On Cauchy problem for nonlinear fractional differential equation with random discrete data

Applied Mathematics and Computation 362 (2019) 124458

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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 Phuonga,b

, Nguyen Huy Tuanc,∗

, Dumitru Baleanud,e

,

Tran Bao Ngocf

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 L2 and Hq (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 ∈ ,

ut(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

∂ω2 u(x,ω)dω, (x,t) ∈  × (0, T ), (3)

∗ Corresponding author.

E-mail addresses: [email protected] (N.D. Phuong), [email protected] (N.H. Tuan), [email protected] (D. Baleanu),

[email protected] (T.B. Ngoc).

https://doi.org/10.1016/j.amc.2019.05.029

0096-3003/© 2019 Elsevier Inc. All rights reserved.