A differential equation along with a set of additional constraints (called boundary conditions) form a boundary value problem. Boundary value problems (linear or nonlinear) are found mostly in engineering, applied mathematics and physical sciences as well. Although, few numerical algorithms are found in solving the higher order boundary value problems using orthogonal polynomials. Since polynomials play a vital role in computing the numerical solutions of the differential equations therefore, the research, conducted during my PhD programme, concerns with the study of the numerical solutions of higher order linear boundary value problems. The main objectives of the research are to develop Galerkin technique for solving special eighth, tenth and twelfth order linear boundary value problems using Legendre polynomials, and numerical approximations for solving the system of Fredholm integro–differential equations and the system of differential equations using Laguerre polynomials. The numerical schemes, developed, have been compared with the existing methods, which shows the higher accuracy of the schemes.
جے توں آویں میرے کول دساں دل دے دکھڑے پھول چھڈ گیوں مینوں یاری لا کے مینوں کردے لوگ مخول ہک واری جے ہس کے بولیں جندڑی دیواں گھول مگھول قدر وفا دی کجھ نہیں کیتی مٹی دتے موتی رول اوڑک اوس نوں کٹنا پوسی جس دے گل وچ پے گیا ڈھول دکھی دی گل دکھی سن دا دکھیاں اگے دِل پھرول جھوٹ نوں پھیتی پھیتی کردا سچی گل دا ہکو بول موت نوں یاد کراں ہر ویلے جندڑی رہندی ڈانواں ڈول
Fatwa is the easiest tool to explore the religious status of any newer problem faced by the public, though it is non-binding theory by its nature still people rely on that and get it from any renowned scholar to for the sake of awareness and to support their religious background. The ultimate result in fatwa depends on the methodology adopted in it. Alike methodologies of tashdeed and tasahul the precautionary methodology has also been practiced in the earlier stages of Islam when the issues were not much complicated but in the era we live in the newer are much complicated which need this methodology to be practiced. This study throws light on the concept of such methodology, on its reasons and principles. It concludes that the precautionary methodology of fatwa is based on shariah rules and its is admissible by its nature but when this methodology becomes the way to save a compulsory thing then it becomes compulsory and when adoption of this methodology becomes to strengthen a Haram action then adoption of this methodology becomes Haram.
Higher education system plays significant role in the development and progress of different economies. Pakistani higher education system is showing lacking performance due to multiple challenges.According to GCI report 2014-15, Pakistan ranked 129 in higher education showing poor performance. Higher performance of HEIs is required as mentioned in the meeting of 3rd Islamic countries vice-chancellors conference held in Islamabad in 2015. There is an urgent need to find how HEIs performance can be improved. The study is performed to find relationship between organizational health (OH) and perceived performance of HEIs of Pakistan. Another objective of the study is to analyze impact of different organizational health dimensions on HEIs performance. Data is collected from top twenty universities as per HEC ranking 2014. Main goal of the study is achieved by finding a significant positive relation between organizational health and organizational performance of HEIs. The study also identified institutional reputation and resource utilization as strongest and weakest dimensions of OH respectively.