{'fraction_non_alphanumeric': 0.017857142857142856, 'fraction_numerical': 0.0, 'mean_word_length': 5.434426229508197, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Comparative Study of Numerical Techniques and Homotopy Perturbation Method for Solving Parabolic Equation and Nonlinear Equations

In this paper, the homotopy perturbation method has been employed to obtain approximate analytic solutions of parabolic (heat) PDE and nonlinear equations. The procedure of the method is systematically outlined. The results obtained by HPM to solve a one-dimensional heat equation have been compared with those obtained by using the method of separation of variables and numerical techniques (Schmidt, Crank-Nicholson and DuFort- Frankel method) available in the literature. The results of HPM are highly efficient. The graphical interpretation for displacement with respect to length with fix time for various techniques is used to solve the problem.