Rothe Time-discretization Method for Caputo Fractional Parabolic Equation

This study is concerned with the existence and uniqueness of weak solutions for a time-fractional parabolic partial differential equation with homogeneous conditions. The results are obtained using Rothe’s method, also called method of lines and by discretizing the time fractional derivative of order \(\boldsymbol{\alpha}\) between zero and one using the Caputo finite difference formula, which is a first-order approximation. By introducing a new function, some a priori estimates are obtained. It turns out that the proposed method is applicable. Consequently, we prove that the \(\boldsymbol{\alpha}\) -Rothe sequence generated by the proposed method converges weakly to a function \(u(x,t)\) that appears to be a solution to the given problem.