Crank–Nicolson method for solving uncertain heat equation

Abstract

For usual uncertain heat equations, it is challenging to acquire their analytic solutions. A forward difference Euler method has been used to compute the uncertain heat equations’ numerical solutions. Nevertheless, the Euler scheme is instability in some cases. This paper proposes an implicit task to overcome this disadvantage, namely the Crank–Nicolson method, which is unconditional stability. An example shows that the Crank–Nicolson scheme is more stable than the previous scheme (Euler scheme). Moreover, the Crank–Nicolson method is also applied to compute two characteristics of uncertain heat equation’s solution—expected value and extreme value. Some examples of uncertain heat equations are designed to show the availability of the Crank–Nicolson method.

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

“…The extensive adoption of the Crank-Nicolson method for solving heat transfer equations is grounded in the observation that, in certain instances, numerical solutions obtained through the explicit finite difference method display instabilities that can be resolved by employing the Crank-Nicolson method for numerical computations. The Crank-Nicolson method has been demonstrated to possess greater stability compared to the explicit finite difference method and is capable of determining both the expected and extreme values within the context of heat transfer equations [11,12,13].…”