Zaman Skalasında Dinamik Denklemlerin Sayısal Çözümleri

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2021

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Mathematics
(2000)
The Atılım University Department of Mathematics was founded in 2000 and it offers education in English. The Department offers students the opportunity to obtain a certificate in Mathematical Finance or Cryptography, aside from their undergraduate diploma. Our students may obtain a diploma secondary to their diploma in Mathematics with the Double-Major Program; as well as a certificate in their minor alongside their diploma in Mathematics through the Minor Program. Our graduates may pursue a career in academics at universities, as well as be hired in sectors such as finance, education, banking, and informatics. Our Department has been accredited by the evaluation and accreditation organization FEDEK for a duration of 5 years (until September 30th, 2025), the maximum FEDEK accreditation period achievable. Our Department is globally and nationally among the leading Mathematics departments with a program that suits international standards and a qualified academic staff; even more so for the last five years with our rankings in the field rankings of URAP, THE, USNEWS and WEBOFMETRIC.

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Bu tezin amacı, zaman skalasında dinamik denklemler için bazı sayısal yöntemleri ¨ incelemektir. Bu nedenle, keyfi zaman skalası için Euler yöntemi ve ikinci mertebe- ¨ den Taylor serisi yöntemi analiz edilmiş¸ ve açıklanmıştır. Her iki yöntem için hata ve ¨ yakınsaklık analizleri de verilmiştir. Trapezoid (Yamuk) kuralı olarak bilinen sayısal yöntem, ikinci mertebeden Taylor serisi yönteminden elde edilmiştir. Her iki yöntem, birinci ve ikinci mertebeden dinamik denklemler için başlangıç değer problemlerine uygulanmıştır. Örneklerin parametreler içermesi sayesinde, başlangıç¸ değer problemlerinin çeşitli zaman skalalarında ve farklı başlangıç¸ değerleri verilerek incelenebilmesi olanağı vardır. Sayısal sonuçlar Matlab kullanılarak hesaplanmıştır ve ilgili yaklaşık ve tam çözümler, hem değerleri tablolanarak, hem de grafikleri çizilerek karşılaştırılmıştır. Son olarak, incelenen yöntemlerle ilgili sonuçlar ve bazı ek yorumlar verilmiştir.
The aim of this thesis is to discuss some numerical methods for solving dynamic equations on time scales. For this purpose, the Euler's method and Taylor series method of order 2 are analyzed and described for an arbitrary time scale. The error and convergence analysis for the two methods are also given. The numerical method known as Trapezoidal Rule is deduced from the Taylor Series method of order2. Both methods are applied to several examples of initial value prob lems associated with first and second order dynamic equations. The examples contain many parameters which makes it possible to investigate one initial value problem on different time scales and impose different initial conditions. The numerical results are computed with Matlab and the related approximate and exact solutions are computed both by tabulating their values and by sketching their graphs. Finally, the conclusion and some additional remarks are given

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Matematik, Mathematics

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109