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تم کو جو ہمیں ملنے کی فرصت نہیں ملتی
تم کو جو ہمیں ملنے کی فرصت نہیں ملتی
لگتا ہے کہ اب ہم سے طبیعت نہیں ملتی
بے چین جو رہتا ہوں تو صد شکر ہے یارو!
درویش کو دنیا میں تو راحت نہیں ملتی
اور ملنا کسی کا بھی ہے دشوار ہی تب تک
جب تک کہ ضرورت سے ضرورت نہیں ملتی
تب تک نہ غمِ ہجر کا کچھ ہو گا مداوا
جب تک کہ ہمیں آپ کی قربت نہیں ملتی
تائب جی تمنائوں کو پڑتا ہے مٹانا
یوں بار گہِ عشق میں عزت نہیں ملتی
حضرت امام ابو حنیفہ کے علمی اسفار
Born to a Muslim family in Kufa, Abu Hanifah (RH) is known to have travelled to the Hejaz region of Arabia in his youth, where he studied under the most renowned teachers of Makkah and Medina. Due to his wisdom and knowledge, he is also considered a renowned Islamic scholar. A question arises that why he has a distinguished name in Islamic history. Firstly, he travelled a lot for the sake of knowledge. He started to travel since 94 A.H but from 104 A.H, he had a consistent travelling to quench his thirst for knowledge. Secondly, he did hi level best to spread this knowledge to others. Even he had devoted his complete life for the sake of it. Amongst Imam Abu Hanifah's teachers was Hammad ibn Sulayman, he joined his circle at the age of 22, having already become a well-known debater and studied with this great teacher until the latter's death, where upon he took over his ‘’majlis’’ (circle) at the age of forty. Many well-known scholars admire and praise Imam Abu Hanifah’s services like Imam as-Shafi (RH) is recorded to have stated: "All men of fiqh are Abu Hanifah's children, " ".I would not have acquired anything of knowledge had it not been for my teacher”.Application of Fractional Calculus to Engineering: A New Computational Approach
In this dissertation, a new heuristic computational intelligence technique has been developed for the solution for fractional order systems in engineering. These systems are provided with generic ordinary linear and nonlinear differential equations involving integer and non-integer order derivatives. The design scheme consists of two parts, firstly, the strength of feed-forward artificial neural network (ANN) is exploited for approximate mathematical modeling and secondly, finding the optimal weights for ANN. The exponential function is used as an activation function due to availability of its fractional derivative. The linear combination of these networks defines an unsupervised error for the system. The error is reduced by selection of appropriate unknown weights, obtained by training the networks using heuristic techniques. The stochastic techniques applied are based on nature inspired heuristics like Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm. Such global search techniques are hybridized with efficient local search techniques for rapid convergence. The local optimizers used are Simulating Annealing (SA) and Pattern Search (PS) techniques. The methodology is validated by applying to a number of linear and nonlinear fraction differential equations with known solutions. The well known nonlinear fractional system in engineering based on Riccati differential equations and Bagley- Torvik Equations are also solved with the scheme. The comparative studies are carried out for training of weights for ANN networks with SA, PS, GA, PSO, GA hybrid with SA (GA-SA), GA hybrid with PS (GA-PS), PSO hybrid with SA (PSO-SA) and PSO hybrid with PS (PSO-PS) algorithms. It is found that the GA-SA, GA-PS, PSO-SA and PSO-PS hybrid approaches are the best stochastic optimizers. The comparison of results is made with available exact solution, approximate analytic solution and standard numerical solvers. It is found that in most of the cases the design scheme has produced the results in good agreement with state of art numerical solvers. The advantage of our approach over such solvers is that it provides the solution on continuous time inputs with finite interval instead of predefine discrete grid of inputs. The other perk up of the scheme in its simplicity of the concept, ease in use, efficiency, and effectiveness.Journals by Discipline
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