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In this paper, the dispersive Schrödinger-Hirota equation with Kerr and parabolic laws in the presence of chromatic and intermodal dispersions were investigated for the first time. The Sinh-Gordon equation expansion method (ShGEEM) was used to get analytical optical solutions of the models. As a result of the application, dark, bright, kink and singular soliton solutions were obtained. 3D, 2D and contour plots of the obtained results were depicted. In addition, the effect of the changing parameter on the wave behavior was investigated and simulated by 2D plots. Due to the fact that the method is applicable and easy to use, no restrictions were encountered. It can be emphasized that the obtained soliton solutions and parameter analyses are innovative for the related investigations in the literature.

Because the importance of the optical wave propagation in fbers, we seek for the optical travelling wave solutions and efect of the third-order dispersion parameter for the extended nonlinear Schrödinger equation (NLSE) which is used to describe the femtosecond(fs) pulse propagation in optical fber. We have used the modifed F-expansion method because it is efective, sufcient and ofers more solutions for this kind of problems. First, we obtain the nonlinear ordinary diferential form (NODE) by using the wave transformation on the investigated extended nonlinear Schrödinger equation. Then, we propose the modifed F-expansion method to get solution of the equation as the NODE form, and by calculating the balancing constant over the NODE form. By substituting the proposed solution function and its derivatives in the NODE form, we obtain an algebraic equation system over this NODE. With the solution of this system, we obtain the appropriate solution sets for the undefned parameters, and then, using the appropriate sets, the proposed solution function, the F-expansion solution functions, and the wave transform, together, we obtain the solution functions of the investigated problem. In order to contribute to the physical interpretation of the problem, 3D, 2D and contour graphs of the obtained solution functions were plotted and interpreted. The problem has not been examined by considering two diferent situations with the efect of third order dispersion parameter as in this study, before. In this respect, the results obtained in the study are new, and it is believed that they will contribute to the studies in this feld.

This article investigates the optical soliton solutions of the time-fractional Biswas–Arshed (BA) equation, which is a new model for soliton transmission through optical fbers where the eigen-phase modulation is negligible and thus removed. We have studied both models including M-truncated or beta derivative operators. Firstly, the time-fractional BA equations have been transformed into a nonlinear ordinary diferential equation using appropriate wave transformations. The singular, singular-periodic, and dark soliton solutions have been derived using the extended rational sine–cosine and sinh–cosh techniques have been applied and the existing conditions for the obtained solutions have been introduced. Utilizing a computer algebraic system, we have verifed that all of the derived solutions satisfy the fractional Biswas–Arshed equation. Additionally, we have compared the results for the beta and M-truncated derivatives and have examined how the equation’s parameters afect the amplitude of the solitons in 2D and 3D graphical demonstrations. The attained solutions may aid in the comprehension of wave propagation in optical fbers and may contribute to the telecommunication sector.

In this paper, we consider the Boussinesq equation which is an important equation and it is widely used in coastal engineering, harbors, shallow seas and water wave to model weakly nonlinear and long wave approximation. Exact traveling wave solutions in such equations are extremely valuable in analytical and numerical theories. To compute the soliton solutions of this model, we employ an enhanced modified extended tanh expansion method. We demonstrate that the characteristics of soliton solutions may reflect the dispersion of propagation on wave fronts and have an acceptable reliance on parameter values. Bright, dark, bright-singular, dark-singular, periodic-bright singular, periodic dark-singular solutions identified and three-dimensional (3D), contour and two-dimensional (2D) layouts are represented graphically.

Purpose: This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. Methodology: After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear partial differential equation problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is formed and by aid of the any symbolic calculation package the unknown parameters are obtained. In order to gain the soliton solutions, the proposed series form, the solution functions of the method, and the cWT are brought together and it is checked that the obtained solutions satisfy the main equation. By considering the functions that satisfy the main equation as solutions, graphical presentations are made and interpreted in order to better understand the obtained results. Findings: By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright–dark have been obtained, and it has been shown that the method can be applied effectively for such equations. Originality: The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright soliton pulse solutions have not been reported before.

This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $ M- $ truncated derivatives. The solutions are compared in the $ 2D $ and $ 3D $ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.

In this paper, we study the novel exact solutions of the (2+1)-dimensional Biswas–Milovic equation for the description of pulse propagation in optical fiber. We successfully constructed some important solutions, such as dark, singular, combined dark-singular soliton, singular periodic wave and rational function solutions, have been analytically obtained using the extended tanh method. These solutions might play an important role in the engineering and physics fields. It is shown that the considered method provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic and other physical phenomena.

In this research paper, a numerical method, named the three-step Ultraspherical wavelet collocation method, is presented for solving some nonlinear multi-dimensional parabolic partial differential equations. The method is third-order accurate in time. In this method, the three-step Taylor method is used to get the time derivative, while the Ultraspherical wavelet collocation method is used to get the space derivatives. Ultraspherical wavelets have good properties which make useful to carry out this aim. The presented method is developed for Burgers’ equation, Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation, and quasilinear parabolic equation. Three illustrative numerical problems are solved to demonstrate the efficiency, simplicity, and reliability of the presented method.

In the article, we are concerned with the oscillatory solutionsof a class of fractional differential equations. By using generalized Riccatifunction and Hardy inequalities, we present some oscillation criterias. As aresult we give some examples that validity of the established results., Bu makalede, kesirli mertebeden diferensiyel denklemlerin birsınıfının salınımlı çözümleriyle ilgilenildi. Genelleştirilmiş Riccatifonksiyonu ve Hardy eşitsizlikleri kullanılarak, baz salınımlılık kriterlerisunuldu. Sonuç olarak, kurulan sonuçları sağlayan bazı örnekler verildi.

By applying the concept (and theory) of fractional $q$ -calculus, we first define and introduce two new $q$ -integral operators for certain analytic functions defined in the unit disc $\mathrm{\scr U}$ . Convexity properties of these $q$ -integral operators on some classes of analytic functions defined by a linear multiplier fractional $q$ -differintegral operator are studied. Special cases of the main results are also mentioned.

Within the context of this study, we have improved a computer program on the syndrome decoding method for correcting codewords received incorrect. This program generates code from given generator matrix and calculates the hamming distance which appertains to this code, also, it finds Slepian(1960) standart array. It gives a list of the decoding table of code and an error pattern relating to be received incorrect codewords. We have used Maple Computer Algebra for calculations [5]. The Algorithm, we have given, reproduces different results from given generator matrix. In here, we have chosen a test problem whose code word’s lengthy is 4. But, it is possible for algorithm that it can make calculations for longer codes.

Bu çalışmada Stokastik analiz, Stokastik diferansiyel denklemler konuları tanıtıldıktan sonra Stokastik Kararalılık kavramı detaylı bir şekilde incelenmektedir. Teorik incelemeler tamamlandıktan sonra bazı stokastik diferansiyel denklem türlerinin bilgisayar cebiri ile basit çözüm algoritmaları geliştirilmiştir. Elde edilen çözümler karşılaştırılmış olup grafiksel gösterimi sağlanmıştır. In this study Stochastic Calculus, Stochastic Differential Equations and the concept of Stochastic Stability issues are introduced after dynamic systems are examined in detail. After presenting theoritical backround we have introduced stability examinitation and exact solution of some class of Stochastic differential equations based on a computer algebra system software. We have developed simple and efficient algorithms to solve Stochastic Differential equations and explanation of stability criterias. At the end of the study, the solution algorithms and outputs are discussed and given in a clean way. 80

Bu tezde, Legendre dalgacığı operasyonel türev matris metodu kullanılarak, yüksek mertebeden ve kesirli mertebeden diferansiyel denklemlerin ve denklem sistemlerinin doğrusal ve doğrusal olmayan formlarının nümerik çözümleri için yeni bir yaklaşım geliştirilmiştir. Öncelikle, operasyonel matris ve kesirli türevi bazı özel koşullar altında, Legendre dalgacığı ve Legendre polinomlarının bazı önemli özellikleri kullanılarak formüle edilmiştir. Sonrasında, yüksek mertebeden ve kesirli mertebeden diferansiyel denklem ve denklem sistemleri bu operasyonel matrisler yardımıyla cebirsel denklem sistemlerine dönüştürülmüştür. Tezde önerilen metot her bölümün sonunda yeterli sayıda aydınlatıcı örnekle test edilmiştir. Sonuç olarak, bazı bilinen metotlarla karşılaştırması gösteriyor ki, Legendre dalgacığı operasyonel türev matrisi metodunun en büyük avantajı sadeliği ve hesaplamalardaki anlaşılabilirliğidir. This thesis introduces a new numerical approach to solve high order and fractional order differential equations of the linear and non-linear forms and systems of such equations utilizing the Legendre wavelet operational matrix method. We first formulated the operational matrix and its fractional derivatives in some special conditions by using some significant features of Legendre wavelets and shifted Legendre polynomials. Then, the high order and fractional order differential equations and systems of such equations were transformed to a system of algebraic equations by using these operational matrices. At the end of each chapter of the thesis, the introduced tecnique is tested on several illustrative examples. Comparing the methodology with several recognized methods demonstrates that the most important advantages of the introduced method are the understandibility of the calculations and its accuracy. 133

Bu tezde çalışmasında Ginzburg-Landau denklemi, Korteweg-de Vries denklemi vb. pek çok öneme sahip doğrusal olmayan kısmi diferansiyel denklemlerin çözüm biçimini kolaylaştırması ve analitik çözüme olan yaklaşımının doğruluğu ile oldukça etkili bir Chebyshev dalgacık sıralama metodu tartışılıp, uygulanmaya çalışılmıştır. Bu metot Cheyshev dalgacık metodu ve Legendre dalgacık sıralama metoduyla kıyaslanmış ve bu metodun ne kadar güçlü doğruluğa sahip olduğu vurgulanmıştır. Öncelikli olarak bu metodun ana fikri, problemlere Chebyshev dalgacık seri açılımları ile yaklaşım sağlamaktır. Chebyshev dalgacık sıralama metodunu uygulayabilmek için ilk olarak integralin operatör matrisleri elde edilmekte ve sonrasında da çözümünün yapılması arzu edilen doğrusal olmayan kısmi diferansiyel denklem bu matrisler yardımı ile daha kolay çözülebilir cebirsel bir denklem sistemine dönüştürülmektedir. Elde edilen bu cebirsel sistem Maple bilgisayar programı kullanılarak çözülmeye çalışılmaktadır.Chebyshev dalgacık sıralama metodu Ginzburg Landau, Korteweg-de Vries ve bazı önemli denklemlere uygulanması sonucunda elde edilen nümerik çözümler ile analitik çözümler birbiri ile karşılaştırılmış ve nümerik çözümün analitik çözüme ne kadar iyi yaklaştığı tablolar ve şekiller yardımıyla gösterilmiştir.Daha sonra, Chebyshev dalgacık sıralama metodunun CWM ve LWCM metoduna göre doğrusal olmayan kısmi diferansiyel denklemlerin çözümü için çok daha uygun olduğu gözlenmiştir. Sonuç olarak bu yöntemin en büyük avantajının diğer göz önüne alınan metotlarla karşılaştırıldığında bunun uygulanabilirliğinin daha basit olduğu ve nümerik çözümünün ise analitik çözüme çok daha iyi yaklaştığı tespit edilmiştir. In this thesis, Chebyshev wavelet collocation method is firstly introduced and also applied to different important nonlinear partial differential equations as Ginzburg- Landau equation, Korteweg-de Vries equation and etc. This method is compared with the each other and showed that how this method powerfull as well as the Legendre wavelet collocation method.Primarily, the main idea of this method is to provide the convergency in terms of Chebyshev wavelet series. To apply the Chebyshev Wavelet collocation method, firstly the operator matrix needs to derived and then transform the nonlinear partial differential equation into a more easily solvable algebraic system by using these matrices. Once obtaining this algebraic system then it may be solved by using Maple packet program.By applying the Chebyshev wavelet collacation method to Ginzburg Landau, Korteweg-de Vries and the other important equation as it mentioned above, obtaining numerical results are compared with the analytical solutions carefully and the results supported by graphs, illustrated by tables. It is finally pointed out that how the numerical solution approximates to the analytical solution.Moreover, we it is observed that the Chebyshev wavelet collocation method is much more suited for solving nonlinear partial diffrential equations than the Chebyshev Wavelet and Legendre Wavelet collocation method. Consequently, it is concluded that the applicability of this method is more simple with respect to the other techniques and also it is seen that the numerical solution has a much better approach to analytical solution. 144

Bu tez dört bölümden oluşmaktadır. Birinci bölüm giriş kısmına ayrılmıştır. Bu bölümde, kesirli türev ve diferensiyel denklemlerin salınımlılığına yönelik tarihsel süreç anlatılmıştır. İkinci bölümde, amaca yönelik temel kavramlara değinilmiştir. Çalışmada kullanılacak olan Modifiye Riemann-Liouville, Liouville kesirli türev operatörü ve Riemann-Liouville kesirli fark operatörü tanımları bu bölümde verilmiştir. Üçüncü bölümde, genelleştirilmiş Riccati dönüşümleri, genelleştirilmiş Philos tipi çekirdekleri ve değişken dönüşümleri kullanılarak kesirli mertebeden denklem sınıflarının çözümleri için salınımlılık kriterleri kurulmuştur. Ayrıca elde edilen salınımlılık kriterlerini aydınlatıcı bazı örnekler verilmiştir. Dördüncü bölümde ise sonuçlar ve bazı öneriler verilmiştir. This thesis consists of four chapters. The first chapter is devoted to introduction. In this section, we present the history of fractional derivatives and oscillation of differential equations. It is mentioned in the second chapter that the fundamental concepts related to the purpose of thesis. Definitions of Modified Riemann-Liouville, Liouville fractional derivative operator and Riemann-Liouville fractional difference operator to be used in the study are given in that section. In the third chapter, by using generalized Riccati transformations, generalized Philos type kernels and variable transformations, we establish oscillation criteria for the fractional differential equations. And also some illustrative examples are given to the obtained criteria. In the fourth section, the conclusions and some suggestions are given. 80

Bu çalışmada kesirli mertebeden lineer adi ve kısmi diferansiyel denklemlerin nümerik çözümlerinin sinc-Galerkin yöntemi ile elde edilmesi araştırılmıştır. Bunun için sinc-Galerkin yöntemi ve yöntemin kesirli mertebeden lineer adi ve kısmi diferansiyel denklemlere uygulanması için gerekli teoremler ifade ve ispat edilmiştir.Sonuç olarak, sinc-Galerkin yöntemi ile elde edilen çözümler diğer nümerik yöntemlerle bulunan çözümlerle grafiksel ve nümerik olarak karşılaştırılmıştır. Bu karşılaştırmalardan sinc-Galerkin yönteminin kesirli mertebeden lineer adi ve kısmi diferansiyel denklemlerin nümerik çözümünü elde etmede çok etkili bir yöntem olduğu görülmektedir. According to the best knowledge of the author, it is not found the study concerning the finding the solutions of the fractional order linear ordinary and partial differential equations by using sinc-Galerkin method in the literature. In this study, the solutions of fractional order linear ordinary and partial differential equations are obtained by means of sinc-Galerkin method. With this aim related teorems with their proofs are given. Obtained solutions are compared by the solutions of other numerical methods, numerically and graphically.As a conclusion sinc-Galerkin method is applied to some fractional order linear ordinary and partial differential equations to obtain their numerical solutions. Numerical comparisons show that sinc-Galerkin method is very effective and aplicaple to solve fractional order linear ordinary and partial differential equations. 96

İki sınır şartı ile verilen 2. mertebeden singüler, Dirichlet-tip sınır değer problemlerini nümerik olarak çözmek için pek çok kullanışlı metod vardır. Bunlardan bir tanesi de Sinc-Galerkin metodudur. Bu metod, Whittaker kardinal fonksiyonu kullanılarak fonksiyonlara ve onların türevlerine yaklaşmaya dayanır. Diferensiyel denklem matris sistemlerini çözmek için gerekli olan hiçbir nümerik integral kullanmadan, iç çarpımların yaklaşımları yardımıyla cebirsel denklem sistemine indirgenir. Bu tezde, böyle problemlerin nümerik çözümünde Sinc-Galerkin metodunun çok etkili bir araç olduğu gösterilmiştir. Tezin sonunda da Sinc-Galerkin metodu, homojen ve homojen olmayan sınır şartlarıyla verilen örneklerle test edilmiştir. There are few methods practicable to numerically solve second-order singular Dirichlet-type boundary value problems with two-point boundary conditions. In this thesis we indicate that the Sinc-Galerkin method is a very powerful tool in numerically solving such problems. The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. The differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products without any numerical integration which is needed to solve matrix system. At the end of the thesis, the method was tested on examples with homogeneous and non-homogeneous boundary conditions. 89