Error Analysis of the Modified Crank–Nicolson Method for 1D Heat Equations
DOI:
https://doi.org/10.55544/ijrah.6.5.20Keywords:
Heat equation, Crank–Nicolson method, finite difference method, Fourier sine series, numerical validation, absolute errorAbstract
This study investigates the numerical solution of one-dimensional heat equations using a Modified Crank–Nicolson method. The method is formulated by discretizing the governing partial differential equation in both space and time, resulting in a stable and efficient finite-difference scheme for obtaining approximate solutions. Several heat-equation problems with prescribed initial and boundary conditions are considered to evaluate the performance of the numerical approach. The accuracy of the numerical solutions is assessed by comparing the computed results with the corresponding exact solutions. Absolute errors are calculated at selected spatial and temporal points and presented in tabular form to provide a quantitative measure of the numerical accuracy. The results show that the Modified Crank–Nicolson method produces numerical solutions that closely agree with the exact solutions, yielding small absolute errors. These findings demonstrate the effectiveness and reliability of the method for the numerical treatment of one-dimensional heat equations.
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