Context : Based on the single degenerate ( SD ) scenario , a super-Chandrasekhar mass model derived from the rapid rotation of a progenitor star may account for the over-luminous type Ia supernovae ( SNe Ia ) like SN 2003fg .
Chen & Li ( 2009 ) calculated a series of binary evolution and showed the parameter spaces for the super-Chandrasekhar mass model .
Pakmor et al .
( 2010 ) developed a equal-mass double degenerate ( DD ) model to explain sub-luminous SNe Ia like SN 1991bg .
But they did not show the evolution of the birth rate of these peculiar SNe Ia or did not compare them with absolute birth rates from observations .

Aims : We want to show the evolution of the birth rates of these peculiar SNe Ia based on their results , and compare the birth rates with observations to check whether these model may account for all the peculiar SNe Ia .

Methods : We carried out a series of binary population synthesis calculations and considered two treatment of common envelope ( CE ) evolution , i.e . \alpha -formalism and \gamma -algorithm .

Results : We found that the evolution of birth rate of these peculiar SNe Ia is heavily dependent on how to treat the CE evolution .
The over-luminous SNe Ia may only occur for \alpha -formalism with low CE ejection efficiency and the delay time of the SNe Ia is between 0.4 and 0.8 Gyr .
The upper limit of the contribution rate of the supernovae to all SN Ia is less than 0.3 % .
The delay time of sub-luminous SNe Ia from equal-mass DD systems is between 0.1 and 0.3 Gyr for \alpha -formalism with \alpha = 3.0 , while longer than 9 Gyr for \alpha = 1.0 .
The range of the delay time for \gamma -algorithm is very wide , i.e .
longer than 0.22 Gyr , even as long as 15 Gyr .
The sub-luminous SNe Ia from equal-mass DD systems may only account for no more than 1 % of all SNe Ia observed .

Conclusions : The super-Chandrasekhar mass model of Chen & Li ( 2009 ) may account for a part of 2003fg-like supernovae and the equal-mass DD model developed by Pakmor et al .
( 2010 ) may explain some 1991bg-like events , too .
In addition , based on the comparison between theories and observations , including the birth rate and delay time of the 1991bg-like events , we found that the \gamma -algorithm is more likely to be an appropriate prescription of the CE evolution of DD systems than the \alpha -formalism if the equal-mass DD systems is the progenitor of 1991bg-like SNe Ia .