Chebyshev sonlu farklar yöntemi ile adi türevli yüksek mertebe başlangıç ve sınır değer problemlerinin çözümü
Solution of initial and boundary value problems of higher order ordinary differential equations with Chebyshev finite difference method
Yazar: SONER AYDINLIK
Danışman: DOÇ. DR. AHMET KIRIŞ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2015
Konu: Matematik = Mathematics

Özet:
Bu çalışmada adi türevli yüksek mertebeden başlangıç ve sınır değer problemlerinin çözülmesi amaçlanmaktadır. Yüksek mertebeden sınır değer problemleri uygulamalı mekaniğin birçok mühendislik probleminde ortaya çıkmaktadır. Adi türevli yüksek mertebe diferansiyel denklemlerin çözümü için global faz-integrasyon, Adomian-ayrıştırma, "The new-iterative", diferansiyel dönüşüm, diferansiyel kuadratik kuralı, homotopi analiz, homotopi pertürbasyon, Spline ve Laplace ayrıştırma gibi çeşitli yöntemler kullanılmaktadır. Bu yöntemlerden Adomian yöntemi karmaşık Adomian polinomlarının hesabını, homotopi yöntemleri sağlanması gereken birçok koşulu ve uygun parametrelerin bulunmasını gerektirmekte ve hemen hepsi Chebyshev polinomlarına göre daha fazla CPU zamanına ihtiyaç duymaktadır.
Chebyshev polinomları sürekli fonksiyonlar uzayı üzerinde tam ortogonal bir küme oluşturmakta verekürsif ilişkileri kolayca elde edilebilmesi nedeniyle özellikle türev ve integralleri istenilen mertebeden rekürsif olarak hesaplanabilmektedir. Chebyshev polinomları aynı dereceden diğer polinomlara göre verilen aralıkta maksimum hatası minimum olan en uygun yaklaşım polinomlarıdır.
Chebyshev sonlu farklar yöntemi diferansiyel denklemlerin sayısal çözümünde sıklıkla kullanılan fonksiyonlardan biridir. Bu yöntemde diferansiyel denklem hangi mertebeden olursa olsun yaklaşım polinomunun denklemde görünen mertebeden türevleri hesaplanmakta ve bu türevler diferansiyel denklemde kullanılarak, denklem, lineerse lineer denklem sistemine, nonlineer ise nonlineer denklem sistemine indirgenmekte, dolayısıyla buradaki tek problem nonlineer denklem sisteminin çözümü olmaktadır. Ayrıca bu yöntem ile birçok sayısal yöntemde olduğu gibi çözüm aralığının sadece belirli noktalarında değil, tüm aralık boyunca geçerli bir yaklaşım polinomu olarak elde edilir.
Chebyshev sonlu farklar yöntemi ile birinci ve ikinci mertebe başlangıç veya sınır değer problemlerinin çözümü literatürde sıklıkla görülmektedir. Nadiren birkaç problemde ise ortaya çıkan üçüncü mertebeden diferansiyel denklemler Chebyshev yaklaşım polinomunun 3. mertebeden türevi için ardışık toplam sembolleri kullanılarak çözülmüştür. Bu mantık  . mertebeden türev için ardışık   tane toplam sembolü kullanılmasını gerektirmekte ve her toplamda içerideki ifade de değişeceğinden, programlamada güçlüklere neden olmaktadır.Yüksek mertebeden diferansiyel denklemler, rekürsif bir ilişkinin var olmasına rağmen bu rekürsif ilişkinin giderek karmaşıklaşması nedeniyle Chebyshev sonlu farklar yöntemi kullanılarak çözülememektedir.
Bu tez çalışmasında Chebyshev polinomlarının her mertebeden türevleri için var olan rekürsif ilişki yerine genel bir formül üretilmiştir. Üretilen bu genel formül ile Chebyshev sonlu farklar yöntemi her mertebeden adi türevli başlangıç veya sınır değer problemine uygulanabilir hale getirilmiş ve bu yöntem uygulanarak problemler lineer veya nonlineer denklem sistemlerine indirgenmiş, bunların çözümünden de diferansiyel denklemin çözümü olarak yaklaşım polinomları elde edilmiştir. Çalışma kapsamında, bu genel türev formülü yardımıyla genelleştirilen Chebyshev sonlu farklar yöntemi farklı yüksek mertebelerden aditürevli lineer sistemlere, başlangıç ve sınır değer, Robin sınır değer, karışık sınır değer problemleri ile nonlineer sistemlere uygulanmıştır. Elde edilen sayısal çözümler ile, analitik çözümler karşılaştırılmış ve yaklaşım polinomunun terim sayısındaki artışla birlikte sayısal çözümlerin hızla analitik çözümlere yakınsadığı grafiklerle gösterilmiştir.

Abstract:
In this work, it is aimed to solve initial and mixed boundary value problems of higher order ordinary differential equations. High-order boundary value problems arise in many engineering problems of aplied mechanics. Many classical engineering problems such as calculus of variations problems, kinematics are modeled successfully with at least second-order differential equations. However, different order differential equations come up when the mathematical constraints are reduced for compliance with physical reality. For example, when non elastic, microstructured material is considered, third-order differential equations arise at the mass flow of a micropolar material. Even for the elastic case, fourth-order differential equations arise in the bending of the linear elastic rod. Modelling of the behavior of the induction motor requires fifth order differential equation, but when the fluid layers are heated and rotated, the resulting classic heat dissipation require sixth order differential equation. Many other examples can be found in the literature, but here the two examples are given lastly for extreme cases; when the magnetic effects are considered in setting of above example, the differential equation is tenth order, and the vibration of a uniform bar governed by eighth-order differential equation is more familiar one.
Various methods such as Global Phase-Integration Method, Adomian-decomposition method, The New-Iterative Method, Differential Transformation Method, Differential Quadrature Rule, Homotopy Analysis Method, Homotopy Perturbation Method, Spline Method and Laplace decomposition methods are used for the solution of high-order ordinary differential equations. Adomian method requires the calculation of complex Adomian polynomials, homotopy methods require several conditions to be satisfied and the presence of appropriate parameters and almost all of them need more CPU time than Chebyshev polynomials.
Chebyshev polynomials create a complete orthogonal set for continuous functions space and specially their derivatives and integrals of any order can be calculated, since their recursive relationships can easily be obtained. Chebyshev polynomials are the most appropriate approximation polynomials, maximum error of which is the minimum on the given interval among the other same degree polynomials.
Chebyshev finite difference method is one of the frequently used functions for numerical solution of differential equations. In this method, no matter what order of differential equations, all order derivatives of the approximation polynomial appearing in the equation are calculated and by using this derivatives in the differential equations, the equation is reduced to the linear system, if equation is linear. Similarly, if the equation is nonlinear, it is reduced to the nonlinear system. Therefore, the only problem remaining here is the solution of the nonlinear equation. In addition, in this method, the solution is not obtained for only at certain points as many numerical methods, but also a valid approximation polynomial as a solution is obtained throughout the entire interval.
The solution of first and second order initial or boundary value problems with Chebyshev finite difference method is studied widely in the literature. As we known, in rarely a few problems, third-order differential equations are solved by Chebyshev finite diffference method. In these studies, the summation symbol is used three times, iteratively for the third-order derivative of the approximation polynomial. This logic requires the use of  the summation symbols    times, consecutively for the   order derivatives, which causes difficulties in programming phases since the expressions inside the summation symbols may change each time. Although there is a recursive relationship between the derivatives of the approximation polynomial, higher order differential equations have not solved using Chebyshev finite difference methods before this thesis because of the increasing complexity of this recursive relationship between derivatives.
In this thesis, a general formula is presented for every order derivatives of Chebyshev polynomials instead of existing recursive relationship of Chebyshev polynomials. Thanks to the obtained general formula, Chebyshev finite difference method is made applicable to the initial or boundary value problems of any order of ordinary differential equations. Thus, the problems are reduced to linear or nonlinear system of equations and the approximation polynomials are obtained as the solutions of the given differential equations by solving this system. In this study, Chebyshev finite difference method with the proposed generalized formula for the any order derivatives is applied to a linear system, some initial and boundary value problems, Robin and mixed boundary value problems and a nonlinear system of different higher order ordinary differential equations. The obtained numerical solutions are compared with analytical solutions and an increase in the number of terms of the approximation polynomial causes rapid convergence of numerical solutions to the analytical solutions, this fact is shown by graphs for all examples.
This thesis consists of four parts.
In the first section, the subject and purpose of thesis is mentioned. 
In the second part of the thesis, Chebyshev Polynomials of the first kind is defined, their properties are given and the recursive relationship between the successive Chebyshev polynomials are shown in detail. Then, the extreme points and the roots of the Chebyshev polynomials were found.
In the third part of the thesis, Chebyshev finite differences method is described in steps. In the first step, assuming that Chebyshev polynomials can be obtained by sum of series, an approximation polynomial which uses the extreme points of Chebyshev polynomials is obtained. In the second step,  a formula was obtained for the coefficients of the approximation polynomial. In the third step, first and second order derivatives of Chebyshev polynomials have been found step by step and then third, fourth and fifth derivatives are calculated in a similar way and by the help of these derivatives, a general formula is obtained for any higher order derivatives of  Chebyshev polynomials. In the last step, with the guidance of this proposed formula, a general formula is obtained for any order of derivatives of approximation polynomial.
In the fourth part of the thesis, some examples are given. First, to illustrate the method,  a basic linear and nonlinear initial value problems are solved. In the second part of this section, various linear examples are solved. Respectively, a second and a sixth order boundary value problems, an eighth order initial value problem, an eighth order Robin boundary value problem, an eighth order mixed boundary value problem, a fourth order system  and lastly a nonlinear initial value problem of second order are given. In the third part of this section, various nonlinear examples are solved. Respectively,  a second and a tenth order boundary value problem, an eighth order initial value problem, a seventh order Robin boundary value problem, a twelfth order mixed boundary value problem, and finally a fourth order system are given.