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Space Spectral Time Fractional Finite Difference Method Along With Stability Analysis for Fractional Order Nonlinear Wave Equations

Thesis Info

Access Option

External Link

Author

Muhammad Sarmad Arshad

Program

PhD

Institute

COMSATS University Islamabad

City

Islamabad

Province

Islamabad.

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10526/1/Thesis%20Sarmad%20Final.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727213908

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Space Spectral Time Fractional Finite Difference Method along with Stability Analysis for Fractional Order Nonlinear Wave Equations In this work, nonlinear partial differential equations governing the obscure phenomena of shallow water waves are discussed. Time fractional model is considered to understand the upcoming solutions on the basis of all historical states of the solution. A semi-analytic technique, Homotopy Perturbation Transform Method (HPTM) is used in conjunction with a numerical technique to validate the approximate solutions. With the aid of graphical interpretation, the favorable wave parameters, to avoid wave breaking are estimated. Afterwards, dynamical analysis of fractional order Schr dinger equation governing the optical wave propagation is reported in detail. The validity criteria for the application of the semi-analytic asymptotic methods are exploited. Comparison between the solutions obtained by the two asymptotic techniques, that is, the Fractional Homotopy Analysis Transform Method and the Optimal Homotopy Analysis Method is performed to select the most accurate technique for the stated problem. Space spectral analysis with integrating factor technique and time fraction finite difference method have been implemented to study the pressure waves propagating in bubbly fluids as well as nonlinear phenomena of plasma waves. Dynamical analysis of acoustic/pressure waves propagating in bubbly fluids is of great significance. Such flows arise in many engineering problems including sonochemistry, sonochemical reactors, cavitation around hydrofoils and ultrasonic propagation in medicine and biology. Fractional approach for modeling the propagation of the pressure waves in liquids containing a large number of tiny gas bubbles is proposed. Moreover, numerical solution of the fractional order Modified Korteweg-de Vries equation governing the dynamics is approximated using a novel space spectral time fractional finite difference tool. A spectral technique for space and a multi-step finite difference scheme for time are designed and implemented. The spatial spectral discretization error and the stability bounds are discussed. The nonlinear phenomena of plasma waves are well demonstrated with the aid of graphical analysis. Stability analysis of integer and fractional order KdV equations have been discussed quantitatively with the help of Evans function approximation.
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ارداس

ارداس
توحید فلک کی تفہیم کرتے ہوئے!
صحرا کے جلال کو تبسم تقسیم کرتے ہوئے!
خزاں کے سینے سے بہار نکال کر!
دشت بیاباں میں علی اصغرؑ مسکرا رہا ہے
تو رات والانجیل کی مثالیں یاد دلا رہا ہے
مدینہ و نجف کے زائروں میں!
سبز موسم کے حسین دائروں میں!
جنوں کی شرطوں میں باب وفا کی تفسیر کرتے ہوئے!
مزاج شمال کے دائروں میں!
انجیر کی گود میں زیتون کا چہرہ دکھا رہا ہے
تورات والانجیل کی مثالیں یاد دلا رہا ہے
تہذیب عشق کی بارگاہ میں!
سرخ موجوں کی روانی میں۔۔۔پیاس کی کہانی میں!
نصاب بیخودی کے یقینی زمانوں کی سبزہ گاہ میں!
ساحلوں پر بکھرے اثاثے کی داستاں سنارہا ہے
ابن حیدر۔۔۔ابن حیدر بن کے مسکرا رہا ہے
تورات والانجیل کی مثالیں یاد دلا رہا ہے
اے حسین ابن علی تجھ پر سلام
اے بنتِ حسینؑ و علی تجھ پر سلام

Kashmir: Occupation and Resistance

Kashmir became a part of the Muslim World in the 14th century C E. Under the local Sultans (1325-1585) , the Mughals (1586-1752) and the Pathans (1752-1819) , it developed into a state and society with its own peculiar Islamic culture. Kashmir was colonised by the Sikhs (1819-46) from the Punjab and the Hindu Dogra Maharajas (1846-1947) . The Muslim Kashmir stood for Islam through the freedom movement during this period. When the movement under the banner of Islam was at its apex and it seemed that the mahkutn (colonised / enslaved) Kashmir was about to achieve the goal, India ( Bharat) , a newly born nation (Aug 15, 1947) , invaded it on Oct. 27, almost 3 months after her own birth. Since then this newly emerging occupying -power through its policies of political deceit, cultural aggression, economic onslaught and delslamizing educational schemes has given rise to the problem of the IOK (Indian Occupied Kashmir) . The paper narrates the story of the past up to 1988-uprising & throws light on the oppression, persecution and discrimination that the Muslim Kashmir has been witnessing. The form offreedom-movement that followed from 1947 to 1989 is also outlined. In 1989 Jihad Kashmir, a new phase of the movement that invited the attention of the world begins. The paper helps to understand the background ofthe genesis ofthis new phase

Determination of Urban Settlement Pattern for Optimal Regional Development in Punjab

Urbanization is an inevitable process of socioeconomic transformation from agrarian to urban economy—relying upon secondary, tertiary, quaternary or even quinary professions—following Fourastie’s envision from traditional to transitional and finally to tertiary societies. But, this inexorable phenomenon is not always bound to have same consequences as urbanization may either be balanced, lognormal, and broad-based in nature or it may evolve into lopsided, dysfunctional, and pseudo urbanization, which is the main argument of the study. The urbanization and the consequent urban settlement pattern have been dealt by two overarching theories, namely; modernization and world system/dependency theory. The modernization approach explains urbanization as an ongoing societal transformation from agrarian to urban occupations, and all countries follow this unilinear growth pattern, though with differential rates and timings. While, world system theory uncovers the competing interests of traditional and tertiary societies. Resultantly, due to unequal or oligopolistic terms of trade between the two, urban settlement pattern in most of the developing countries has emerged in the form of primate versus skeletal towns/regions. Unfortunately, both regions have suffered from these imbalanced terms of trade. At one end, the skeletal regions have been deprived of their due share from available resources and at the other end, primate regions have suffered from spiraling levels of congestions, crime, terrorism, environmental degradation, and poor urban governance. The study examines the validity and applicability of both theories by observing the urbanization process based on time series data for more than one hundred years in Punjab, Pakistan. Considering the existing institutional and policy framework in the province, the study analyzes urbanization trends using a number of urban settlement and regional analytical techniques such as rank size, rank mobility, and temporal rank order analysis, form ratio, compaction index, centrality functional index, beta index, network shape index, distance matrix and urban density analysis in the province. The study explores that the urbanization process in Punjab is steady and observing high urbanization growth rate—conforming to the modernization theory. The study further confirms that the observed patterns of urbanization are concentrated and the urban base has shrunken over the years—indicating signs of lopsided or pseudo urbanization— conforming to the presence of dependency conditions in the province. viiTouching upon overarching theoretical underpinnings, the study suggests a two pronged strategy to deal with the urban challenge in the province. At one end, the study recommends fostering urbanization process in the province to accommodate the layoff population of rural regions. As a one possible strategic intervention, it recommends to upgrade those villages into urban centers having suitably higher threshold values in terms of functional and population criteria. This strategic intervention is not only expected to augment the urbanization process in the province, but will also transform the existing lopsided urban settlement pattern into more broad-based and a balanced pattern in the province. At the other end, the study advocates to curtail the over accumulation of investments into large urban centers. Again, as one possible strategic intervention, the public sector investments may be redirected from large urban centers to skeletal towns and regions i.e., placing both primate and skeletal regions at their right place along the decentralization continuum. So, with this interplay of modernization and dependency urbanization processes, the existing urban settlement pattern may be transformed into an efficient, equitable, and ecologically balanced system—invigorating homogeneity, parity, and balance in the urban settlement pattern and for having optimal regional development in the province.