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The Numerical Solution of Integral and Integro-Differential Equations by Wavelet Collocation Method

Thesis Info

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Author

Khan, Imran

Program

PhD

Institute

University of Peshawar

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11130/1/Imran_Khan_Maths_HSR_2018_UoP_Peshawar_13.09.2018.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727587040

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This thesis presents some new algorithms for the numerical solution of IEs, IDEs and PIDEs. These new algorithms are based on WCM. The focus is upon two types of wavelets namely HW and LLMWs both having compact support. HW has already been applied for numerical solution of IEs and IDEs by a few re- searchers and the numerical results can be found in the existing literature. One draw back of the HWCM is its slow convergence. In order to improve the convergence rate we have applied LLMWCM to IEs and IDEs. In most of the cases of the IEs and IDEs we obtained better results with LLMWCM as compare to HWCM. We also extended the existing HWCM for IEs and IDEs in the literature as well as the newly developed LLMWCM for IEs and IDEs in this thesis to numerical solution of some particular type of PIDEs. Both the HWCM and LLMWCM are thoroughly investigated in this thesis for numer- ical solution of di erent types of IEs, IDEs and PIDEs including Fredholm, Volterra and Fredholm-Volterra IEs, IDEs and PIDEs. The methods are also investigated for higher- order IDEs. An important characteristics of the method is that it can be applied to both linear and nonlinear problems. In the present work, in case of solving linear IEs, IDEs and PIDEs, the resulting systems will be solved by using Gauss elimination method, while for nonlinear case, we use Newton''s or Broyden''s method. All methods are implementing and testing by a computer programming software MATLAB. Several test problems are performed in order to verify the accuracy and e ciency of the present methods. The experimental rates of convergence and MAEs have been calculated for di erent number of CPs. The approximate solution of these test problems are compared with exact solutions and also the numerical results are compared with other well established methods to show better e ciency, accuracy and simple applicability of the newly developed methods. A comparative study of HWCM and LLMWCM is also perform in this thesis.
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Chapter 2: Services of Theologians of Sindh

Introduction

In the Indo-Pak sub-continent, Sindh is the first of the zones where the advent of Islām was the earliest. Historically, the scholars here did a great work of intellect and research. In the year 270 A. H/883A. D, due to internal disturbance, so many small states came into being. [1]

            Out of these, the state of Manṣūrah was taken over by the Ḥubārī tribe. Muslim ruler ‘Abdallāh bin ‘Umar Ḥubārī of Manṣūrah had sent a scholar to Hindū Rājāh of Alvar on his request who presented the translation of the Qur’ān in his court. This incident narrated by the famous historian Buzurg Bin Shehr Yār has also been endorsed by renowned historians of the sub-continent. [2]But nothing with certainty can be said about the symmetrical order and the dictation of this translation, however, it is certain that the honor of translating the Qur’ān the first of all lies with Sindh. [3]

            In the era of Arghaun and Tarkhān, Makhdūm Nūḥ Hālāe’ (d:998 A. H/1589A. D) properly translated the Qur’ān in Persian which has become available in secured form despite the varying times. [4]

            This translation, with the efforts and research of ‘Allāma Muṣṭafā Qāsmī (d:1424A. H/2003A. D) has been published by Sindh Literary Board Ḥyderābād. This translation was compiled even earlier than the translation of famous researcher Shāh Walī Allāh(d:1176A. H/1762A. D). In this regard, we are justified to opine that in the sub-continent the honor of the first translation in the Persian language also goes to Sindh. In that period, a movement among the Sindhī Scholars emerged that the Islamic literature and art be documented in the mother tongue Sindhī so that a...

Determining the Occupational Health and Safety of Workers in The Food Manufacturing Sector of Pakistan

Background: For organizations, conducting work in a safe environment has become mandatory under the provisions and guidelines provided by international and local laws. Occupational health and safety (OHS) standards have become a requisite in almost every industry these days; however, many organizations in developing countries are negligent at adhering to the safety measures, and there is also an absence of proper monitoring. Objective: Our objective was to determine the occupational health and safety standards that are prevailing in the food manufacturing sector of Pakistan. For this, we approached several food manufacturing companies and conducted interviews with quality assurance managers and health and safety executives to evaluate and analyze the workplace conditions to understand their OHS practices. Methods: We conducted 7 interviews with quality assurance managers in the food manufacturing and services companies of Pakistan. The interview guide was used for eliciting responses and a coding method was used to arrive at the themes of the data gathered. Results: The interviews conducted with quality assurance managers and those working to ensure that safety measures are practiced in their company, showed that the export-oriented food manufacturing companies in Pakistan are following OHS standards, as it has become a necessity, with numerous benefits such as ensuring worker safety, maintaining company reputation. However, the extent to which these measures are implemented and strictly followed varies, as there are certain issues that arise when companies try to follow and implement OHS practices The problems vary from company to company however, most of them relate to worker adaptability, costs, and training needs.  OHS situation in local firms is concerning. Most of them do not have HSE officers, not the concept of toolbox talk, no preventive and safety measures.  These poor practices make the situation unsafe, which lead to fatalities Conclusion: By establishing a national framework for policies and legislations as per the conventions of ILO for a better workplace environment along with proper law enforcement improvement in occupational health and safety domain can be achieved.      

Right of Religious Freedom in Shariah and Law: A Case Study of Pakistan

The question of religious freedom and human rights has always been a corner stone of wide ranging discussions and an issue of immense importance in 20th century .Once viewed in overall realm of history, it predominantly figures out that issue offreedom of religion and worship always remained universal, self evident and inalienable. Notwithstanding the upheavals of civilizations liberty and choice of religion and dogmas perpetually remained consistent and irrevocable. Nature conferred liberty leaving at the absolute discretion of human beings to decide the discourse of life they wish to adopt. As a consequence of relentless and undaunted struggle by the flag bearers of human Rights, the right of freedom of religion is universally acclaimed through various international Conventions and Charters. Practicing a particular religion and worshipping God of own choice without any impediment is deemed an inalienable right of every human being. However in stark contrast to internationally accepted norms and principles, flagrant infringement to this unquestionable right in some of the intolerant civilizations and states portrays a disgusting picture. Believably lack of tolerance stems out of rigidity, myopic mindedness, and stagnancy. Those societies which do not let vent in the fresh ideas and appreciate divergent thoughts consequently plunge into deep ignominy of ignorance and darkness. As history bears testimony to the fact that unrelenting fanaticism and prejudices prevalent in various systems of mythologies, schools of thoughts and religions commonly stems out of intolerance and outright stubbornness subsequently leading to indiscriminate persecution and ultimate destruction of societies. Glaring examples of current history are State of India and Burma, where people are being indiscriminately persecuted purely due to divergence of faith. In sharp contrast to the intolerance, it is an irrefutable fact that Islam and its laws are pluralistic and coexistent. It fervently supports the freedom of religion, beliefs and diversifying theological systems. In an Islamic society, the life, property and honour of a Covenant holder (Dhimmi) is fully protected and respected exactly like that of a Muslim. Covenant holders (Dhimmis) will have full freedom of conscience and belief and will be at an absolute liberty to perform their religious rites and ceremonies in their own way. Truly, Islam and equity are synonymous to each other as indicative of the spirit, which it echoes, when even a non Muslim could not only conceptualizeits own ideas and beliefs but is fully permitted to positively advance dissenting ideas with decency and while remaining within the precincts of morality. The notion being perpetrated against Islam that it advocates extermination of every non Muslim from the surface of earth is nefariously aimed at tarnishing the true image of most benevolent and humane religion of Islam. Citing the precedent set forth in Pakistan, it is an undeniable fact that Constitution of the Islamic Republic of Pakistan explicitly recognizes the rights of Religious Minorities, while fully guaranteeing them the freedom of belief and faith, protecting and maintaining their places of worship and religious institutions. In consonance of various Constitutional Articles there shall be no discrimination (irrespective of being Muslim or non Muslim) on the grounds of race, religion, caste, sex etc. State shall be bound to safeguard their legitimate rights and their representation in Federal and Provincial legislative bodies. There is no, however, denying the fact that quite frequently grievances against the Misuse of Blasphemy Laws come to surface thereby Minorities misconstrue to be marginalized by the predominant Muslim Population. Going deep down to the problem, it appears to be a misperception. Both, state and saner elements of the society, firmly believe in equitable and just treatment to be meted out to Minorities and no discrimination and misuse under the garb of Blasphemy law is ever permitted. In an attempt to unleash negative propaganda against Muslims and Islam vociferous voices to repeal the Blasphemy Laws are raised which is totally irrational, illogical and unacceptable to majority Muslim Population. The contention against discriminate use of law is the utmost responsibility of state to effectively put in force the criminal justice system to obviate the chances of injustice and abuse of law. This has very rightly been pointed out and enforced by the superior judiciary through their various unambiguous, binding and authoritative judgments.