مولانا ابو یوسف محمد شریف محدث کوٹلوی کی خدمات حدیث شریف

In this paper, based on the primary sources, an attempt has been made to discuss the contribution and services of Mawlānā Abū Yūsuf Muḥammad Sharīf (d. January 1951) commonly known as Faqīh-i-Ā‘ẓam and Muḥaddith-i-Koṯalwī in the field of Ḥadīth, the second most important source of Islamic Law. Mawlānā Muḥaddith-i-Koṯalwī having a deep insight in the Science of Ḥadīth has written extensively on this facet of Islamic learning. Mawlānā Abū Yūsuf Muḥammad Sharīf Muḥaddith-i-Koṯalwī through his fatāwā, articles, sermons and books done a great service for the Science of Ḥadīth. He has explained the five pillars of Islam and other rituals in the light of different āḥādīth. Being a Ḥanafī ‘ālim, Mawlānā Muḥaddith-i-Koṯalwī also proved that the Fiqh-i-Ḥanafī, the most popular fiqh among the South Asian Muslims, is in accordance with the authentic āḥādīth. He has also discussed the difference between a Ḥadīth and Sunnah.

The Numerical Solution of Integral and Integro-Di Erential Equations by Wavelet Collocation Method

This thesis presents some new algorithms for the numerical solution of IEs, IDEs and PIDEs. These new algorithms are based on WCM. The focus is upon two types of wavelets namely HW and LLMWs both having compact support. HW has already been applied for numerical solution of IEs and IDEs by a few re- searchers and the numerical results can be found in the existing literature. One draw back of the HWCM is its slow convergence. In order to improve the convergence rate we have applied LLMWCM to IEs and IDEs. In most of the cases of the IEs and IDEs we obtained better results with LLMWCM as compare to HWCM. We also extended the existing HWCM for IEs and IDEs in the literature as well as the newly developed LLMWCM for IEs and IDEs in this thesis to numerical solution of some particular type of PIDEs. Both the HWCM and LLMWCM are thoroughly investigated in this thesis for numer- ical solution of di erent types of IEs, IDEs and PIDEs including Fredholm, Volterra and Fredholm-Volterra IEs, IDEs and PIDEs. The methods are also investigated for higher- order IDEs. An important characteristics of the method is that it can be applied to both linear and nonlinear problems. In the present work, in case of solving linear IEs, IDEs and PIDEs, the resulting systems will be solved by using Gauss elimination method, while for nonlinear case, we use Newton''s or Broyden''s method. All methods are implementing and testing by a computer programming software MATLAB. Several test problems are performed in order to verify the accuracy and e ciency of the present methods. The experimental rates of convergence and MAEs have been calculated for di erent number of CPs. The approximate solution of these test problems are compared with exact solutions and also the numerical results are compared with other well established methods to show better e ciency, accuracy and simple applicability of the newly developed methods. A comparative study of HWCM and LLMWCM is also perform in this thesis.