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The Noether Symmetries and Invariants of Some Partial Differential Equations

Thesis Info

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External Link

Author

Adnan Aslam

Program

PhD

Institute

National University of Sciences & Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2014

Thesis Completion Status

Completed

Subject

Physics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2872/1/2245S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727586512

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A connection is obtained between isometries and Noether symmetries for the area-minimizing La- grangian. It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an (n − 1)-area enclosing a constant n-volume in a Euclidean space is so(n) ⊕s Rn and in a space of constant curvature the Lie algebra is so(n). Here for the non-compact space this has to be taken in the sense of being cut at a fixed boundary that respects the symmetry of the space and is not a volume enclosing hypersurface otherwise. Further if the space has one section of constant cur- vature of dimension n1 , another of n2 , etc. to nk and one of zero curvature of dimension m, with n≥ k j=1 nj + m (as some of the sections may have no symmetry), then the Lie algebra of Noether symmetries is ⊕k so(nj + 1) ⊕ (so(m) ⊕s Rm ). j=1 For a subclass of the general class of linear hyperbolic systems, obtainable from complex base hy- perbolic equation, semi-invariant and joint invariants are investigate by complex and real symmetry analysis. A comparison of all the invariants derived by complex and real methods is presented here which shows that the complex procedure provides a few invariants different from those extracted by real symmetry analysis for a linear hyperbolic system. The equations for the classification of symmetries of the scalar linear elliptic equation are obtained in terms of Cotton’s invariants. New joint differential invariants of the scalar linear elliptic equations in two independent variables are derived, in terms of Cotton’s invariants by application of the infinitesimal method. Joint differential invariants of the scalar linear elliptic equation are also derived from the bases of the joint differential invariants of the scalar linear hyperbolic equation under the application of the complex linear transformation. We also find a basis of joint differential invariants for such equations by utilization of the operators of invariant differentiation. The other invariants are functions of the bases elements and their invariant derivatives. Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by splitting the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables.
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پروفیسر فضل الرحمن فریدی

پروفیسر فضل الرحمن فریدی مرحوم
۲۶؍ جولائی کے اخبار میں ڈاکٹر شرف الدین اصلاحی کے انتقال کی خبر کے ساتھ ڈاکٹر فضل الرحمن فریدی کی وفات کی بھی خبر تھی، غم دوگنا ہوگیا، دنیائے علم کی ویرانی سی ویرانی ہے، اس کیفیت خزاں میں شجر زندگی کے اوراق زرد ہوتے جاتے ہیں۔
ڈاکٹر صاحب شیراز ہند جونپور کے مردم خیز قصبہ مچھلی شہر میں پیدا ہوئے، الہ آباد اور علی گڑھ میں اعلیٰ تعلیم حاصل کی، معاشیات کے موضوع میں اختصاص کیا، پہلے مسلم یونیورسٹی اور بعد میں سعودی عرب کی ملک عبدالعزیز یونیورسٹی میں اسی کادرس دیا، اﷲ نے قلب و ذہن کو پاکیزگی بخشی، اسلام کے نظریۂ معاشیات کو عصری نظام سرمایہ داری اور قمار و سود کی گرم بازاری میں یقین و اعتماد کے ساتھ پیش کرکے اس کی بہتری اور برتری ثابت کرنا، اس دور کا فرض کفایہ تھا جس کو پورا کرنے والوں میں فریدی مرحوم کا حصہ بڑا نمایاں ہے۔
تدریس کے ساتھ انہوں نے تصنیف و تالیف کا عمل جاری رکھا، جماعت اسلامی سے متاثر تھے اسی لیے جماعت کے انگریزی ترجمان ’’ریڈینس‘‘ کی ادارت اور دوسری انتظامی ذمہ داریاں بھی وقتاً فوقتاً انجام دیتے رہے، لیکن رسالہ ’’زندگی نو‘‘ ان کے افکار و نظریات کا سب سے موثر ترجمان رہا، وہ اس کے مدیر تھے اور اشارات میں ان کی ادارتی تحریریں اشارات سے زیادہ بینات کی صورت سامنے آتی رہیں۔ خصوصاً معاشی موضوعات پر نہایت معلومات افزا ہوتیں، ان کے افکار کی تہہ میں صرف یہ جذبہ پنہاں ہوتا کہ اسلام کی معاشی تعلیمات کی برکتوں کا اندازہ کرنے کے لیے موجودہ زمانہ کا ماحول سب سے سازگار ہے لیکن ہماری معلومات صرف روایتی مذہبی تعلیمات تک محدود ہیں، آئی ایم ایف جیسے مالیاتی اداروں کو ان کے اسلوب میں بتانے کی ضرورت ہے کہ قرضوں کی...

The Dress Code for Muslim Women: A Linguistic Analysis of the Qurānic Verses and the Prophetic Traditions

It is not uncommon to find cases of Muslim women being harassed or bullied in many of the Muslim-minority countries because of their dress. These Islamophobic attacks, unfortunately, are not merely conducted by radicalised individuals; but the subjugation of the rights of Muslim women also comes from institutional bodies and governments. Secular nations, such as France, Germany, Italy, Belgium, Netherlands, Bulgaria, Switzerland, USA, UK, Canada, China, and Russia have either imposed restrictions on Muslim women regarding their dress code. They see veil as a non-acceptance of progressive or cumulative values which is unsurprisingly not welcomed by the Muslim community. In such environment, it is inevitable for the Muslims to understand what the Qur’ān and Sunnah really say about the dress code for Muslim women in order to explain what their religion really requires from them and to communicate it appropriately to the government officials, journalists, politicians, and other relevant stakeholders. It is also essential from the perspective of segregating cultural aspects from the religious aspects. Many of the commonly used words for the dressing of Muslim women are more rooted in culture than the religion. It is accordingly vital to understand what the Qur’ān and Sunnah really command about the women dressing and how it has been interpreted in various Islamic societies and cultures. This paper accordingly presents an analysis of all the relevant Qur’ānic verses and the prophetic traditions (from the 6 most renowned books of ahadith). The linguistic analysis employed in this paper results in the identification of items of dress that were worn by Muslim women to safeguard their modesty during the times of Prophet Muhammad (ﷺ). The same principles are relevant for today’s age and time and the Muslims can use those guidelines to delineate cultural practices from the religious injunctions.

A Pplications Pplication of E Volutionary C Omputing Ing T Echniques to N on -L Inear S Ystems

This dissertation presents an application of heuristic computational intelligence for the solution of non-linear systems in engineering. The design scheme is comprised of mathematical model based on feed-forward artificial neural network (ANN). The linear combination of these networks defines the unsupervised error for the system. The most suitable weights to minimize the error are obtained by training the networks employing stochastic solvers. These techniques are based on nature inspired heuristics including Pattern Search (PS), Simulated Annealing (SA), Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithms. Rapid local convergent algorithms such as Interior Point (IP) and Active Set (AS) methods are hybridized with these global search techniques. To validate the scheme, a number of linear and non-linear initial and boundary value problems have been solved. The design methodology is also applied to a number of problems having special applications in engineering including, singular systems based on non-linear Lane Emden Fowler equation, non-linear van der Pol oscillator with stiff and non-stiff conditions and systems with high nonlinearity governed by Painlevé transcendent I. In addition to that, the scheme also provides an alternate solution for biomedical application like model of heart for low, high and normal blood pressure. It is found that the proposed results are in good agreement with available exact solution and numerical solvers like Adomian decomposition method, Homotopy Perturbation method, Homotopy analysis method and Optimal Homotopy asymptotic method, ODE15i and Runge Kutta method. The comparative studies of stochastic solvers are carried out under a stringent criterion of accuracy, effectiveness, reliability and robustness of the results based on Monte Carlo simulation and its analysis. The solvers based on SA, PS, GA, PSO, GA and PSO hybrid with IP or AS algorithms are used for optimization of neural network. It is found that the GA-IP, GA-AS, PSO-IP and PSO-AS algorithms are the best stochastic optimizers. The other perk up of the scheme have in its simplicity of the concept, ease in use, efficiency and unlike other numerical techniques, it provides the solution on continuous inputs with finite interval instead of predefine discrete grid of inputs.