Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to fractional single-phase lag heat equation. This model contains terms $\cal{D}_t^\alpha(u_t)$ $\cal{D}_t^\alpha u $ (with $\alpha \in(0,1)$), where $\cal{D}_t^\alpha$ denotes Caputo derivative in time constant order $\alpha\in(0,1)$. We consider homogeneous Dirichlet boundary data for temperature. rigorously show unique under low regularity assumptions on data. Our main strategy is use variational formulation semidiscretisation based Rothe's method. obtain priori estimates discrete solutions convergence Rothe functions solution. The approach employed problem. also one-dimensional problem derive representation formula establish bounds explicit its by extending properties multinomial Mittag-Leffler function.