A New Fractional Order School Academic Performance Model and Numerical Solutions

Abstract

Academic achievement is defined as the degree to which a student has achieved a learning goal. It is typically measured through the utilisation of examinations, continuous assessments and grade point averages. The student’s apprehension of failure can result in the accumulation of stress over time, which can consequently lead to a decline in academic achievement. Conversely, factors such as inadequate cognitive abilities, negative parental influence, familial circumstances and the physical and mental health of the child have been identified as the primary contributors to academic achievement. The present study proposes a novel fractional order mathematical model of academic achievement, comprising three compartments: students with above average achievement (S ), students with average achievement (M) and students with below average achievement (B). The Caputo derivative definition was employed as the fractional derivative and a stability analysis of the fractional model was conducted. Numerical solutions were obtained via the Generalized Euler Method and their graphs were drawn.

Keywords

• Fractional Order School Academic Performance Model
• Mathematical Modelling
• Generalised Euler Method
• Caputo Derivative
• Stability Analysis.

References

1. Al Khatib S.A., Meta-Cognitive Self-Regulated Learning and Motivational Beliefs as Predictors of College Students’ Performance, International Journal for Research in Education, 27, 57-72, 2010.
2. An X., Hannum E., Sargent T., Teaching quality and student outcomes: Academic achievement and educational engagement in rural northwest China, China: An International Journal, 5(2), 309-334, 2007.
3. Acay B., Inc M., Mustapha U.T., Yusuf A., Fractional dynamics and analysis for a lana fever infectious ailment with Caputo operator, Chaos, Solitons and Fractals, 153, 111605, 2021.
4. Alderman M.K., Motivation for Achievement: Possibilities for Teaching and Learning, Routledge, 2013.
5. Alivernini F., Lucidi F., Relationship between social context, self-efficacy, motivation, academic achievement, and intention to drop out of high school: A longitudinal study, The Journal of Educational Research, 104(4), 241-252, 2011.
6. Bansal S., Thind S.K., Jaswal S., Relationship between quality of home environment, locus of control and achievement motivation among high achiever urban female adolescents, Journal of Human Ecology, 19(4), 253-257, 2006.
7. Barron K.E., Harackiewicz J.M., Achievement Goals and Optimal Motivation: A Multiple Goals Approach. In C. Sansone and J.M. Harackiewicz (Eds.), Intrinsic and Extrinsic Motivation: The Search for Optimal Motivation and Performance, Academic Press, 229-254, 2000.
8. Bilgil H., Yousef A., Erciyes A., Erdin¸c ¨U., ¨ Ozt¨urk Z., A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event, Journal of Computational and Applied Mathematics, 425, 115015, 2023.

9. Bong M., Between-and within-domain relations of academic motivation among middle and high school students: Self-efficacy, task-value, and achievement goals, Journal of Educational Psychology, 93(1),23-34, 2001.
10. Bouffard T., Bouchard M., Goulet G., Denoncourt I., Couture N., Influence of achievement goals and self-efficacy on students’ self-regulation and performance, International Journal of Psychology, 40(6), 373-384, 2005.
11. Brauer F., Castillo-Chavez C., Feng Z., Mathematical Models in Epidemiology, Springer, 2019.
12. Kermack W.O., McKendrick A.G., A contribution to the mathematical theory of epidemics, in: Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 115 (772), 700-721, 1927.
13. Öztürk Z., Bilgil H., Sorgun S., Stability analysis of fractional PSQp smoking model and application in Turkey, New Trends in Mathematical Sciences, 10(4), 54-62, 2022.
14. Öztürk Z., Bilgil H., Sorgun S., Application of fractional SIQRV model for SARS-CoV-2 and stability analysis, Symmetry, 15(5), 1048, 2023.
15. Öztürk Z., Bilgil H., Sorgun S., A new application of fractional glucose-insulin model and numerical solutions. Sigma Journal of Engineering and Natural Sciences, 42(2), 407-413, 2024.
16. Öztürk Z., Yousef A., Bilgil H., Sorgun S., A Fractional-order mathematical model to analyze the stability and develop a sterilization strategy for the habitat of stray dogs, An International Journal of Optimization and Control: Theories and Applications, 14(2), 134-146, 2024.
17. Podlubny I., Fractional Differential Equations, Academy Press, 1999.
18. Sharma P., Sharma M., Relationship between self-esteem and academic achievement of secondary school students, Elementary Education Online, 20(1), 3208-3212, 2021.
19. Singh H., Kumar D., Baleanu D., Methods of Mathematical Modelling: Fractional Differential Equations, CRC Press, 2019.
20. Usaini S., Mustapha U.T., Sabiu S.M., Modelling scholastic underachievement as a contagious disease, Mathematical Methods in the Applied Sciences, 41(18), 8603-8612, 2018.
21. Uzun K., Karataş Z., Predictors of academic self efficacy: Intolerance of uncertainty, positive beliefs about worry and academic locus of control, International Education Studies, 13(6), 104-116, 2020.
22. Uzun P.Y., Uzun K., Koca İ., The effect of fractional order mathematical modelling for examination of academic achievement in schools with stochastic behaviors, An International Journal of Optimization and Control: Theories and Applications, 13(2), 244-258, 2023.
23. Veeresha P., Ilhan E., Prakasha D.G., Başkonuş H.M., Gao W., A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease, Alexandria Engineering Journal, 61(2), 1747-1756, 2022.
24. Yaro D., Omari-Sasu S.K., Harvim P., Saviour A.W., Obeng B.A., Generalized Euler method for modeling measles with fractional differential equations, International Journal of Innovative Research and Development, 4(4), 358-366, 2015.
25. Yusuf A., Mustapha U.T., Sulaiman T.A., Hincal E., Bayram M., Modeling the effect of horizontal and vertical transmissions of HIV infection with Caputo fractional derivative, Chaos, Solitons and Fractals, 145, 110794, 2021.
26. Yusuf A., Mustapha U.T., Sulaiman T.A., Hincal E., Bayram M., Modeling the effect of horizontal and vertical transmissions of HIV infection with Caputo fractional derivative, Chaos, Solitons and Fractals, 145, 110794, 2021.
27. Yusuf A., Qureshi S., Mustapha U.T., Musa S.S., Sulaiman T.A., Fractional modeling for improving scholastic performance of students with optimal control, International Journal of Applied and Computational Mathematics, 8(1), 37, 2022.