhe Basismentioning

confidence: 99%

“…In this case, there is no practical exact solution and there will be a need for an approximate and/or numerical solution. The two models discussed in this section have multiplicative fractional noise and hence require some additional efforts compared with the additive noise models that are usually Gaussian and straightforward to be analyzed using FWHE as shown in [2].…”

Section: Examplesmentioning

confidence: 99%

“…The classical Brownian motion (BM) is commonly used in modeling and analysis of such models. However, stochastic processes with history-dependency or long-distance correlations cannot be modeled with the classical BM and hence the fBM is developed to model processes with heavy tails such as internet traffic, biological systems, financial markets, and hydrology [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…There is much research work conducted for studying the stochastic and random effects in the case of models with integer-order and/or fractional-order derivatives. For techniques based on solving the Fokker-Planck equation, analytical solutions, numerical approaches, and transformations, see [2,8] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Spectral techniques are good alternatives with higher efficiency than EM-based techniques. Examples include using Haar wavelets [12] and the recently developed FWHE that uses Hermite functionals of the fractional noise [2]. The spectral techniques have many advantages such as obtaining analytical exact or approximate statistics for many models with high convergence rates, sometimes exponential, without generating the time-consuming samples.…”

Section: Introductionmentioning

confidence: 99%

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Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

2023

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Biological and financial models are examples of dynamical systems where both stochastic and historical behavior are important to be considered. The fractional Brownian motion (fBM) is commonly used, sometimes with fractional-order derivatives, to model the combined stochastic and fractional effects. Recently, spectral techniques are used to analyze models with fBM using, e.g., iterated Itô fractional integrals such as the fractional Wiener-Hermite (FWHE). In the current work, FWHE is generalized and adapted to be consistent with the Malliavin calculus approach. The conditions for existence and uniqueness are outlined in addition to the proof of convergence. The solution algorithm is described in detail. Using FWHE, the stochastic fractional model is replaced by a deterministic fractional-order system that can be handled using well-known mathematical tools to evaluate the solution statistics. Analytical solutions can be obtained for many important models such as the fractional stochastic Black–Scholes model. The convergence is studied and compared with the exact solution and high convergence is noticed compared with other techniques. A general numerical algorithm is described to analyze the resultant deterministic system in the case of no feasible analytical solutions. The algorithm is applied to study and simulate the population model with nonlinear losses for different values of the Hurst parameter. The results show the efficiency of FWHE in analyzing practical linear and nonlinear models.

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Section: Construction Of Fwhe Basismentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

2023

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A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

El-Beltagy

Mahdi

Noor

2023

Axioms

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In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs.

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On the series solution of the stochastic Newell Whitehead Segel equation

Hussain

2023

MATH

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<abstract><p>The purpose of this paper is to present a two-step approach for finding the series solution of the stochastic Newell-Whitehead-Segel (NWS) equation. The proposed two-step approach starts with the use of the Wiener-Hermite expansion (WHE) technique, which allows the conversion of the stochastic problem into a set of coupled deterministic partial differential equations (PDEs) by components. The deterministic kernels of the WHE serve as the solution to the stochastic NWS equation by decomposing the stochastic process. The second step involves solving these PDEs using the reduced differential transform (RDT) algorithm, which enables the determination of the deterministic kernels. The final step involves plugging these kernels back into the WHE to derive the series solution of the stochastic NWS equation. The expectation and variance of the solution are calculated and graphically displayed to provide a clear visual representation of the results. We believe that this two-step technique for computing the series solution process can be used to a great extent for stochastic PDEs arising in a variety of sciences.</p></abstract>

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Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

Cited by 3 publications

References 24 publications

Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

On the series solution of the stochastic Newell Whitehead Segel equation

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