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Wavelets and Radial Basis Functions in Scienti C Computing

Thesis Info

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Author

Aziz, Imran

Program

PhD

Institute

University of Engineering and Technology

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11342/1/Imran_Aziz_Maths_HSR_2015_UET_Peshawar_10.04.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727668079

Similar


The present work is an application of wavelets and radial basis functions to numerical computing. More specifically, we have used Haar and Legendre wavelet for applications of wavelets and multiquadric for applications of radial basis functions. The application areas considered in this thesis are the numerical solution of Integral Equations (IEs), various order Integrodifferential Equations (IDEs), systems of IEs, Elliptic Partial Differential Equations (EPDEs), Parabolic Partial Differential Equations (PPDEs) and highly oscillatory integrals. A few theoretical results are proved for efficient evaluation of some particular systems that arise when we apply one- or two-dimensional Haar wavelet in the wavelet collocation method. Based on these theoretical results new numerical methods based on Haar wavelet are developed for solution of IEs, IDEs and systems of IEs. EPDEs are solved numerically using collocation methods with Haar and Legendre wavelet. Legendre wavelet is also applied for the numerical solution of PPDEs. A new method based on multiquadric radial basis functions is introduced for numerical solution of highly oscillatory integrals. While applying Haar wavelet to numerical solution of IEs we have considered both nonlinear Fredholm and nonlinear Volterra IEs of the second kind. Similarly in case of IDEs a Haar wavelet based method is applied to find numerical solution of first and higher orders nonlinear Fredholm and nonlinear Volterra IDEs. The main advantage of this method is that it is generic as it can be applied to IEs, IDEs and systems of IEs. More specifically the new approach aims at the numerical solution of Fredholm, Volterra and Volterra-Fredholm types of IEs, IDEs and IDEs of higher orders including initial- as well as boundary-value problems. With a slight modification the method can also be applied to find numerical solution of two-dimensional IEs, system of IDEs and partial IDEs. Another distinguishing feature of themethod is that unlike many other existing methods in the literature it does not use any intermediate technique for numerical integration of the kernel function in IEs or IDEs. We have developed two new types of collocation methods based on Haar wavelet and Legendre wavelet for numerical solution of EPDEs. A modification of the collocation method based on Haar wavelet for elliptic differential equations is also introduced that improves the efficiency of the method. The collocation method based on Legendre wavelet is extended to find numerical solution of PPDEs. An advantage of the proposed methods is that it can be applied to different types of boundary conditions (BCs) with slight modifications. For highly oscillatory multidimensional integrals a new Levin’s type method based on multiquadric radial basis functions is developed. Levin method converts the numerical integration problem of highly oscillatory multidimensional integral to a PDE which is subsequently solved using meshless method. The proposed methods are validated on a variety of problems as well as numerical results of the proposed methods are compared with several existing methods from the literature. The numerical results show better performance of the proposed methods for several benchmark problems.
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المبحث الثالث: العيون الحزينة

المبحث الثالث: العيون الحزينة

قصيدة " إلی عيني الحزينتين " لنازک الملائكة[1]


عینيّ، أيّ أسیً یرین[2] علیکما
ویُثیر[3] في غسق[4] الدجی[5] دمعیکما؟

إني أری خلف الجفُون ضراعۃ[6]
تستنطق[7] الکون[8] العریض المبھما[9]

أفقان تحت اللیل ألمحُ فیھما
فطرات ضوءٍ یرتشفن[10] الأنجما

ألکونُ مبتسمُ فأیّۃُ لوعۃٍ[11]
یا مقلتيّ[12] تلوح[13] في جفنیکما؟

مسکینتانِ، رأیتما ما لا یَرَی
جیلٌ أقام علی الضلال[14]، وحوّما [15]

جھل الحقائق في الحیاۃ، فلم یُطق[16]
عن زیفھا ھرباً وعاش مھوما [17]

مسکینتان کتمتما حمم[18] الأسیٰ[19]
فأبی[20] تأوّہ[21] خافقي[22] أن تکتما

فاذا الدموع غشاوۃ[23] رفّت[24] علی
جفنیکما، ، سیلاً سخیناً مفعما[25]

ورأیتما، خَلَلَ الدّموعِ، مفاتن ال
ماضي وطاف الشوقُ فی أفقیکما

عبثاً تصوغان[26] التوسّل في الدجی،
قلبُ القضاء[27] قضی[28] بألا تَنعما

عبثاً ، فیا عینيّ لا تتضرّعا[29]
لا شيء، یرجعِ بالجمال إلیکما

حسبي[30] وحسبکما الرضوخُ[31] لما قضّی
قلب اللیالي فارضخا[32] واستسلیما

کم حالمٍ من قبلنا فقد المنّی

تفسیر القرآن از سر سید احمد خان کا تحقیقی و تنقیدی جائزہ

Sir Syed Aḥmed Khān belonged to a famous family of the subcontinent during the late Mughal and early British colonial period. He was famous for his close relations with the colonial government. He served many years in the judiciary. In recognition of his services, he was conferred upon with various titles such as Sir, The Imperial Advisor, etc. He is the founder of the educational campaign which was later known as the Aligarh movement. He was worried about the future of Muslims in India. This worry forced him to produce various literary and Islamic books to uplift the political, cultural, educational and social status of the Indian Muslims. One of his famous contribution to Islamic literature of Quranic exegeses is his Tafsīr al-Qur’ān. His tafsīr is influenced by western thoughts. He, instead of following the traditional methodology of Quranic exegeses, tried to understand the Quranic verses rationally. This led him to deviate from many established concepts of Islamic doctrines. He went against the Muslims’ affirmed beliefs in his exegesis. He mistrusted some of the basics of Islamic thoughts and tried his best to make new parameters of writing & reading of the Quranic exegesis on human logics. In addition, some of his views show certain relevance to the Mu'tazilites school of thought. The aim of this paper is to present an analytical and a critical evaluation of the exegetical opinions of Sir Syed Aḥmed Khān, particularly on the issues where he deviated from the mainstream Islamic thoughts in his exegesis, Tafsīr al-Qur’ān.

Adaptive Soft Computing Synergistic Paradigms for Vsc Based Facts Damping Controls

Adaptive Soft Computing Synergistic Paradigms for VSC Based FACTS Damping Controls Since their inception, damping of Low Frequency Oscillations (LFOs) has been a critical issue in electric power systems. Voltage Source Converter (VSC) based Flexible AC Transmission Systems (FACTS) have a renowned capability for rapid regulation of various network quantities, thus being a serious candidate for future power system control and smart grids. VSC based FACTS have built in capability of absorbing or delivering reactive power. FACTS controllers when equipped with efficient supplementary damping control combat LFOs. Although, a large research investment in efficient damping control for FACTS and advancement in the field of Artificial Intelligence (AI) has led to more robust controls. Even then, there is a growing realization that the contribution to damping performance enhancement should be more rigorously addressed for different tradeoffs, such as complexity and control effort smoothness. This dissertation puts forth the claim that efficient damping control strategy to improve application quality in terms of damping performance, control effort smoothness and execution time is essential for a high performance FACTS supplementary control. This work is a design paradigm shift from conventional Takagi Sugeno Kang (TSK) based control to advanced control based on hybrid Soft Computing (SC) techniques. SC techniques are the most lucrative choice to supplementary damping control design for their optimal performance delivery in critical applications with low complexity and high precision. The direct focus of this research is to exploit the potential of hybrid SC paradigms, obtained from diversified domains such as signal processing (Fourier and wavelets), applied mathematics (Bsplines and polynomials) and AI (neural and fuzzy), to name a few. A modular approach for optimization of overall Multiple Input Single Output (MISO) TSK structure to speculate optimal combination of antecedent and consequent parts has been proposed. The contributions of this framework are the ixdamping performance improvement with smooth control effort and improved convergence speed. The work, proposed in this thesis, has further been extended to Multiple Input Multiple Output (MIMO) structure. The parameters of controller are updated online, using gradient descent based backpropagation algorithm, to avoid offline training overhead. These synergistic paradigms are later used with indirect Multiple Input Single Output (MISO) and MIMO control to ruminate optimal control schemes. The convergence analysis based on Lyapunov stability criteria has been used to ensure the stability of control scheme and to derive an upper bound on the learning rates. In case of MIMO control, convergence is guaranteed using Adaptive Learning Rates (ALRs). These schemes are applied successively to Single Machine Infinite Bus System (SMIB) with single FACTS, multimachine system with single FACTS and multimachine system with multitype FACTS controllers. The proposed control schemes have been tested for different contingencies and various operating conditions. Finally, the qualitative behavior of all the control paradigms has been quantified using different performance indices that support the nonlinear time domain simulation results. The statistics also support the claim that an ideal performance of a supplementary damping control demands a perfect match between antecedent and consequent part of the NeuroFuzzy network. The frequency domain analysis using Wigner-Ville Distribution (WVD) has also been carried out to analyze the frequency spectra of control effort for smoothness validation. The proposed hybrid control paradigms give better performance in transient and steady-state regions.