General method to solve the heat equation

General solutions of the heat equation are presented in terms of the Koopman-Darmois family of exponential functions, which include both the separable solution and the fundamental solution. In particular, we derive a new closed-form solution, which may not be obtained via the separation of variables or via an integral transform. It is demonstrated that the new solution describes the time evolution of the distribution of random walkers under an absorbing boundary.