Kesir mertebeli türeve sahip bazı lineer olmayan fiziksel denklemlerin çözümleri üzerine
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Tarih
2022
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Ege Üniversitesi, Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Son yıllarda, doğa bilimleri ve mühendislik gibi uygulamalı alanlarda,
birçok problemin kesirli mertebeden türeve sahip lineer olmayan diferansiyel
denklemler ile modellenmesi ve bu formdaki denklemlerin gerçeğe daha yakın
sonuçlar vermesi, kesirli analiz konusunun önemini hızla arttırmıştır.
Bu tezin amacı; karmaşık olguları ifade eden kesir mertebeli lineer olmayan
denklemlerin yeni tam çözümlerini araştırmak, ulaşılan çözümlerin fiziksel
davranışlarını daha iyi anlamak için çözüm tiplerini belirlemek, bazı kesirli
türev tanımlarını karşılaştırmak ve literatürde popüler olan analitik çözüm
yöntemlerini değerlendirmektir.
Bu çalışmada; lineer olmayan zaman-kesirli yaklaşık uzun su dalgası denklemi,
lineer olmayan birleştirilmiş zaman-kesirli Boussinesq-Burger denklemi,
lineer olmayan (2 + 1)-boyutlu uzay-zaman kesirli genelleştirilmiş Nizhnik-
Novikov-Veselov denklemi, lineer olmayan uzay-zaman kesirli düzenli uzun
dalga denklemi, uzay-zaman kesirli simetrik düzenli uzun dalga denklemi,
zaman-kesirli (2 + 1)-boyutlu lineer olmayan Heisenberg ferromanyetik spin
zinciri denklemi ve zaman-kesirli iyon sesi ve Langmuir dalgaları denklemi
uyumlu kesirli türev tanımına göre ele alınmıştır. Ek olarak, uzay-zaman
kesirli eşit genişlikli dalga denklemi, uzay-zaman kesirli (2 + 1)-boyutlu Boussinesq
dinamik denklemi, uzay-zaman kesirli birleştirilmiş Boussinesq denklemi
ve zaman-kesirli biyolojik popülasyon denklemi; modifiye edilmiş Riemann-
Liouville, uyumlu kesirli türev, beta türev ve kesilmiş M-kesirli türev tanımları
kullanılarak ifade edilmiştir. Sığ su dalgaları, iyonik akustik dalgalar, uzun dalgaların yayılımı, modern mıknatıs teorisi, plazma fiziği ve biyoloji gibi alanlarda karşımıza çıkan bu
modellerin tam çözümleri; modifiye edilmiş basit denklem yöntemi, üstel
rasyonel fonksiyon yöntemi, genişletilmiş
G1/G2-genişleme yöntemi, modifiye edilmiş genişletilmiş tanh-fonksiyon yöntemi ve geliştirilmiş tan φ/2-genişleme
yöntemi ile araştırılmıştır. Ulaşılan çözümlerin dalga tipleri belirlenmiştir ve
bazı fiziksel yorumlara yer verilmiştir. Ayrıca, elde edilen dalga çözümlerinin
doğruluğunun ve geçerliliğinin kontrolünde, bazı hesaplamalarda ve çeşitli
formlardaki (üç boyutlu, iki boyutlu, kontur ve yoğunluk) grafiklerin çiziminde
Mathematica paket programı kullanılmıçtır. Önerilen denklem ve denklem
sistemlerinden ulaşılan; tekil, tekil-periyodik, karanlık, parlak, birleşik-tekil,
bükülme ve rasyonel gibi anlamlı dalga formlarının, günümüzde pek çok fiziksel
modelin geliştirilmesinde, yorumlanmasında ve gerçek dünya problemlerine
uyarlanmasında etkili olacağı düşünülmektedir.
In recent years, in applied fields such as natural sciences and engineering, the modelling of many problems with nonlinear differential equations having fractional derivatives and the fact that equations in this form give more realistic results have rapidly increased the significance of fractional analysis. The aim of this thesis is to investigate for new exact solutions of fractional nonlinear equations expressing complex phenomena, to determine the solution types to better understand the physical behaviour of the solutions, to compare some fractional derivative definitions and to evaluate the analytical solution methods popular in the literature. In this study, the nonlinear time-fractional approximate long water wave equation, the nonlinear coupled time-fractional Boussinesq-Burger equation, the nonlinear (2 + 1)-dimensional space-time fractional generalized Nizhnik- Novikov-Veselov equation, the nonlinear space-time fractional regularized long wave equation, the space-time fractional symmetric regularized long wave equation, the time-fractional (2 + 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation, the time-fractional ion sound and Langmuir waves equation are handled according to the definition of conformable fractional derivative. In addition, the space-time fractional equal width wave equation, the space-time fractional (2 + 1)-dimensional Boussinesq dynamical equation, the space-time fractional coupled Boussinesq equation, and the time-fractional biological population equation are expressed using modified Riemann-Liouville fractional derivative, conformable fractional derivative, beta derivative, and truncated M-fractional derivative definitions. The exact solutions of these models, which are encountered in fields such as shallow water waves, ionic acoustic waves, propagation of long waves, modern magnet theory, plasma physics, and biology are investigated with the modified simple equation method, the exponential rational function method, the extended G1/G2-expansion method, the modified extended tanh-function method, and the improved tan φ2-expansion method. The wave types of the solutions are determined and some physical interpretations are given. Moreover, the Mathematica package program is used to check the accuracy and validity of the obtained wave solutions, for some calculations, and for drawing graphics in various forms (three dimensional, two dimensional, contour, and density). It is thought that meaningful wave forms such as singular, singularperiodic, dark, bright, combined-singular, kink, and rational, obtained from the proposed equations and equation systems, will be effective in the development, interpretation, and adaptation of many physical models to real-world problems.
In recent years, in applied fields such as natural sciences and engineering, the modelling of many problems with nonlinear differential equations having fractional derivatives and the fact that equations in this form give more realistic results have rapidly increased the significance of fractional analysis. The aim of this thesis is to investigate for new exact solutions of fractional nonlinear equations expressing complex phenomena, to determine the solution types to better understand the physical behaviour of the solutions, to compare some fractional derivative definitions and to evaluate the analytical solution methods popular in the literature. In this study, the nonlinear time-fractional approximate long water wave equation, the nonlinear coupled time-fractional Boussinesq-Burger equation, the nonlinear (2 + 1)-dimensional space-time fractional generalized Nizhnik- Novikov-Veselov equation, the nonlinear space-time fractional regularized long wave equation, the space-time fractional symmetric regularized long wave equation, the time-fractional (2 + 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation, the time-fractional ion sound and Langmuir waves equation are handled according to the definition of conformable fractional derivative. In addition, the space-time fractional equal width wave equation, the space-time fractional (2 + 1)-dimensional Boussinesq dynamical equation, the space-time fractional coupled Boussinesq equation, and the time-fractional biological population equation are expressed using modified Riemann-Liouville fractional derivative, conformable fractional derivative, beta derivative, and truncated M-fractional derivative definitions. The exact solutions of these models, which are encountered in fields such as shallow water waves, ionic acoustic waves, propagation of long waves, modern magnet theory, plasma physics, and biology are investigated with the modified simple equation method, the exponential rational function method, the extended G1/G2-expansion method, the modified extended tanh-function method, and the improved tan φ2-expansion method. The wave types of the solutions are determined and some physical interpretations are given. Moreover, the Mathematica package program is used to check the accuracy and validity of the obtained wave solutions, for some calculations, and for drawing graphics in various forms (three dimensional, two dimensional, contour, and density). It is thought that meaningful wave forms such as singular, singularperiodic, dark, bright, combined-singular, kink, and rational, obtained from the proposed equations and equation systems, will be effective in the development, interpretation, and adaptation of many physical models to real-world problems.
Açıklama
Anahtar Kelimeler
Lineer Olmayan Kesirli Kısmi Diferansiyel Denklem, Kesirli Türev, Analitik Çözüm Yöntemi, Tam Dalga Çözümü, Nonlinear Fractional Partial Differential Equation, Fractional Derivative, Analytical Solution Method, Exact Wave Solution
