As we known, the classical heat transfer equation ∂u/∂t=∆u describes heat transfer in a homogeneous medium. Meanwhile, the time-fractional diffusion equation ∂_t^α u=∆u where α ϵ (0,1) and ∂_t^α is the fractional derivative in the Caputo sense, can be used to represent  the singular diffusion (small diffusion pattern) caused by particle adhesion and trapping. Probabilistically, this equation concerns non-Markov memory processes. Our aim is to study the properties of solutions of time-fractional diffusion equations such as existence, uniqueness of solutions, numerical simulations and study of asymptotic properties of solutions.

- Showed the asymptotic behaviour of solutions to time-fractional elliptic equations driven by a multiplicative white noise in the mean square sense.
- Showed the asymptotic stability of the trivial solution of scalar nonlinear fractional differential equations with linearly dominated delay.
The first result above is a main content in the published paper:
[1] Hoàng Thế Tuấn, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete and Continuous Dynamical Systems - Series B, 26 (2021), no. 3, pp. 1749-1762.

The second result above is a main content in the published paper:
[2] Hoàng Thế Tuấn, Stefan Siegmund, Stability of scalar nonlinear fractional differential equations with linearly dominated delay. Fractional Calculus and Applied Analysis, 23 (2020), no. 1, pp. 250-267.

- By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov-Perron operator defined from this representation of solutions, we have shown the asymptotic behaviour of solutions to time-fractional elliptic equations driven by a multiplicative white noise in the mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of solutions to their equilibrium point.
- We have established estimations for general Mittag-Leffler type functions
and use them to describe the asymptotic behavior of solutions of scalar fractional order differential equations with delays in both cases (the vector field is linear or nonlinear) by the linearization method.

Scientific papers in referred journals: 02 published papers are
[1] Hoàng Thế Tuấn, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete and Continuous Dynamical Systems - Series B, 26 (2021), no. 3, pp. 1749-1762.
[2] Hoàng Thế Tuấn, Stefan Siegmund, Stability of scalar nonlinear fractional differential equations with linearly dominated delay. Fractional Calculus and Applied Analysis, 23 (2020), no. 1, pp. 250-267.
- Educational activities: Supervising sucessfully 02 mather students are
Le Thi Phuong Thuy and Tong Thu Trang (Thai Nguyen University of Education-Thai Nguyen University)
- A report on the contents and results of the project.