Doğrusal olmayan denklemlerin varyasyonel iterasyon, homotopi pertürbasyon ve varyasyonel homotopi pertürbasyon yöntemleri ile çözümleri
Solutions of non-linear equations by variational iteration,  
homotopy perturbation and variational homotopy perturbation
Yazar: AYŞE DEMİRTAŞ
Danışman: PROF. DR. EMANULLAH HIZEL
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2014
Konu: Matematik = Mathematics

Özet:
Bu çalışmanın amacı, son zamanlarda yaygın olarak kullanılan varyasyonel iterasyon, homotopi pertürbasyon ve varyasyonel homotopi pertürbasyon metodlarını kullanarak bazı doğrusal ve doğrusal olmayan problemlerin çözümlerini elde etmek ve  bu çözümlerin analitik çözümler ile karşılaştırmasını yapmaktır.
Doğrusal olmayan diferansiyel denklemin integrallenebilmesi veya analitik çözümünün varlığının gösterilebilmesi için bazı ölçütler geliştirilmiştir. Bu çalışma süresince bu ölçütlerden en çok kullanılanlardan olan varyasyonel iterasyon, Homotopi pertürbasyon ve varyasyonel homotopi pertürbasyon yöntemleri incelenmiştir.
Varyasyonel iterasyon yöntemi doğrusal olmayan problemlerin yaklaşık çözümünü bulmada kullanılır. Bu metotta problemler başlangıç koşulları ile birlikte verilir. Varyasyonel teorisi yoluyla Lagrange parametresi belirlenerek çözüme ulaşılır. Homotopi pertürbasyon doğrusal ve doğrusal olmayan adi ve kısmi diferansiyel denklemlerin, integral denklemlerinin çözümü için uygulanabilmektedir. Bu yöntemde Homotopi tekniğine göre   parametresi ile homotopi kurulur. Bu parametre küçük bir parametre olarak düşünülür. Bilinen Pertürbasyon yöntemlerini ve topolojideki homotopinin avantajlarını kapsayan bu yöntem, kolaylıkla çözülebilecek basit problemlere dönüştürülerek çözüm elde edilebilmektedir. Bu yöntem doğrusal olmayan dalga denklemlerine, başlangıç değer problemlerine, doğrusal olmayan salınım denklemlerine ve integral denklemlerine uygulanabilmektedir. Pek çok durumda bu metot çabuk bir şekilde seri çözüm vermektedir.
Son zamanlarda geliştirilen varyasyonel homotopi pertürbasyon metodu ise  varyasyonel iterasyon ve homotopi pertürbasyon yöntemlerinin birleştirilmesinden oluşan bir yöntemdir. Bu yeni yöntem ile doğrusal ve doğrusal olmayan denklemlerin analitik veya yaklaşık çözümleri için herhangi bir kısıtlayıcı varsayım ve doğrusallaştırma gerektirmeden analitik çözüme yakın çözümler bulmak mümkündür.
Bu tez çalışması 4 bölümden oluşmaktadır. Tezin ilk bölümü giriş ve varyasyonel iterasyon metodu, homotopi pertürbasyon metodu  ve varyasyonel homotopi pertürbasyon yöntemi ile ilgili literatür araştırmalarından oluşmaktadır. 2.bölümde varyasyonel iterasyon, homotopi pertürbasyon ve varyasyonel homotopi  pertürbasyon metodlarının   teorik altyapısına değinilmiş ve bu metodlarla ilgili örneklere yer verilmiştir.
 3. bölümde ise Benjamin-Bona-Mahony (BBM) denkleminin varyasyonel iterasyon ve varyasyonel homotopi pertürbasyon metodları ile çözümüne ayrıca (2+1) boyutlu Broer-kaup Sistemi ve Soliter Dalga çözümlerinin  homotopi pertürbasyon yöntemi ile Sagamath  programını kullanarak yapmış olduğumuz çalışmalara yer verilmiştir. 4.bölüm bu yöntemlerle yapmış olduğumuz çalışmalar sonrasındaki çıkarımlarımızdan oluşan sonuç bölümüdür.

Abstract:
Engineering and physical sciences in many problems resulting from mathematical models, nonlineer includes differential equations. It is therefore essential differential equation nonlineer analytical or numerical solutions are extremely important in order to obtain. This type of equation, except for a very limited number of those who have a large majority of his analytical solutions. This is why solutions nonlineer equation numerical methods or analytical approach using methods check. Nonlineer problems to find solutions for about a variety of numerical and analytical approach procedures have been developed. But numerical methods, general solution does not enough information about analytical approach methods, there has been to the fore.

The purpose of this study, the variational iteration is widely used recently, homotopy perturbation and the variational perturbation methods using some homotopy linear and non-linear problems, solutions, and these solutions is to make a comparison of the analytical solutions.

Non-linear integral differential equation can be obtained or developed some criteria in order to be represented in the presence of the analytical solution. This work is the most used this criteria for the duration of the variational Iteration, homotopy perturbation and variational homotopy pertürbation.

Variational iteration method is a new technique developed in 1997. The solution of non-linear partial differential equations, especially very effective. This is supplied with the initial conditions problems's notwhere. The variational theory by determining a solution reached through Lagrange parameter. The selection of initial conditions will affect the rate of the analytical solution, it is important to approach. 

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modem mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions.

The variational iteration method is particularly suitable for solving this kind of problems. Approximate initial/boundary conditions and point boundary initial/conditions are also discussed, with the variational iteration method being capable of recovering the correct initial/boundary conditions and finding the solutions simultaneously. 

The variational iteration method is remarkably effective for solving differential  equations of the fifth order. Four iterations are enough to obtain very highly accurate solution. The results show that the variational iteration method is a powerful mathematical tool for finding the numerical solutions of differential equations.

Homotopy perturbation method (HPM), founded in 1998 by Ji-Huan He. He, and the concept of homotopy perturbation technique when creating a method of nonlinear problems, easy solution linear problems. It is well known that these methods, in the 1990s the resulting methods are basically serial solutions. Be in the form of a series of solutions and, in some cases, solutions can be obtained, this closed forms, methods and popular among scientists working in different branches have different interpretations of the solutions a huge advantage. Homotopy perturbation linear and non-linear ordinary and partial differential equations, integral equations to solve can be applied. According to this method in the Homotopy technique with the parameter homotopy is established.

The homotopy perturbation method (HPM) is a series expansion method used in the solution of nonlinear partial differential equations. The method employs a homotopy transform to generate a convergent series solution of differential equations. This gives flexibility in the choice of basis functions for the solution and the linear inversion operators while still retaining a simplicity that makes the method easily understandable from the standpoint of general perturbation methods. 

This parameter is considered to be a small parameter. Known advantages of this method of covering the methods and Perturbation in the topology can be fixed easily, homotopy's simple problems in the converted solution can be achieved. This method of nonlinear wave equations, initial value problems, nonlinear oscillation equations and integral equations can be applied. In many cases, this method provides a quick way to serial solution.

The homotopy perturbation method (HPM) is a series expansion method used in the solution of nonlinear partial differential equations. The method employs ahomotopy transform to generate a convergent series solution of differential equations. This gives flexibility in the choice of basis functions for the solution and the linear inversion operators (as compared to the Adomian decomposition method), while still retaining a simplicity that makes the method easily understandable from the standpoint of general perturbation methods. The HPM was introduced by Ji-Huan He of Shanghai University in 1998. The HPM is a special case of the homotopy analysis method (HAM) developed by Liao Shijun in 1992. The HAM uses a so-called convergence-control parameter to guarantee the convergence of approximation series over a given interval of physical parameters.

Recently developed the variational homotopy perturbation method  is a method of combining variational Iteration and homotopy  perturbation Methods. This new method with linear and non-linear equations to approximate solutions to analytical or any restrictive assumption that it is possible to find solutions to the analytical solution without requiring close and linearization.

We use this method for solving higher dimensional initial boundary value problems with variable coefficients. The developed algorithm is quite efficient and is practically well suited for use in these problems. 

Variational homotopy perturbation method has a very simple solution procedure and absorbs all of the positive features of variational iteration and homotopy perturbation methods and is highly compatible with the diversity of the physical problems. In this work, we will use variational homotopy perturbation method to solve equations with initial and boundary conditions. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. This paper considers the effectiveness of the variational homotopy perturbation method in solving equations.

This thesis consists of four parts.  The first part of the thesis contains  entry and variational iteration,  Homotopy perturbation and Variational homotopy perturbation method with the corresponding literature surveys. 2. section variational iteration, homotopy perturbation and the variational homotopy perturbation methods theoretical framework are addressed and these methods are relevant examples. 3. section is Benjamin-Bona-Mahony equation (BBM) variational iteration and variational perturbation methods, with the resolution of the homotopy and     (2 + 1)-dimensional Broer-Kaup Solitary Wave solutions of the system and the homotopy perturbation method with the Sagamath using the program that we have done the studies. 4. we have done studies of these methods after the chapter conclusions of the result.