Recovery of weight function of one-dimensional mixed distributed-fractional problem

Gaussian models, traditionally employed to describe transport phenomena, often fail to capture anomalous processes, prompting the need for advanced mathematical frameworks. The continuous-time random walk (CTRW) approach, utilizing distributed order fractional (DOF) derivatives defined as $\mathbb{D}^\nu g(t) = {\int}_0^1 \partial^{\alpha}_t g(t)\nu(\alpha)d\alpha$, provides a robust alternative for modeling anomalous transport. This paper addresses the reconstruction of mixed DOF weight functions $\nu(\alpha) = \sum_{j = 1}^I p_j \delta(\alpha - \alpha_j) + \mu_\nu(\alpha), I \in \mathbb{N}$, combining discrete and continuous components, in fractional partial differential equations. We propose an algorithm to identify $\nu$ and the function $u_\nu$ satisfying $\mathbb{D}^\nu u_\nu = \frac{\partial^2 u_\nu}{\partial x^2}$, with boundary conditions $u_\nu(0,t) = g(t)$, $u_\nu(1,t) = h(t)$, initial condition $u_\nu(x,0) = 0$, and overdetermined condition $u_\nu(x_0,t) = \phi(t)$. We establish the uniqueness of $\nu$ and use the nonlinear Tikhonov regularization method and develop numerical schemes to recover it from noisy data $g_\varepsilon$, $h_\varepsilon$, and $\phi_\varepsilon$. Our method enhances the modeling of complex anomalous transport, such as ultraslow diffusion, with numerical results validating its effectiveness.