Search or add a thesis

Advanced Search (Beta)
Home > Efficacious Algorithms for the Solutions of Fractional Differential Equations

Efficacious Algorithms for the Solutions of Fractional Differential Equations

Thesis Info

Access Option

External Link

Author

Tariq, Hira

Program

PhD

Institute

University of the Punjab

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11284/1/Hira%20Thesis-PhD.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726033251

Asian Research Index Whatsapp Chanel
Asian Research Index Whatsapp Chanel

Join our Whatsapp Channel to get regular updates.

Similar


Fractional calculus is the generalization of integer-order calculus to non-integer order. In recent decades, fractional calculus has re-attracted the attention of scientists and engineers. Due to the nonlocal property, fractional operators give a perfect aid to characterize the memory and hereditary properties of various processes and materials. As a result, and motivated by the increasingly important role played by fractional calculus, mathematicians are constantly developing algorithms for solving fractional differential equations. The objective of this work is to develop the efficacious algorithms for the solutions of both ordinary and partial fractional differential equations. For solving both linear and non-linear fractional differential equations, spline method and residual power series method are used. Matlab, Maple and Mathematica programmes are developed to compute the numerical solutions. The simplicity, efficiency and accuracy of the presented methods are demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.
Loading...
Loading...

Similar Books

Loading...

Similar Chapters

Loading...

Similar News

Loading...

Similar Articles

Loading...

Similar Article Headings

Loading...

اردو فکشن میں جدید اصناف

اُردو فِکشن میں جدید اصناف

(مائیکرو فکشن،افسانچہ،فلیش فکشن کا تحقیقی جائزہ)

منیرعباس سپرا،پی ایچ ڈی سکالر

انسان کو شروع سے ہی کہانی سے دلچسپی رہی ہے کہانی کی تاریخ بھی اتنی ہی قدیم ہوگی جتنی انسان کی ہے کہانی کی تاریخ انسانی زندگی اور اس کی بڑھتی ہوئی پیچیدگیوں کی تاریخ ہوتی ہے جس میں انسان کے کارناموں کی روداد ہوتی ہے جس میں اس نے اپنے ماحول معاشرے کی کسی قوت سے مقابل اگر کامیابی حاصل کی ہو ۔یہی کہانی انسان کے احساس برتری کی تسکین کا ذریعہ بنتی ہے اور اس طرح کہانی کا تعلق ہماری فطرت سے بھی ہے۔

کہانی دنیا کا قدیم ترین فن ہے انسانی قوت تخلیق نے سب سے پہلے قصہ کی تخلیق کی اور داستان قصہ گوئی کی شکل میں موجود تھی پھر قصے کہانیوں نے جدید عہد کے علائق تک انسان کے ساتھ ہی کہانی نے بھی بہت سے مراحل طے کیے داستان گوئی سے ناول ،ناولٹ اور افسانے سے مختصر کہانیوں (مائیکرو فکشن ،افسانحہ ،فلیش فکشن) کی طرف اردو فکشن کا یہ ارتقائی سفر جارہی ہے کیونکہ فکشن میں یہ تغیر تبدل ادب کے ارتقاء کیلئے ناگزیر ہے ارتقاء کا یہی سفر سماج میں بھی جاری رہتا ہےپہلے کی نسبت لوگوں کے پاس فرصت کے لمحات کم ہیں مشینی، صنعتی اور ٹیکنالوجی کی ترقی نے لوگوں کو مصروف کردیا ہےاسی مصروفیت کی حالت میں لوگ ادب سے بھی لطف اندوز ہونا چاہتے ہیں اسی لیے عام قارئین داستان ناول وغیرہ کی ضخیم کتب کی بجائے کم وقت میں کہانیوں سے وہی لذت کشید کرنے کیلئے مختصر کہانیوں( مائیکرو فکشن ،ا فسانحہ ،فلیش فکشن وغیرہ) کی طرف راغب ہو رہے ہیں آج کے ٹیکنالوجی ،مشینی اور گلوبلا ئزیشن کے دور میں معمولیات زندگی میں...

Disparity in Political Participation on Social Media Public Sphere among Male and Female Students

Social media has become a key term in new participatory political discourse. Previous studies argued that youth is least interested in democracies and politics; hence, this study aims to explore the role of social media in increasing political participation among male and female university students. Data was collected from a sample of 340 students, chosen from equal gender proportion of five universities with purposive sampling technique to conduct quantitative survey research. The findings of study revealed a positive relationship between political participation and social media usage of students. It also confirms that majority respondents were using social media for political information but the trend is more popular among the male students as compared to females. Male participants were more involved in discussing politics and posting political content on social media, while females were more active in actionable politics like casting vote. The study concludes that social media is playing a significant role in enhancing political participation among university students and predicts a better future of democracies in social media world as new technology provides the language that young voters understand. It also realized that measures are required to attract female students towards participatory politics in Pakistan. ______

A Study of Polluted Eco-System of Industrial Areas Caused by the Industrial Effluents

The adverse effects of industrial pollution are becoming a challenge for scientists and environmentalists around the globe. The management of the pollution is imperative to improve the human health, economy, aquatic life and to protect from further deterioration of the environment. The leading intend of the present work was to evaluate trace elemental contaminations in agricultural soil, crops and vegetables being irrigated with industrial effluents and their treatment to reduce the pollution. This research will be beneficial to decrease the industrial pollution by the immobilization of the toxic constituents in the effluents and will provide database pertaining to the concentration of metals in the industrial effluents and their accumulation in soil, crops and vegetables. The data will assist to identify the trends, nature, and sources of pollution and will aid in the formulation of legislation related to the controlled release of industrial effluents into the environment. Moreover, present data for nutrition can be useful for nutritionists and food technologists for the formulation of diet menu for the inhabitants of the respective regions with adequacy/ safety viewpoint for balance intake of essential and toxic trace elements. For this research, more than 500 samples of vegetables (brinjal, baffle gourd, ridged gourd, tomato, pumpkin, bitter gourd, cabbage, mustard, spinach, potato, turnip, radish & carrot), crops (millet, maize, rice & wheat), effluents (ceramics, pulp/paper & textile/yarn industries) and soils (top & sub-surfaces) have been collected from the vicinity of industrial zones of Faisalabad and Gujranwala areas. Each species of vegetable and crop plants was separated into its fruits (edible portion), flowers, leaves, stems and roots to evaluate the bio-distribution of trace elements, in each portion. Neutron Activation Analysis (NAA) and Atomic Absorption Spectroscopic (AAS) techniques have been utilized to analyze the selected samples for the quantitative determination of more than 36 trace and toxic elements. Accuracy and precision have been ensured by comparing with five different certified reference materials (CRMs) and by making replicate measurements for each sample. Moreover, the Z-score method was also applied to assess the discrepancy between the measured and the certified values. Ultra-filtration membrane therapy (UFMT), which is a separation technique, was used for the reduction of toxic level in industrial effluents. Various runs have been conducted on samples of the effluents by using a lab-scale UFMT unit, which was fitted VIIwith a Polyethylene tere phthalate (PETP) membrane. This filtration technique is very effective, reliable and economical for the quantitative separation of suspended particles from the effluents. The effects of temperature and pressure on flow rates of the effluents have been investigated. The parameters such as flux, temperature, applied pressure, filtration velocity, density, concentration of the effluents and their relationships have been illustrated. Spectro-photometric analyses prove the effectiveness of UFMT system in removing dissolved coloured species and chromate ions also. The pollution parameters such as colour/ dyes, biochemical oxygen demand (BOD), total suspended solids (TSS), total dissolved solids (TDS), turbidity, oil/ grease/ fat etc., have been reduced quantitatively up to 96% in the post filtration effluents. Moreover, in the absence of other electrolytes, the chromate removal up to 98.9% from effluents has also been achieved. Arsenic, chromium and iron metals have also been successfully removed from the industrial effluents, on laboratory scale, by using husk of sweet peanut. In this regard, optimize experimental parameters have been established for smooth/reliable performance. The analytical results for the concentrations of 36 minor, major, rare earth and toxic elements in each sample of vegetables, cereal, soil and effluents are presented in tables 6.1 to 6.12. Moreover, the evaluated concentrations of some selected trace elements have been presented in figures 7.4 ¾ 7.41 for their comparison patterns with each other. The results of physico-chemical analysis and trace elemental concentrations showed that all untreated effluents were un-fit for irrigation purposes due to the higher values of metals as compared to the NEQS values. Effluents vary in quality for textile, pulp, and ceramics industries and are specific for each industry. The effluent contamination has been decreased in the following pattern. Textile/ Yarn > Pulp/ Paper > Ceramics Faisalabad industrial area was divided into four zones (i.e. F-1, F-2, F-3 & F-4). Zone F-1 represents the area of Industrial Estate, F-2 represents the area of Ghulam Muhammad abad, F-3 represents the area of Peoples Colony and F-4 represents the area of Sitara Colony. According to the high concentration of the elements, the intensity of toxicity in the specified soils of Faisalabad is decreased in the following order. F-1 > F-2 > F-3 > F-4 VIIISimilarly, Gujranwala industrial area was divided into four zones (i.e. G-1, G-2, G-3 & G-4). Zone G-1 represents the area of Dhula, G -2 represents the area of Garjakh, G -3 represents the area of Small Industrial Estate and G-4 represents the area of Muhammad Nagar. Moreover, due to the high concentration values of concerned elements, the intensity of the toxicity in the specified soils of Gujranwala shows the following decreasing sequence. G-4 < G-3 < G-2 < G-1 Leaching tendency of some selected trace elements was observed for Faisalabad and Gujranwala soils. The elements (i.e. Ba, Cr, As, Na, Cl, K, Br & Mg) move from topsoil (S t ) to sub-soil (S s ) very easily as compared to other elements (i.e. Mn, Sb, Sc, Co, Se, Fe & Zn) due to high leaching tendency. The same behaviour was observed in both soils of Faisalabad and Gujranwala. Therefore, the quantities of the elements (i.e Ba, Cr, As, Na, Cl, K, Br & Mg) are higher in sub-soils as compared to the topsoil. This behavior was also confirmed by the evidence of observed high electrical conductivity (EC) values (5.6-4.3 mS cm -1 ) at sub-soil as compared to topsoil (4.1-3.1 mS cm -1 ) values. According to the concentrations of the trace elements, the industrial (Gujranwala & Faisalabad) and non-industrial (Rawalpindi & Islamabad) national soils are arranged in the following descending series. Gujranwala > Faisalabad > Rawalpindi > Islamabad A comparison was made among the national soils (i.e. Faisalabad & Gujranwala) and international soils (i.e. Norway & India). All soils samples were analyzed using NAA technique. According to the high concentrations of the trace elements, generally all zones are arranged in the following sequence. Gujranwala > Faisalabad > Norway > India Vegetables are staple part of food and are widely consumed in all over the world. The determination of metal contents in vegetables is significant from the viewpoint of crop-yield technology, food nutrition and health impacts. The differences for the accumulation of mineral/ metal contents in the edible portions of vegetables depend upon the soil compositions and the rate of uptake of minerals/ metals by each plant. Results showed that different vegetables had different abilities to take up heavy metals. IXHowever, the general trend shows that the maximum concentration of the trace elements is accumulated in roots while their least concentration is found in fruits i.e. edible part of the vegetables and are arranged in the following decrasing sequence. Roots > Stems > Leave > Fruits (Edible portion of vegetables/ crops) All over the world, about 70% of human diet consists of cereals and legumes. In case of edible portion of cereals the toxic activity decreases in the following sequence, which indicates that wheat crop is the least affected by the industrial effluents as compared to other cereal crops. Millet > Maize > Rice > Wheat It was observed that the concentrations of all elements are high in the wheat of Faisalabad and low in the wheat of Kashmir. The order of toxicity decreases as following: Faisalabad > Gujranwala > Islamabad > Kashmir The concentrations for majority of elements are high in the rice of Faisalabad and low in Kashmir. The order of toxicity decreases in the following sequence. Faisalabad > Islamabad > Gujranwala > Kashmir Similarly, the concentrations for majority of elements are high in the vegetables of Faisalabad and low in Islamabad. The order of toxicity decreases as under: Faisalabad > Gujranwala > Kashmir > Islamabad Regular monitoring for further assessment as to ascertain the quality of the foodstuffs and the origin of trace metal distribution is a pre-requisite. In order to obtain consolidate achievements numerous analyses of various species are required where seasonal and regional variations need to be studied in detail." xml:lang="en_US