Multiple studies have addressed the blow-up time of the Fujita-type equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In this paper, we propose a method for obtaining a computable and mathematically rigorous inclusion of the L^2(varOmega ) blow-up time of a solution to the Fujita-type equation subject to initial and Dirichlet boundary conditions using a numerical verification method. More specifically, we develop a computer-assisted method, by using the numerically verified solution for nonlinear parabolic equations and its estimation of the energy functional, which proves that the concerned solution blows up in the L^2(varOmega ) sense in finite time with a rigorous estimation of this time. To illustrate how our method actually works, we consider the Fujita-type equation with Dirichlet boundary conditions and the initial function u(0,x)=frac{192}{5}x(x-1)(x^2-x-1) in a one-dimensional domain varOmega and demonstrate its efficiency in predicting L^2(varOmega ) blow-up time. The existing theory cannot prove that the solution of the equation blows up in L^2(varOmega ). However, our proposed method shows that the solution is the L^2(varOmega ) blow-up solution and the L^2(varOmega ) blow-up time is in the interval (0.3068, 0.317713].