One of the most used branches of image processing is image improvement through restoration and denoising. The goal of image denoising is to removing the noise from the observed noisy image and to get some images as close as possible to the original image. Besides noise removal ability, another important requirement for image denoising procedures is to preserving the original image structures in the whole of the process. There are several tools used for image denoising which we focus on one the most important approaches i.e. a method based on the partial differential equation (PDE). Among various PDEs, some evolution equations such as parabolic equations were more used in the denoising process. However, hyperbolic equations can improve quality of the revealed edges and thus quality of the image is preserved better than parabolic equations. Perona and Malik were the first ones who offered the anisotropic diffusion equation for the image denoising process. After a while, Catté et al provided a well-defined version of that equation by changing the diffusion function. The problem with this type of equation was the “staircase effect” on the image. In this regard, a fourth-order PDE was suggested by You and Kaveh. But some new problems were emerged by solving that equation. Some small white and black spots were appeared on the image. After a gap, Ratner and Zeevi considered the image as an elastic sheet placed in a damping environment and provided the telegraph-diffusion equation which is a parabolic-hyperbolic one.The advent of the fractional derivatives leads to generate a new calculus entitled “fractional calculus”. Then fractional partial differential equations were developed and applied to model some problems. Bai and Feng were the first ones used fractional PDEs for image denoising. Recently, Zhang et al proposed a space-fractional telegraph equation for image denoising which we focused on it in this thesis. We provide some preliminaries from the real analysis, functional analysis and fractional calculus which we need them in the sequel. Then, we investigate an initial boundary value problem on the basis of a telegraph equation i.e. a parabolic-hyperbolic partial differential equation. Existence and uniqueness theorems related to the solution of that problem is represented and dealt with. This problem is applied for image denoising by using an explicit finite difference scheme. After that we deal with the effect of the fractional derivatives by applying a space fractional telegraph-diffusion equation. This equation interpolates between the second and the fourth order anisotropic diffusion equations using spatial fractional derivatives. We represent a well-posedness theorem related to the existence, uniqueness, and stability of the solution of the model. A numerical method based on the finite difference method and fast Fourier transform is used for solving the problem and denoising some images. Some numerical tests in comparison with the literature imply that the space fractional telegraph-diffusion equation can be considered as a valuable tool for image denoising.