{'fraction_non_alphanumeric': 0.016166281755196306, 'fraction_numerical': 0.0, 'mean_word_length': 5.2, '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}}
Numerical solution of one-dimensional biharmonic equations using Haar wavelets

In this paper, an operational matrix of integration based on Haar wavelets is introduced, and a procedure for applying the matrix to solve biharmonic equations is formulated. The technique can be used for solving boundary value problems of one-dimensional biharmonic equations. The efficiency of the proposed method is tested with the aid of an example.