This research aims to solve the nonlinear parabolic differential equation approximately which has not been previously solved analytically using the explicit forward time-centered space and the implicit generalized finite difference method which may give an actual and reliable approach for approximating the solution. In this paper, the approximate solution will be the first test of this type of equation, where, the results show that different cases depend on the convergent condition for each method. This approximate solution can be utilized to understand and analyze the properties of the equation and study the abilities to reach the optimal solution. Moreover, it can also be employed to examine the behavior of equations and to recognize their stability properties. The two methods show their accuracy through the numerical results that allow the solution for different cases: unsteady, singular, and blow-up solutions. Two examples are examined to present these types of solutions and study their behavior with illustrations of the methods supported by figures generated by MATLAB, along with the possibility the comparison of the results. Finally, the approximate solutions which are obtained from the implemented methods indicate that the technique used to determine the equation is accurate, leading to more accurate and efficient equation determination.