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Home > Analysis and Application of Optimal Homotopy Asymptotic Method and the Use of Daftardar- Jaffery Polynomials

Analysis and Application of Optimal Homotopy Asymptotic Method and the Use of Daftardar- Jaffery Polynomials

Thesis Info

Access Option

External Link

Author

Javed Ali

Program

PhD

Institute

Islamia Collage Peshawar

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2013

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1358

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725475244

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In this thesis, we use the standard optimal homotopy asymptotic method for the analytical solution of higher order ordinary differential equations with two or multi- point boundary conditions, a nonlinear family of partial differential equations with an initial condition and Integro-differential equations. The results obtained are compared with the results obtained by the application of Adomain decomposition method, homotopy perturbation method, variational iteration method, differential transform method and homotopy analysis method etc. The optimal homotopy asymptotic method uses a flexible auxiliary function, which controls the convergence of the solution and allows adjustment in the convergence region where it is needed. Moreover, the procedure of this method is simple, clearly well-defined and the built- in recursive relations are explicitly worked out for easy use. Numerical results obtained by the optimal homotopy asymptotic method reveal high accuracy and excellent agreement with the exact solutions. Apart from the application of optimal homotopy asymtotic method, we develop a new scheme by using Daftardar-Jafari polynomials in the homotopy of optimal homotopy asymptotic method to solve nonlinear problems more effectively. This scheme is almost as simple as the optimal homotopy asymtotic method but its results are more accurate than the usual optimal homotopy asymptotic method. To show the effectiveness of this scheme, we solve different bench mark problems from literature and compare the results with those obtained by the standard optimal homotopy asymptotic method. To determine the convergence control parameters of the auxiliary function, two well known methods are followed: The Least squares and the Galerkin methods. We also explore and use a variety of the forms of auxiliary functions that results more accuracy and shows flexibility and reliability of the methods. For symbolic computation, we use Mathematica 7. The work presented in chapters 2, 3 and 4 of this thesis has been published in different reputed international journals and the work presented in chapters 3, 5, 6 and 7 is submitted for possible publication. The details of published/submitted work are included in the list of publications.
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والدین کی عظمت

والدین کی عظمت
نحمدہ ونصلی علی رسولہ الکریم امّا بعد فاعوذ بااللہ من الشیطن الرجیم
بسم اللہ الرحمن الرحیم
معزز اسا تذہ کرام اور میرے ہم مکتب ساتھیو!
آج مجھے جس موضوع پر گفتگو کرنی ہے وہ ہے’’عظمت ِوالدین‘‘
صاحب ِصد ر!
والدین ایک عظیم نعمت ہیں ، ان کا وجود مسعود رحمت ہی رحمت ہے۔ ان کی موجودگی اولاد کے لیے باعث ِبرکت ہے، ان کی رضاء میں اللہ کی رضا ہے، ان کی ناراضی میں خدا کی ناراضی ہے۔ حدیث پاک میں ہے کہ جس نے یہ دیکھنا ہو کہ مجھ پر اللہ تعالیٰ راضی ہے یا ناراض تو وہ اپنے والدین کا چہرہ دیکھ لے اگر والدین خوش ہیں تو رب تعالیٰ بھی خوش ہے اگر والدین ناراض ہیں تو رب تعالیٰ بھی ناراض ہے۔
صدرِمحترم!
قرآنِ پاک میں نصِّ قطعی سے ثابت ہے کہ اپنے والدین کے ساتھ بھلائی کرو، ان کا احترام کرو، اُن کے ساتھ اچھے طریقے سے گفتگو کرو، ان کو اُف تک نہ کہو ، والدین ایک ایسی ہستی ہیں کہ یہ اولاد کا کبھی بھی بُر انہیں سوچتے ،ان کے لیے مشکلات برداشت کرتے ہیں، پریشانیوں کو سینے سے لگاتے ہیں، اُن کی اعلیٰ تعلیم وتربیت کا انتظام کرتے ہیں۔
صدرِذی وقار!
والدین کے چہرہ کو دیکھنا ایک عبادت ہے اللہ تعالیٰ مقبول حج کا ثواب عطا فرماتا ہے۔ حدیث پاکؐ میں ہے کہ کسی شخص نے پوچھا کہ مجھ پر سب سے زیادہ حق کس کا ہے آپؐ نے فرمایا کہ تیری ماں ، پھر پوچھا گیا آپؐ نے فرمایا کہ تیری ماں، پھر پوچھا تو آپؐ نے فرمایا کہ تیراباپ۔
معزز سامعین!
ماں کے بارے میں کہا گیا کہ’’ جنت ماں کے قدموں کے نیچے ہے‘‘ ماں ایک ایسی نعمت غیر مترقبہ ہے کہ جس کا بدل دنیا میں ناممکن ہے۔ ماں آنکھوں کو...

Portrayal of Positive Psychological Capital in Quran

The present study aims at exploring positive psychological capital in the verses of Qur’an. Positive psychology is the latest advancement in the field of psychology which focuses on improving the well-being of society. Positive psychological capital refers to the combination of overall qualities of positive psychology that contributes to the well-being and mental health. The present study is based on the content analysis of the verses of Qur’an. Content analysis comprises of three steps including identifying the categories or themes, dividing the information into units or parts and finally rating all the themes in all units. In the first step the researchers identified 41 themes from Qur’an by using committee approach and reading the verses between the lines. All these categories were identified by keeping in view the underlying themes of positive psychology. In the second step 30 units were devised from Qur’an by considering each Part as a single unit. The categories included behavior modification, belief in divine help, brotherhood, bravery, contentment, civility, credibility, encouragement of virtue, emotional regulation, excellence, forgiveness, generosity, gratitude, honesty, hopefulness, humility, justice, knowledge, lawful spending, learning, meaningfulness, mindfulness, moderation, obedience, patience, peace, determination, positivity, prosperity, repentance, resilience, reward, self-actualization, self-awareness, self-control, sincerity, social leadership, truthfulness, trust, and wisdom. Results suggest that the most prominent category in Qur’an is the reward. Validity of the study was maintained through the selection of the themes with the help of committee approach. Reliability of the scoring system was maintained through partial inter-rater reliability. Overall the present research has many implications in the positive psychology of religion.    

Preparation, Characterization, in Silico Modelling and Bioavailability of Nanosuspention for Poorly Water Soluble Drugs Dexibuprofen and Domperidone

The aim of this study was to prepare and evaluate the impact of polymers on fabricating stable dexibuprofen (Dexi) and domperidone nanocrystals with enhanced therapeutic potential, using a low energy, anti-solvent precipitation method (APSP). Combinations of Hydroxypropyl methyl cellulose-Polyvinyl pyrrolidone and Hydroxypropyl methyl cellulose-Eudragit were shown to be very effective in producing stable dexi-nanocrystals with particle sizes of 85.0±2.5 nm and 90±3.0 nm, and polydispersity of 0.179± 0.01, 0.182± 0.02 respectively. Produced nanocrystals of DOMP and Ethucel combination, the average particle size and polydespersity index were found to be approximately 130.00 ± 3.0 nm, 0.15 ± 0.01, moreover, the combination of hydroxyl propyl methyl cellulose and polyvinyl alcohol also found very affective to produce DOMP nanocrystals with small particle size (200.0 ±3.5nm) and PDI (0.2±0.02). The stability studies conducted for 90 days demonstrated that nanocrystals stored at 2-8°C and 25°C were more stable than those at 40°C. The maximum recovery of dexi-nanocrystals was observed from the formulations using the combination of Hydroxypropyl methylcellulose- Polyvinyl pyrrolidone and Hydroxy propyl methyl cellulose- Eudragit, which equated to98% and 94% of the nominal active drug content respectively. In case of DOMP nanosuspensions stored at 2-8 ºC and room temperature (25 ºC) exhibited higher stability compared to the samples stored at 40 ºC. Crystallinity of the processed and unprocessed dexi and DOMP particles was confirmed using Differential Scanning Calorimetry, Powdered X-ray Diffraction and Transmission Electron Microscope. FTIR studies for both Dexi and DOMP nanocrystals confirmed that the process did not affect chemical integrity of DOMP. The DOMP nanoparticles exhibited significantly enhanced dissolution rate (P<0.05) compared to the raw counterpart. The saturation solubility of the ix dexinanocrystals was substantially increased to 270.0±3.5 µg/mL compared to the raw dexi in water (51.0± 2.0μg/mL) and stabilizer solution (92.0 ± 3.0 μg/mL). Enhanced dissolution rate (P<0.05) was observed for the dexi-nanocrystals compared to the unprocessed drug substance and marketed tablets. The saturation solubility of the DOMP nanocrystals was significantly increased to 2700.0± 3.5 µg/mL compared to the raw dexi in water (952.0± 2.0μg/mL) and stabiliser solutions which include HPMC (1253.0± 3.5) and HPMC-PVA (1133.0 ± 3.5 µg/ml). DOMP nanocrystals resulted in enhanced dissolution rate (P<0.05) compared to the unprocessed drug substance. Molecular Modeling studies underpinned and substantiated the experimental studies, including the impact of polymers on nanoparticle sizes, nanocrystals recovery and stability studies of the produced dexinanocrystals. The highest potential was shown by combination ofHPMC-PVP- Dexi (-4.7) and HPMC EUD- DEXi (-4.6) . In case of DOMP, Molecular Modelling studies underpinned the molecular level understanding of the DOMP-Polymer nanocrystal interaction and substantiated the experimental studies, including the impact of polymers on nanocrystals sizes, and stability studies of the produced DOMP nanocrystals. Ethocel as single polymer showed highest binding potential (-27.26±0.24 ) for DOMP nanocrystals and HPMC-PVA was found the best dimer to effectively bind (-25.22 ± 0.79) with the nanocrystals with subsequent small particle size and high stability profile. The findings of the nociceptive assay showed that the dexinanocrystals exhibited significant analgesia, compared to the raw dexibuprofen and the control standard diclofenac sodium. The analgesic effect was, however, produced at much lower doses (5mg/kg) than that of control standard, diclofenac sodium (20mg/kg) and dexibuprofen counterparts (40 mg/kg). While theactivity results in anti-emetic assay confirmed that the nano-formulation provided significant anti-emetic effect (P< 0.05) x at a dose of 0.5 mg/kg, comparable to that of the unprocessed DOMP and the control standard Metoclopramide administered at the doses of 1.0m/kg and 30mg/kg respectively. However, DOMP nanoparticles showed the anti-emetic effect at lower doses compared to that of control standard and unprocessed DOMP. Furthermore, there was observed a substantial % increase of antiemetic potential for DOMP nanoparticles compared to the unprocessed DOMP and the standard metoclopramide.