*1.1. Fractional Diffusion Equations*

A fractional diffusion equation of order *α* ∈ (0, 1) is obtained by rewriting a normal diffusion equation in integral form as

$$u(\mathbf{x},t) + \int\_0^t A u(\mathbf{x},t)dt = \mathbf{u}\_0 + \int\_0^t f(\mathbf{x},t)dt, \quad (\mathbf{x},t) \in \Omega \times (0,T). \tag{1}$$

Then, replacing the first of right-hand side (RHS) integral of Equation (1) by a Riemann-Liouville fractional integral, *I<sup>α</sup>* of order 0 < *α* < 1, we get

$$u(\mathbf{x},t) + \int\_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} A u(\mathbf{x},t) dt = u\_0 + \int\_0^t f(\mathbf{x},t) dt, \quad (\mathbf{x},t) \in \Omega \times (0,T).$$

Now, differentiating the above equation on both sides with respect to *t*, we get the following fractional diffusion equation:

$$\begin{aligned} \partial\_t u + \partial\_t^{1-\mathfrak{a}} Au &= f(x,t) & \text{in} \quad \Omega \times (0,T), \\ u &= 0 & \text{on} \quad (\Gamma = \partial\Omega) \times (0,T), \\ u(\cdot,0) &= u\_0 & \text{in} \quad \Omega. \end{aligned} \tag{2}$$

If *α* = 1, then Equation (2) is a classical diffusion equation. Equation (2) with 0 < *α* < 1 is called the fractional diffusion equation. These equations appear in the model of anomalous diffusion in heterogeneous media. Anomalous diffusion is one of the most ubiquitous phenomena in nature; it has been observed in various fields of physical sciences, for example, surface growth, transport of fluid in porous media, two-dimensional rotating flow and diffusion of plasma. Because of such anomalies, the classical diffusion models can not be used to study the dynamics of such systems. In this situation, fractional derivatives extend the help and play a crucial role in characterizing such diffusion. The model corresponding to such derivative is called a fractional partial differential equation. From the continuous time random walk (CTRW) model, Metzler and Klafter [1] derived Equation (3) with 0 < *α* < 1 as a macroscopic model.

## *1.2. Impulsive Partial Differential Equations*

Impulsive partial differential equations are a very important class of differential equations. These equations arise from the modelling of various real world processes having memory and are subject to short time fluctuations. The theory of impulsive differential equation is very rich and wide. It is mainly due to the fact that the it inherit intrinsic difficulties of the problems. These kinds of equations have lots of applications in different branches of Science and Engineering. These kinds of equations arise naturally from several physical and natural processes like earthquakes and pulse vaccination strategy. For more information, we refer to [2–4] and references therein. For more theoretical work, one can see the interesting book by Bainov and Simeonov [5]. The authors Shun et al. in [6] consider second-order impulsive Hamiltonian systems and established the existence of infinitely many solutions.
