Lineer olmayan kısmi diferensiyel denklemler ve tam çözümleri
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Date
2020-02-07
Authors
Çankal, Pelin Doğan
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Publisher
Bursa Uludağ Üniversitesi
Abstract
Bu tezde, bazı oluşum tipi ((1)+(1), (2)+(1) ve (3)+(1) boyutlu) lineer olmayan reel ve kompleks kısmi türevli diferansiyel denklemler ele alındı. Bu denklemler çeşitli bilim dallarındaki birçok fiziksel olayı betimleyen lineer olmayan matematiksel modellerdir. Söz konusu denklemlerin analitik tam çözümleri elde edilmeye çalışıldı. Bu bağlamda literatürde yoğun bir şekilde incelenmekte ve geliştirilmekte olan exp , Kudryashov ve tanh fonksiyon metotları ele alındı. Bu yöntemler göz önüne alınan fiziksel modellere ayrı ayrı tatbik edildi. Bunun yanında lineer olmayan denklemlerin bilineerleştirilmesine dayanan Hirota yaklaşımı incelendi. Bu yaklaşımdaki bazı zorlukları hafifletmek için önerilen basitleştirilmiş Hirota metodu kullanılarak çoklu soliton çözümlerin nasıl elde edildiği araştırıldı. Çalışmamızın son kısmında ise denklemin mertebesi, derecesi veya lineerlik özelliklerine herhangi bir kısıtlama yapılmasına gerek bırakmayan Lie grup yaklaşımı ele alındı. Elde edilen sonuçların kıyaslamaları, çözüm tiplerinin fiziksel anlamları ve çözümlerin grafiksel yapıları gösterildi.
In this thesis, some evolution type ((1)+(1), (2)+(1) and (3)+(1) dimensional) nonlinear real and complex partial differential equations are considered. These equations are nonlinear mathematical models that describe many physical phenomena in various disciplines. Analytical exact solutions of these equations were tried to be obtained. In this context, Kudryashov and tanh function methods, which are extensively studied and developed in the literature, are discussed. These methods were applied separately to the physical models considered. In addition, the Hirota approach based on the bilinearization of nonlinear equations was examined. In order to alleviate some of the difficulties in this approach, we investigated how multiple soliton solutions were obtained by using the proposed simplified Hirota method. In the last part of our study, Lie group approach, which does not require any restriction on the order, degree or linearity properties of the equation, is discussed. Comparisons of the obtained results, physical meanings of the solution types and graphical structures of the solutions were also demonstrated.
In this thesis, some evolution type ((1)+(1), (2)+(1) and (3)+(1) dimensional) nonlinear real and complex partial differential equations are considered. These equations are nonlinear mathematical models that describe many physical phenomena in various disciplines. Analytical exact solutions of these equations were tried to be obtained. In this context, Kudryashov and tanh function methods, which are extensively studied and developed in the literature, are discussed. These methods were applied separately to the physical models considered. In addition, the Hirota approach based on the bilinearization of nonlinear equations was examined. In order to alleviate some of the difficulties in this approach, we investigated how multiple soliton solutions were obtained by using the proposed simplified Hirota method. In the last part of our study, Lie group approach, which does not require any restriction on the order, degree or linearity properties of the equation, is discussed. Comparisons of the obtained results, physical meanings of the solution types and graphical structures of the solutions were also demonstrated.
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Keywords
Tam çözüm, Exact solution, Lie simetrisi, Soliton, Oluşum türü denklemler, Lie symmetry, Evolution type equations
Citation
Çankal, P. D. (2020). Lineer olmayan kısmi diferensiyel denklemler ve tam çözümleri. Yayınlanmamış yüksek lisans tezi. Bursa Uludağ Üniversitesi Fen Bilimleri Enstitüsü.