Strongly magnetized stellar objects have magnetospheres characterized by such activitiesrnthat define the geometry of the inner edge of the disk as well as control the inflow of matterrnto the NS surface itself. Of all possible components of the surface magnetic field of the centralrnobject (neutron star-NS), we have only considered the quadrupole term to investigaternwhat ever is going on nearer to the surface of the NS. There is a highly important differencernof the accretion flow in a quadrupolar and dipolar magnetic fields. The dipolar magneticrnfield will in the end always present a barrier to the accretion flow since the field lines arernperpendicular to the plane of the disk, but the quadrupolar magnetic field will in the simplestrncase lie in the plane of the disk, and thus it will rather channel the accretion flow allrnthe way down to the stellar equator. This work involves a mathematical treatment of anrnaccretion disk around a magnetized star. In order to define the disk structure magnetohydrodynamicrn(MHD) equations are solved in cylindrical coordinates. For the detailed resultsrnan ordinary differential equation (ODE) derived from the angular momentum equation isrnnumerically solved. So, both Keplerian and non-Keplerian cases of thin accretion disk arernsolved. Further, introductory work on slim disk is included as a part of this work. Thernresults of our analysis indicate the existence of two different regions: a super-Keplerian innermostrnregion and a broader sub-Keplerian outer region. The effects of stellar and toroidalrnmagnetic fields on the variations of viscosity, temperature and density have also been studied.rnWe have identified the nature of the inner portion of an accretion disk. The velocityrnof the transition varies from corotating magnetospheric boundary to super-Keplerian forrnlow density inner most portion of accretion disk, that extends from 0:5RM to the peak andrnthen to sub-Keplerian. Our results are applicable to accreting astrophysical systems suchrnviiirnixrnas neutron stars (NSs) and white dwarfs (WDs). It can also explain observational resultsrnnot yet fully backed with theories