تعلیل احادیث صحیحین: امام ابو حاتم وأبو زرعہ كی آراء كا تنقیدی جائزہ

It is unique and high distinction of the Muslim Ummah that they have saved the teachings of Prophet Muhammad (PBUH). There have been high profile scholars who had dedicated their lives for the investigation of veracity and authenticity of Ḥadīths. They have been sifting the narrators of Ḥadīths through the myriad of resources to endorse or reject the authenticity of Ḥadīths. Amongst these eminent scholars, Imām Abū Ḥātim and Imām Abū Zur'ah, who made great contribution in the field of Elal Ḥadīth. Both criticized Aḥādīth some of which are quoted in Sahih Imām Bukhārī and Sahih Imām Muslim. Both books have always been taken as an authentic source for the veracity of Aḥādīth. However, they have pointed out a few such aspects which are vital for keeping the veracity and authenticity of Aḥādīth. This article is an attempt to study those Aḥādīth as quoted in the two above mentioned books and were criticized by Imām Abū Ḥātim and Abū Zur’ah in their book Elal Ḥadīths, by comprising with sayings of the concerned scholars. The most accurate aspect has been elaborated.

Family of Numerical Methods for the Solutions of Two Point Boundary-Value Problems Using Non-Polynomial Spline Functions

The research presented in this dissertation is aimed at the development of family of numerical solutions of Two Point Boundary-Value Problems (BVP) in all the branches of engineering sciences with higher precision. In such engineering related boundary-value problems the boundary conditions are specified at two points. The core of this dissertation is focused on the development of a new technique based on quartic non-polynomial spline functions, connecting spline functions values at "mid knots" for the numerical approximations to engineering BVPs and their corresponding values of the fourth-order derivatives. This new approach, which is recently being cited, is so developed that it leads to a family of numerical methods that may be used for computing approximations to the solution of a system of boundary-value problems of third and fourth-order associated with contact and sandwich beam problems. This research work is furthered on the development of a family of higher order numerical solutions to special non-linear third-order boundary-value problems. It is shown that the developed family of higher order methods gives better approximations in contrast to the existing numerical methods. It has further been shown that the existing second, fourth- order and higher order finite-difference and spline functions based methods become special cases of our new technique developed at mid knots presented in this dissertation. Results from the numerical experimentation of renowned problems are also given to illustrate applicability and efficiency of the newly developed family of numerical methods.