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Research Papers

Analysis and Numerical Solutions for Fractional Stochastic Evolution Equations With Almost Sectorial Operators

[+] Author and Article Information
Xiao-Li Ding

Department of Mathematics,
Xi'an Polytechnic University,
Xi'an, Shaanxi 710048, China
e-mail: dingding0605@126.com

Juan J. Nieto

Departamento de Análisis Matemático,
Facultad de Matemáticas,
Universidad de Santiago de Compostela,
Santiago de Compostela 15782, Spain
e-mail: juanjose.nieto.roig@usc.es

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 27, 2019; final manuscript received April 30, 2019; published online June 10, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(9), 091001 (Jun 10, 2019) (12 pages) Paper No: CND-19-1032; doi: 10.1115/1.4043725 History: Received January 27, 2019; Revised April 30, 2019

Fractional stochastic evolution equations often arise in theory and applications. Finding exact solutions of such equations is impossible in most cases. In this paper, our main goal is to establish the existence and uniqueness of mild solutions of the equations, and give a numerical method for approximating such mild solutions. The numerical method is based on a combination of subspaces decomposition technique and waveform relaxation method, which is called a frequency decomposition waveform relaxation method. Moreover, the convergence of the frequency decomposition waveform relaxation method is discussed in detail. Finally, several illustrative examples are presented to confirm the validity and applicability of the proposed numerical method.

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Figures

Grahic Jump Location
Fig. 1

Euler approximation solution and analytical solution for Eq. (42)

Grahic Jump Location
Fig. 2

Implicit Euler–Maruyama approximation solution for α = 0.5, λ = 2, μ = 1, and T =1

Grahic Jump Location
Fig. 3

Analytical solution and implicit Euler–Maruyama numerical solution for Eq. (44) for α = 1, λ = 2, μ = 1, u0 = 1, and T =1

Grahic Jump Location
Fig. 4

Numerical solution obtained by implicit Euler–Maruyama method for α = 0.6

Grahic Jump Location
Fig. 5

The error between waveform relaxation solution and the numerical solution obtained by implicit Euler–Maruyama method for α = 0.6 with iterative numbers k

Grahic Jump Location
Fig. 6

Solutions of system (47) with standard Brownian noise

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