TERMINAL VALUE PROBLEM FOR STOCHASTIC FRACTIONAL EQUATION WITHIN AN OPERATOR WITH EXPONENTIAL KERNEL

Nguyen Duc Phuong, Luu Vu Cam Hoan, Dumitru Baleanu and Anh Tuan Nguyen
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Nguyen Duc Phuong: Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
Luu Vu Cam Hoan: Faculty of Basic Science, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam
Dumitru Baleanu: Department of Mathematics, Cankaya University, Ankara, Turkey4Institute of Space Sciences, Magurele–Bucharest, Romania5Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
Anh Tuan Nguyen: Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam7Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

FRACTALS (fractals), 2023, vol. 31, issue 04, 1-16

Abstract: In this paper, we investigate a terminal value problem for stochastic fractional diffusion equations with Caputo–Fabrizio derivative. The stochastic noise we consider here is the white noise taken value in the Hilbert space W. The main contribution is to investigate the well-posedness and ill-posedness of such problem in two distinct cases of the smoothness of the Hilbert scale space Wν (see Assumption 3.1), which is a subspace of W. When Wν is smooth enough, i.e. the parameter ν is sufficiently large, our problem is well-posed and it has a unique solution in the space of Hölder continuous functions. In contract, in the different case when ν is smaller, our problem is ill-posed; therefore, we construct a regularization result.

Keywords: Ill-Posed Problem; Fractional Stochastic Equation; Hilbert Scales; Caputo–Fabrizio Derivative (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23400625

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