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Stability Estimates for the Elliptic and Parabolic Obstacle Problems

Thesis Info

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Author

Naveed Ahmad

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1783

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727230873

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Stability of the solution of the multidimensional unilateral obstacle problem with respect to the variations of the coefficients of the corresponding second order partial differential operator is studied in the present thesis. A H ̈lder type stability estimate in the second order Sobolev space of functions is o established for the solution of the elliptic obstacle problem. Rodrigues ([25, 1987]) proved the stability result with respect to bounded external force functions under the nondegeneracy condition. We generalize this result to the case of arbitrary functions belonging to the space Lp (D), p ≥ 2. Further parabolic obstacle problem is studied with different principal parts in the parabolic differential operator and applied to the American put option problem with various local volatility functions. The estimate of the area between the early exercise boundaries through the uniform distance between the local volatility functions is obtained.
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مفتی محمد شفیع ؒ

مفتی محمد شفیع ؒ

 مفتی شفیعؒ کا آبائی وطن دیوبند ہے، جو ضلع سہارنپور یوپی میں برصغیر کا مشہور ترین قصبہ ہے یہیں آپ کی ولادت۱۳۱۴ھ میں ہوئی شمسی حساب سے یہ جنوری ۱۸۹۷ء تھا آپ کے والد محمد یسین بہت بڑے عالم دین تھے آپ کے دادا نے آپکا نام محمد مبین رکھا لیکن آپ کے والد بزرگوار محمد یسین نے ولادت کی اطلاع کا خط اپنے شیخ گنگوہیؒ کو لکھا تو انہوں نے جواب میں نام محمد شفیع تجویز کیا۔

تعلیم و تربیت

”قرآن کی تعلیم سے فراغت کے بعد دارالعلوم ہی میں خط و املاء کی مشق اور فارسی کی تمام مروجہ کتابوں کی تعلیم اپنے والد سے حاصل کی حساب اور فنون ریاضی اقلیدس وغیرہ اپنے چچا منشی منظور احمد ؒ سے پڑھے عربی نحو و صرف اور فقہ کی ابتدائی کتابیں بھی فصول اکبری،ہدایۃ النحو اورمنیۃ المصلی اپنے والد سے ہی پڑھیں اس طرح تعلیم قرآن کے بعد تقریباً پانچ سال فارسی، ریاضی وغیرہ کی پوری تعلیم اور عربی کی ابتدائی کتب میں صرف ہوئے۔“ [[1]]

۱۳۳۰ھ میں سولہ سال کی عمر میں اصول فقہ اور ادب کی کتابیں دارالعلوم کے درجہ عربی میں باقاعدہ داخلہ لے کر شروع کیں۔

شیخ الہند سے استفادہ اور اصلاحی تعلق

 مفتی شفیع اپنے والد کے ساتھ بچپن سے ہی ان کو خدمت میں حاضر ہوا کرتے تھے۔

 

انور شاہ کشمیری سے استفادہ اور دورہ حدیث

 ۱۳۳۵ھ میں مفتی شفیع نے دورہ حدیث انور شاہ کشمیری ؒ کے سامنے کیا فلسفہ کی بعض کتابیں بھی انہی سے پڑھیں رد قادیانیت کا جو کام شاہ صاحب نے شروع کیا اس میں بھی بڑھ چڑھ کر حصہ لیا اور اسی موضوع پر کئی کتابیں ختم نبوت مسیح و موعود کی پہچان بھی تالیف...

RESILIENCE AS A MEDIATOR IN RELATIONSHIP BETWEEN ANXIETY AND PERSONALITY AMONG PHYSIOTHERAPISTS-A CROSS-SECTIONAL STUDY

Background and Aims: The COVID-19 outbreak is the biggest global crisis in generations having severe and far-reaching repercussions for the health system, creating high prevalence of severe posttraumatic stress symptoms for physical therapists, highlighting the need for psychological help. Therefore, this study aimed to investigate how resilience functions as a mediator in the relationship between anxiety and personality among Karachi based physiotherapists.         Methodology: A cross-sectional study was conducted among physiotherapists of Karachi using a convenience sampling technique from August to December 2021. The data was collected using a 10-Item Personality Inventory, Brief Resilience Scale, and Hamilton Anxiety Rating Scale questionnaire. Results: Among 70 participants, the emotional stability trait (7.6±1.4) had higher mean values on the TIPI. On the HAM-A scale, 81.4% physiotherapists had moderate to severe anxiety followed by 15.7% mild to moderate anxiety whereas 92.9% were shown to have normal resilience. Despite all correlations being negligible, only the association between agreeableness and resilience was non-significantly higher (r=0.83). Conclusion: It was concluded that there was no association between the anxiety state and resilience among physiotherapists, although there was a non-significantly higher relationship between agreeableness and openness to experiences personality traits. A high prevalence of moderate to severe anxiety was noted while using a standard resilience strategy.

Distribution, Food Preference and Endo-Parasites in Common Mallard Anas Platyrhynchos Collected from Wetlands of the Punjab, Pakistan

The population survey was conducted to evaluate the population of common mallard (Anas platyrhynchos) at ten wetlands of the Punjab. The mean population of ten year at Chashma Barrage (65651464), Taunsa Barrage (1913191), Ucchali Lake (885232), Head Qadirabad (24972), Head Marala (41852), Head Rasool (350), Khabbeki lake (9242), Head Sulaimanki (1515166), Jhalar Lake (10321), and Head Islam (21739) was calculated and recorded. The population of Chashma barrage was statistically highly significant (P<0.01) as compared to other wetlands. Toevaluate the food preference, among total 40 gut samples, total weight of gut was (102.583.451g), total weight of gizzard (39.92 1.279 g ), weight of food material (4.340.170), gizzard weight without food material (35.33 1.208), weight of seed (1.62 0.092), weight of vegetation (1.960.087), weight of other material was (0.940.069). The comparison between male and female showed that the total weight of gut, total weight of gizzard, weight of food material, gizzard weight without food material and weight of seed was statistically highly significant (P<0.01), weight of other material was statistically significant (P<0.05) and weight of vegetation was statistically non-significant (P >0.05). The total 11 species of plants were identified from the gut analysis which were, Ipomoea aquatic, Hydrilla verticillata, Trapa natans, Oryza sativa, Vallisneria spirallis, Schenoplectus litoralis, Nymphaea nouchali, Typha angustifolia, Potamogeton nodosus, Eichhornia crassipes and Cyperus alopecuroides. The total 14 samples were found infected with different gastro- intestinal parasites. The six species of parasites found were Amidostomum anseris, Hystrielus tricolor, Notocotylus attenuatu, Hymenolepis gracilis, Diacranterian introversa, and Diorchis stefanskii. Four species were recorded at Chashma Barrage, Taunsa Barrage and Head Sulaimanki. Three species were recorded at water-logged area of Bahawalnagar and Faisalabad and two species were recorded at Head Marala. Hymenolepis gracilis, Dicranotaenia introversa and Diorchis stefanskii was recorded from the four wetlands.Hystrielus tricolor and Notocotylus attenuates was recorded from the three wetlands and Amidostomum anseris was recorded from the two wetlands. The current study has provided the valuable baseline information for the management planning, conservation, improvement of habitat, host-parasite interactions, fitness, and health status of this species.