Exponential Convolution Quadrature for Nonlinear Subdiffusion Equations with Nonsmooth Initial Data

Li, Buyang; Ma, Shu

doi:10.1137/21m1421386

Cited by 29 publications

(8 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we analyze the following time-fractional diffusion equation [20][21][22][23][24][25][26][27][28] equipped with variable-order fractional substantial derivative…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

2022

View full text Add to dashboard Cite

A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed.

show abstract

“…In this paper, we analyze the following time-fractional diffusion equation [20][21][22][23][24][25][26][27][28] equipped with variable-order fractional substantial derivative…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

2022

View full text Add to dashboard Cite

show abstract

“…(1.3) Although high-accuracy numerical methods have been used to solve the deterministic counterpart of the model (1.1) (see e.g., [5,17,24]), the accuracy of these numerical methods will be destroyed when encountering the interaction between the fractional derivative and stochastic noise, which motivates the studies on numerical methods for the model (1.1) or its special cases. When α = 1, the model (1.1) degenerates into a space-fractional SPDE, and the semi-implicit Euler method is modified in [19] for the case γ = 0.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and Mittag--Leffler Euler integrator for space-time fractional SPDEs with fractionally integrated additive noise

Dai¹,

Hong²,

Sheng³

2022

Preprint

View full text Add to dashboard Cite

This paper considers the space-time fractional stochastic partial differential equation (SPDE, for short) with fractionally integrated additive noise, which is general and includes many (fractional) SPDEs with additive noise. Firstly, the existence, uniqueness and temporal regularity of the mild solution are presented. Then the Mittag-Leffler Euler integrator is proposed as a time-stepping method to numerically solve the underlying model. Two key ingredients are developed to overcome the difficulty caused by the interaction between the time-fractional derivative and the fractionally integrated noise. One is a novel decomposition way for the drift part of the mild solution, named here the integral decomposition technique, and the other is to derive some fine estimates associated with the solution operator by making use of the properties of the Mittag-Leffler function. Consequently, the proposed Mittag-Leffler Euler integrator is proved to be convergent with order min{ 1 2(Ω, H)-norm for the nonlinear case. In particular, the corresponding convergence order can attain min{α + αr 2β + (γ − 1 2 ) + − ε, α + γ − ε, 1} for the linear case.

show abstract

“…The subdiffusion equations, which can model the sublinear growth of mean squared particle displacement in transport processes, have attracted much interests of physicists, engineers and applied mathematicians in developing highly accurate and efficient computational methods and numerical analysis. The regularity of the solution of (1) has been fully understood in literature [14,16,29]. In this paper, we assume that the solution u(t) = u(•, t) of (1) satisfies the following finite regularity:…”

Section: Introductionmentioning

confidence: 99%

“…where C > 0, σ > 0, 0 ≤ ℓ ≤ p + 1, p and q are nonnegative integers depending the smoothness of the initial data and the source term f (x, t), and u (ℓ) (t) denotes the ℓ-th order time derivative of u(t) = u(•, t); see [16], in which σ = α can be obtained. From (3), one will find that there exists the initial singularity of the solution to (1).…”

Section: Introductionmentioning

confidence: 99%

Core Content Selection and Textbook Design for High School Mathematics

Zhang¹,

Zhou²,

Wang³

et al. 2021

School Mathematics Textbooks in China

View full text Add to dashboard Cite

This paper is devoted to studying the following fractional L 2 -critical nonlinear Schrödinger equation≥ 0 is an external potential. We obtain normalized L 2 -norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold a * > 0 such that (1.4) has minimizers for 0 < a < a * , and minimizers do not exist for any a > a * . For the case of a = a * , it gives a fact that the existence and non-existence of minimizers depend strongly on the value of V (0). Especially for V (0) = 0, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as a ր a * . Applying implicit function theorem, the uniqueness of minimizers is also proved for a > 0 small enough.

show abstract

Exponential Convolution Quadrature for Nonlinear Subdiffusion Equations with Nonsmooth Initial Data

Cited by 29 publications

References 31 publications

Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

Well-posedness and Mittag--Leffler Euler integrator for space-time fractional SPDEs with fractionally integrated additive noise

Core Content Selection and Textbook Design for High School Mathematics

Contact Info

Product

Resources

About