Applied Mathematics and Computation 362 (2019) 124458 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On Cauchy problem for nonlinear fractional differential equation with random discrete data Nguyen Duc Phuong a,b , Nguyen Huy Tuan c,∗ , Dumitru Baleanu d,e , Tran Bao Ngoc f a Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam b Department of Mathematics and Computer Science VNUHCM - University of Science, 227 Nguyen Van Cu Str., Dist. 5, HoChiMinh City, Vietnam c Applied Analysis Research Group Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam d Department of Mathematics, Cankaya University, Ankara, Turkey e Institute of Space Sciences, Magurele, Bucharest, Romania f Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam a r t i c l e i n f o Keywords: Fractional derivative ill-posed problem Elliptic equation Random noise Regularized solution a b s t r a c t This paper is concerned with finding the solution u(x, t) of the Cauchy problem for non- linear fractional elliptic equation with perturbed input data. This study shows that our forward problem is severely ill-posed in sense of Hadamard. For this ill-posed problem, the trigonometric of non-parametric regression associated with the truncation method is applied to construct a regularized solution. Under prior assumptions for the exact solu- tion, the convergence rate is obtained in both L 2 and H q (for q > 0) norm. Moreover, the numerical example is also investigated to justify our results. © 2019 Elsevier Inc. All rights reserved. 1. Introduction In this work, we focus on finding the solution for the following time fractional elliptic equation ∂ α t u + u = f (x, t , u), (x, t ) ∈  × (0, T ), (1) with the Cauchy condition and initial conditions  u(x, t ) = 0, (x, t ) ∈ ∂  × (0, T ), u(x, 0) = ρ (x), x ∈ , u t (x, 0) = ξ (x), x ∈ . (2) where  = (0, π ) ⊂ R is a bounded connected domain with a smooth boundary ∂ , and T is a given positive real number. The time fractional derivative ∂ α t u is the Caputo fractional derivative of order α ∈ (1, 2) with respect to t defined in [1–3] as follows ∂ α t u(x, t ) = 1 (2 − α )  t 0 (t − ω) (1−α) ∂ 2 ∂ω 2 u(x, ω)dω, (x, t ) ∈  × (0, T ), (3) ∗ Corresponding author. E-mail addresses: nguyenducphuong@iuh.edu.vn (N.D. Phuong), nguyenhuytuan@tdtu.edu.vn (N.H. Tuan), dumitru@cankaya.edu.tr (D. Baleanu), tranbaongoc@hcmuaf.edu.vn (T.B. Ngoc). https://doi.org/10.1016/j.amc.2019.05.029 0096-3003/© 2019 Elsevier Inc. All rights reserved.