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Haar Wavelet Approach for Numerical Solution of Ordinary, Partial and Fractional Differential Equations With Delay

Thesis Info

Access Option

External Link

Author

Rohul Amin

Program

PhD

Institute

University of Peshawar

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/8184/1/Rohul_Amin_Maths_2017_HSR_UoP_Peshawar_22.06.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726353996

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In this thesis, the main emphasis is on collocation technique using Haar wavelet. A new method based on Haar wavelet collocation is being formu- lated for numerical solution of delay differential equations, delay differential systems, delay partial differential equations and fractional delay differential equations. The numerical method is applied to both linear and nonlinear time invariant delay differential equations, time-varying delay differential equa- tions and system of these equations. For delay partial differential equations two methods are considered: the first one is a hybrid method of finite differ- ence scheme and one-dimensional Haar wavelet collocation method while in the second method two-dimensional Haar wavelet collocation method is ap- plied, and a comparative study is performed between the two methods. We also extend the method developed for delay differential equations to solve nu- merically fractional delay differential equations using Caputo derivatives and Haar wavelet. Here we consider fractional derivatives in the Caputo sense. Also we designed algorithms for all the new developed methods. The imple- mentations and testing of all methods are performed in MATLAB software. Several numerical experiments are conducted to verify the accuracy, ef- ficiency and convergence of the proposed method. The proposed method is also compared with some of the existing numerical methods in the literature and is applied to a number of benchmark test problems. The numerical re- sults are also compared with the exact solutions and the performance of the method is demonstrated by calculating the maximum absolute errors, mean square root errors and experimental rates of convergence for different number of collocation points. The numerical results show that the method is simply applicable, accurate, efficient and robust
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بدعنوانی کے خاتمے میں معاشرے کا کردار

بدعنوانی کے خاتمے میں معاشرے کا کردار
کوئی چیز بھی اللہ تعالیٰ نے بے مقصد پیدانہیں فرمائی ، ہر چیز کی تخلیق میں کوئی نہ کوئی غرض و غایت ضرور کارفرما رہی ہے، لیکن انسان چونکہ ظلوما ًجہو لا کے مصداق تخلیق کردہ اشیاء میں کوئی نہ کوئی تبدیلی کا مرتکب ہوتا رہتا ہے اور اس چیزکی تخلیق کا جوعظیم مقصد ہے وہ پس پردہ چلا جا تا ہے۔ اور یوں کائنات کی رنگینیوں ، رعنائیوں اور دل آویزیوں کے آفتاب نصف النہار کو گرہن لگ جاتا ہے۔ کسی چیز کی اصل ہیئت کو تبدیل کرنے کا نام بدعنوانی ہے۔
مجاہد سرحدپر بجائے حفاظت کے جاسوسی کررہا ہے تو یہ بدعنوانی ہے۔ معلم مسند تدریس پرمتمکن ہو کر تشنگان علم کی پیاس بجھانے میں تساہل اور غفلت کا شکار ہے تو یہ تدریسی بدعنوانی ہے۔ مسیحاجب اپنے پیشے سے وفا نہیں کر رہا اور اس کے زیرعلاج مریض کے مرض میں اضافے کا سبب اس کی نا اہلی اور نا تجر بہ کاری ہے تو یہ گھناؤنا جرم اور کرپشن ہے۔ جس معاشرے میں ابتداء سے لے کر انتہاء تک بدعنوانی ہی بد عنوانی ہو اس معاشرے کی فضاء میں محو پرواز طائر خوش الحان بھی اپنی پرواز کوتا دیر قائم نہیں رکھ سکتا۔ ایسے معاشرے کی مسموم فضاء اس کے تنزل کا باعث بنتی ہے۔
ہم اگر بد عنوانی اور کرپشن کے خلاف قدم نہیں بڑھائیں گے تو اس کی جڑیں طول پکڑتی جائیں گی اور پھر اس ناسور پر نشتر چلانے کے لیے تا دیر ہوم ورک کرنا پڑے گا۔ اس مرغِ بسمل کی طرح تڑپاتے ہوئے زہر ہلا ہل کے لیے کسی تریاق کی اشد ضرورت ہے۔
آج ضرورت اس امر کی ہے کہ ہر شعبہ سے تعلق رکھنے والاشخص اس کومنطقی انجام تک پہنچانے کے لیے کمر بستہ ہوجائے ، صحافی اپنے اخبار...

عصر حاضر میں حصول انصاف میں مشکلات مسلم فکر کا تجزیاتی مطالعہ

Islam has given a complete code of life to run the system of this universe, which started with the arrival of Hazrat Adam (A.S) in this world and was completed with the Prophet (P.B.U.H). It is a comprehensive and system in which the individual and the rights and duties of both congregations are based on the concept of justice and fairness. Under this system of Islam, the supreme command on earth is only Allah’s persons, the position of man is that of vicegerent and caliph, mans responsibility is to carry out all the affairs of life according to the will of Allah. The earth and its resources will created for the benefits of all mankind. All human beings have the right to use them according to their needs, but no individual has the right to seize or hoard these resources to the detriment of others. Just as every person has rights, such as protection of life and property, honor and reputation, so it is also his duty to provide comfort and relief to no other people and not to cause any harm to anyone.

Synthesis and Characterization of Copper I Complexes With Thiones

A series of Copper(I) complexes with thiones of the general formula [CuaLbScXd](H2O)e and [CuaLbSc]Xd.(H2O)e (where a = 1 - 6, b = 1 - 6, c = 0 - 6, d = 0 - 6, e = 0 - 1 and X = Cl, Br & I) were prepared and characterized by elemental, IR and NMR (1H and 13C) spectroscopic methods. Crystal structures of nine complexes, [Cu(Metu)4]Cl 1, [Cu4(Dmtu)4(S)(μ-S)Cl]Cl.(H2O) 2, [Cu(DeTu)Cl]3 3, [Cu(Metu)4]Br 4, [Cu(Dphtu)2Br]H2O 5, ([Cu(Metu)]I)6 6, [Cu(Mtu)4]I 7, [Cu(Detu)3I] 8 and [Cu(Dbtu)3I]0.6(H2O) 9 were determined by single crystal x-ray diffraction. An upfield shift in the >C=S resonance of thiones in 13 C-NMR and downfield shift in the N-H resonance in 1H-NMR were consistent with the sulfur coordination of thiones to Copper(I). The crystal structures of 1, 4, 7, 8 and 9 show that they are monomeric having tetrahedral or distorted tetrahedral geometries at Copper(I) center. The crystal structure of 2 showed that it consists of a tetrameric cationic complex having terminal as well as doubly bridged Dmtu and S ligands. The complex has two kinds of coordination environments around four Copper atoms, a distorted tetrahedral and a trigonal planer. [Cu(Detu)Cl]3 is a trinuclear complex consisting of Cu3S3 six-membered ring possessing chair conformation. The geometry around each Cu center was nearly trigonal planar. In 5, the Cu(I) atom adopts a slightly distorted trigonal planar coordination. The X-ray studies of 6 showed that the complex was hexanuclear consisting of [Metu-Cu-I] units. The geometry around each Cu center was distorted tetrahedral. In all these complexes intra or/and intermolecular hydrogen bonding is present in 2-D or 3-D network. The antimicrobial activities of Copper(I) bromide complexes were evaluated by minimum inhibitory concentrations (MIC) and they showed wide range of activity against the test isolates.