Nonlinear partial differential equations are often used to understand and model nonlinear processes arising in many branches of science and engineering. For most of partial differential equations a general closed-form analytical solution is not available and therefore use of numerical methods always remains an important alternative for the solution of partial differential equations. Several numerical methods are developed for the solution of partial differential equations including finite difference methods, finite element methods, spectral methods and spline methods. However numerical methods posses some limitations such as mesh generation, slow rate of convergence, spatial dependence, stability, low accuracy and difficult to implement in complex geometries. One of domain type methods is known as radial basis functions method, which is a truly meshless method, infinitely differentiable, numerically accurate, stable, very high rate of convergence, spatial independence and flexible with respect to complex geometry. The main difference between the mesh free radial basis functions method and classical mesh-based methods is that the radial basis functions can be extended to the entire domain of influence without diving into elements. In this thesis, we present mesh free radial basis functions method based on collocation principle for numerical solution of various time dependent nonlinear partial differential equations namely, Regularized Long Wave (RLW) equation, Modified Regularized Long Wave (MRLW) equation, Modified Equal Width Wave (MEW) equation, Klein- Gordon Schrödinger (KGS) equations, Klein-Gordon Zakharov (KGZ) equations, Two dimensional Coupled Burgers’ equations and Two dimensional Reaction-Diffusion Brusselator equations. Different radial basis functions are used for this purpose. First order forward and second order central difference approximation is employed to the time derivative. The elementary stability and convergence of the proposed method are discussed. Accuracy of the method is assessed in terms of various error norms, number of nodal points and time step size. Performance of the proposed method is validated through examples from literature. Apart from ease of implementation, better accuracy is obtained. Comparison with existing methods such as finite difference methods, finite element methods, boundary element methods and spline methods is made to show the superiority and simple applicability of the mesh free method.
زندگی میں عمل کی اہمیت زندگی میں عمل کی اہمیت کی وضاحت کچھ اس طرح ہے عمل کے بغیر تصویر زیست ممکن ہی نہیں۔ زندگی حرکت و عمل کا دوسرا نام ہے اور بے عملی یا جمودکا دوسرا نام موت ہے۔ عمل سے زندگی کا بگاڑ ہے،عمل ہی سے زندگی سنورتی ہے، زندگی جنت بھی عمل سے ہی بنتی ہے اور اس عمل ہی کی وجہ سے جہنم بنتی ہے۔ جام زندگی کے دوام کا راز گر دشِ پیہم میں پوشیدہ ہے، بے عملی نہ صرف انسان کوسست ،کاہل اور کمزور بناتی ہے بلکہ بے یقین اور بزدل بھی بناتی ہے۔ اس کے برعکس عمل انسان کومستعد، معتمد اور معزز بناتا ہے۔ باعمل اور محنتی لوگوں کو اپنی ذات پر اعتماد اور بھروسا ہوتا ہے۔ جب زندگی میں عمل پیہم کی عادت پڑ جاتی ہے تو اس میں اعتماد اور بھروسہ شامل ہو جا تا ہے۔ اور یہی اعتمادان سے بڑے بڑے کارہائے نمایاں انجام دلواتا ہے اور ان کو ترقی اور خوشحالی کے زینے پر چڑھاجاتا ہے بے عمل انسان میں ایک قسم کی بزدلی اور مایوسی پیدا ہو جاتی ہے۔ جو اس کے جسم و جاں کو مزید زنگ لگا دیتی ہے نتیجتاً سست ہو جا تا ہے اور پھر اس کے لیے زندگی میں مشکلات اور پریشانیوں کا درواز ہ و اہوجا تا ہے اور اس کے حوصلے پست اور روح پژمردہ ہو جاتی ہے۔ جو انسان صاحب عمل ہوتا ہے وہ اپنے کسی کام کے لیے دوسرے کا محتاج نہیں ہوتا وہ اپنا کام خود کرتا ہے اور اس سے وہ مسرت و شادمانی حاصل کرتا ہے۔ اس سے نہ صرف یہ کہ وہ دوسروں کا شرمندہ احسان نہیں ہوتا۔ بلکہ مسرت عمل بھی حاصل کرتا ہے۔ حضور صلی اللہ علیہ و آلہٖ وسلم مجسم عمل تھے اور اپنا کام خود اپنے دستِ...
The freedom of thought and expression has acquired a diversity of meanings and limitations determined by the evolving socio-historical contexts of various civilizations through the history. This piece of research briefly investigates the history of evolution of this concept in Islamic and Western intellectual circles and then intends to focus upon the elaboration of this idea by Imam Badi-uz-Zaman An-Naurasi. Iman Naurasi supports the human right of thinking and expressing ideas freely but at the same time recognizes the need of divine intervention in delimiting its boundaries. Humans, if left entirely unrestricted to make their choices in life, would only bring about disastrous results and thus lead to cessation of a collaborative social life. Therefore, Imam proposes that humanity should abide by some universal principles and high moral values inspired by the divine revelation so that the society could collectively ensure the pursuit of human progress and well-being. Imam proves his point through the instances from the Islamic history that divinely guided freedom of thought and expression provided the world with a vivid model of life which provided people the rights of choosing everything in their life ranging from faith to marital and economic decisions.
Reading is an interactive- constructive process in which readers comprehend, interpret, and respond to the text according to what they already know. Effective readers have personal expectations about what they will get from a selection, and they bring those expectations to bear as they read by predicting and testing their predictions. The children actively create meaning by constructing, or generating relationship between what is within the text and what they already know. Some researchers describe reading as a ;transaction, (Rosenblatt 1988) in which meaning emerges from a give and take relationship between the reader and the print on the page. The interaction between the reader and the text allowed reader to construct their own meaning background Knowledge and experience according to there. This research was therefore conducted to identify strategies that were being implemented by the teacher to develop reading comprehension in students. And how, the teachers could be facilitated to improve their teaching strategies to enhance students reading comprehension skills. This study was divided into two phases. Phase 1 dealt with implementing strategies that could develop comprehension skills through stories in an ESL classroom. Story helps to develop language in the classroom, as Meek (1988) points out, ;Stories encourage them to take on new meaning regularly in a context of shared enjoyment. They learn about language and the different ways it is used. (P.37) They learn new language of stories, which is different from everyday spoken language. Gradually children take possession of the language of books and they begin to learn vital lessons about the reading process Phase II was an exploration of how to facilitate the teacher in her professional development. For enhancing her knowledge about implementing effective strategies like predicting, questioning, silent reading and group work, to develop children reading comprehension. The findings revealed the teachers understanding and her changes in her perception throughout the course of the study. And how this study helped her to professionally develop her style of teaching in order to get the desired children's reading comprehension. The findings also revealed that as a result of teacher development and reflection on practice, the teacher could change her conception and her approaches to the teaching of reading comprehension skills.