ہیں فلک کی صدا یہ دو آنکھیں
رنگ، خوشبو، ادا یہ دو آنکھیں
شوخ آنکھوں کے وار کون سہے
دھڑکنوں کی قضا یہ دو آنکھیں
مفلسِ شہر پر نگہ ہو کوئی
آنسوئوں کی صدا یہ دو آنکھیں
ہیں زمانے حصار میں اِن کے
آئو دیکھو فضاؔ یہ دو آنکھیں
Dr Ghulam Mustafa Khan [d. 2005] can truly be said to be a teacher of teachers [ustadh al-asatadha]. In his long and distinguished academic career spanning over six decades, he brought into existence literally hundreds of teachers in his subject Urdu and also created in many of them a taste for critical research that resulted in the existence of a whole corpus of standard critical material. This will endure in academic circles for a long time. It was his common habit to encourage his students, especially those of an academic bent of mind, to enrol for a Ph.D. In the course of which he would proffer them all kinds of assistance even if he were not their research supervisor. As such, the names of Dr Najmul Islam, who succeeded him as Chairman, Department of Urdu, University of Sindh, stands out. Dr Najmul Islam was the editor of a scholarly research journal, Tehqiq, which appeared in over 20 volumes in his own lifetime. Each issue contained scores of critical essays of a very high academic standard including his own [Dr N. Islam’s]. There is also the name of Dr Hasrat Kasganjvi, who emerged as a creative artist and critic of merit later on authoring dozens of books of high academic standard. The list is long and distinguished and contains the names of the major teachers of Urdu language and literature at the main Pakistani universities. Apart from Urdu teachers, the thousands of persons who had the benefit of attending his lectures and courses are also evidence of his intellectual wide-ranging scholarship. Then there is the select circle of his murids who formed his circle of spiritual knowledge and who had proffered their religious devotion at his hands. To these he gave the benefit of his spiritual counsel and advice as well as the vast knowledge he possessed of the Islamic sciences in his table-talk. Among these persons, the name of ex-President Ziaul Haq is most prominent.
In this thesis a series solution to linear or nonlinear problems has been considered which are arising in the field of applied mathematics. In the first chapter of this dissertation a brief history of our problems and methods is given. In the second chapter, some basic definitions and concepts are given which have been used in our work. In the third chapter, we examined the motion of an incompressible unidirectional magneto-hydrodynamics (MHD) thin film flow of a third grade fluid over an oscillating inclined belt embedded in a porous medium. Moreover, heat transfer analysis has also been discussed in the present work. We modeled the proposed physical problem in terms of PDEs. The equations obtained through PDEs were then solved by two different methods i.e., Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM). Comparisons of these two methods for different time level were analyzed graphically. Comparison of the results shows that both the methods are in full agreement. The effects of various non-dimensional parameters have also been linerated graphically. In the fourth chapter of this thesis the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt is given. The proposed problem was modeled in terms of PDEs together with physical boundary conditions. Two different cases were discussed. The first one is the lift case, while the second one is the drainage case. Two different techniques, namely the Adomian Decomposition Method (ADM) and the Optimal Homotopy Asymptotic Method (OHAM) were used for finding the analytical solutions of the above mentioned cases. These solutions were compared and found in excellent agreement. For the physical analysis of the problem, graphical results were provided and discussed for various embedded flow parameters. In the fifth chapter, the Optimal Homotopy Asymtotic Method (OHAM) [45-46] was used for approximate solution of singularly perturbed two-point boundary value problems. We concluded that the proposed method gives efficient results. We solved some numerical examples, that include initial and boundary value problems. In the sixth chapter, the solution of higher order boundary value problems by our proposed method “Modified Optimal Homotopy Perturbation Method” (MOHPM) has been given. A homotopy with an embedding parameter and Daftardar-Jafari polynomials were used. To control the convergence of solution, some auxiliary functions which depended upon variables and some constants were used. The proposed method was simple, rapid, effective and accurate. The accuracy has been proved by comparing our results with the solutions of optimal homotopy perturbation method (OHPM), optimal homotopy asymptotic method (OHAM), variational iterative method (VIM), variational iteration method using He’s polynomials (VIMHP), homotopy perturbation method (HPM) and Adomain decomposition method (ADM).