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The purpose of the present article is to obtain the similarity solution for the shock wave generated by a piston propagating in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows.

The Lie group theoretic method given by Sophus Lie is used to obtain the similarity solution in the present article.

Similarity solution with exponential law shock path is obtained for both ideal and nonideal gas cases. The effects on the flow variables, density ratio at the shock front and shock strength by the variation of the shock Cowling number, adiabatic index of the gas, gravitational parameter and nonidealness parameter are investigated. The shock strength decreases with an increase in the shock Cowling number, nonidealness parameter and adiabatic index, whereas the strength of the shock wave increases with an increase in gravitational parameter.

Propagation of shock wave with spherical geometry in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows has not been studied by any author using the Lie group theoretic method.

The purpose of this study is twofold. Firstly, the study has investigated the changing scenario of gender bias in households' education expenditure and the socioeconomic factors responsible for it. Secondly, the study has estimated the inequality in education expenditure for the male and female students and determined the significance of socioeconomic variables in gender discrimination.

To address the above-mentioned issues, this paper has used the unit-level data of NSSO 52nd, 64th, 71st and 75th rounds from 1995–1996 to 2017–2018. The log linear regression model is applied to estimate factor impending average education expenditure dynamics. The Oaxaca–Blinder Decomposition method has been employed to measure gender discrimination, and the Lorenz curve and Gini coefficient are used to assess inequality among girls experiencing prejudice.

The study has discovered an gender bias in education expenditure against females during the study period in India. Further, it has been found that gender discrimination against girl students is decreasing. Moreover, the factors such as age, religion, castes, MPCE (income quantile), type of institution, present enrolment and type of education are responsible for this gender differences.

This paper uses 20 years of household-level data for study and suggests that discriminatory behaviour of households and credit constraints of the underdeveloped countries prevent investment in girl's education. Therefore, the state must pay for education of girls by offering scholarships and free or heavily subsidized education. In addition to this, awareness programs for gender equality should also be implemented by the government, especially in rural areas.

The peer-review history for this article is available at: https://publons.com/publon/10.1108/IJSE-08-2022-0537.