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A numerical method for solving a class of fractional partial differential equations with variable coefficients based on Legendre polynomials is proposed. A fractional order operational matrix of Legendre polynomials is also derived. The initial equations are transformed into the products of several matrixes by using the operational matrix. A system of linear equations is obtained by dispersing the coefficients and the products of matrixes. Only a small number of Legendre polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.

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The revolution in web-based technologies has enriched pedagogical practices and motivated scholars to address learners’ positive and negative emotions in the web-based language learning environment. In this study, we first examined the psychometric properties of the Foreign Language Enjoyment Scale (FLES) and then developed the Online Foreign Language Enjoyment Scale (OFLES). We adopted a mixed-method approach using a sample of 383 first language Chinese EFL undergraduates. In stage one of the research, exploratory factor analysis and confirmatory factor analysis supported an 11-item and 4-factor OFLES structure with an ideal model fit. The four sub-domains of the new enjoyment construct were OFLES- Teacher, Private, Interaction, and Competence. The Teacher domain accounted for most variance. In stage two, the qualitative analyses of feedback on an open-ended question concerning enjoyable episodes from 56 of the 383 participants revealed various sources of enjoyment for Chinese university EFL learners attending online English courses. Our findings contribute to an emerging wave of research examining the cross-contextual application of the FLES and provide important pedagogical implications for L2 practitioners and researchers. We discuss suggestions for future research.

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