Modeling of viscoelastic flows in terms of fractional calculus is an inquisitive area in engineering and industry. The physical models with fractional operators in nonNewtonian fluid mechanics represent more realistic behavior of flows involving nonlinear complex dynamics. The efficiency is because of freedom to choose derivative of any order in the mathematical formulation of the flows. Non-integer derivatives are important for the systems having hereditary behavior as they depend on the past conditions along with the local conditions. Viscoelastic fluids keep memory of old deformations and their behavior is related to these deformations. The fractional derivatives are more adequate in predicting the characteristics of viscoelastic fluids than the ordinary derivatives. In literature linear flow problems, with non-integer derivatives are solved by classical transforms methods. Unfortunately, most of the viscoelastic fluids unlike Newtonian fluids, are not characterized by only one relation. Mathematical equations, for the viscoelastic fluid flows are highly nonlinear in nature. In most of situations, highly nonlinear PDEs cannot be solved exactly, by existing techniques. Literature survey indicates, that appropriate consideration is not given to the numerical solutions, of anomalous nonlinear flow problems with noninteger derivatives. In this study, we have modeled the viscoelastic flow problems via fractional calculus approach and considered numerical techniques using finite difference approximations along with ”L1 algorithm”to discretize non-integer derivatives of time and finite element, discretization is used for space variables, in order to solve the governing fractional viscoelastic models. Finally we have predicted the behavior of viscoelastic fluids that can be used directly for the simulations of industrial processes.
وہی ہوا کہ تماشا لگانا پڑ گیا ناں بڑا تھا شوق تمھیں سیدھی رہ دکھانے کا سمجھ میں آیا کہ ہم لوگ چیختے کیوں تھے جو لوگ دوستی کو بزدلی سمجھتے تھے ستارا وار ابھرنے لگا ہوں آنکھوں سے یہ لوگ موت سے کم پر تو خوش نہیں ہوتے
تمھیں بھی دکھ کو صحافت بنانا پڑ گیا ناں تمام شہر مخالف بنانا پڑ گیا ناں تمھیں بھی خوف رجسٹر کرانا پڑ گیا ناں اب امن کے لیے جرگہ بٹھانا پڑ گیا ناں دیے کی لو کو بڑھانا گھٹانا پڑ گیا ناں سو زندہ رہنے کو مر کر دکھانا پڑ گیا ناں
Almighty sent his chosen prophets to every Ummah to teach them the lesson of goodness and to avoid evil. The purposes of the Prophet (peace and blessings of Allah be upon him) were reform processes: to teach and guide, and pushed the people to Almighty Allah. According to the narrative in Islamic law, everyone's either Muslim or non-Muslim, relative or/neighbor has determined the right to live things. And good conduct is that because Abdullah bin Mubarak said That is, "Meeting old habits, doing righteousness, and forbidding people to hurt you". If human behavior is good then its own life will also pass through fortune and joy, and it will also be the mercy and chain of things for others. Collective relations, interacting with the recipients, and the Family system, is proven to be a source of intelligence. The result of the happy moral is Reality and Heaven of Arman al-Rahman, and the fulfillment of the Prophecy is the wrath of Lord Koran and the Fire of Hell.
"Sihr-ul-Biyan" is an evergreen and living Urdu Masnavi. It was edited and published in 1805 for the first time in
Fort William College, Calcutta. While preparing this edition, Sher Ali Afsos took into consideration the needs and
requirements of the English officers and students. We can guess easily from the texts of the Masnavi, discovered
later on, that some changes were made just to meet the academic needs of the English students. Hence, it was
awfully required to have a comprehensive edition of the Masnavi based on an authentic text. Different texts of
this Masnavi were found in different times and the scholars continued their efforts to search original text of Mir
Hassan.
Fort William College did not publish autographic text of Mir Hassan but we persume that there might be a text
contemporary to Mir Hassan. All later editions from 1844 to 1947 do not reflect clearly that which text is their
exempler. In 2000, seven years ago, a renowned Urdu scholar and outstanding expert of textual criticism, Rashid
Hassan Khan has edited "Sihr-ul-Biyan" but it is based on the edition of Fort William College. Therefore it has no
authority of any specific contemporary text of Mir Hassan or any other authentic text.
The present edition is based on the text written in 1215 Hijrah. It appeared fourteen years after the death of Mir
Hassan. It clearly shows that the edition of 1215 Hijra is based on some contemporary text of Mir Hassan. The
very purpose of this research article is to provide the future generation with a comprehensive edition of the
Masnavi "Sihr-ul-Biyan" based on an authentic text.