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Optical soliton pulses or molecules with the Sasa-Satsuma equation have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model in birefringent fibers is examined to uncover quite important optical solitons in (2+1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Triki-Biswas equation that is a new model for soliton dynamics of sub pico-second pulses in a monomodal optical fiber with non-Kerr dispersion is considered to get optical solitos of its in this paper. Optical bright and singular solitons are given by trial equation approach. Also, singular periodic solutions are located.

The Chen–Lee–Liu equation in birefringent fibers is examined to uncover quite important optical solitons of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Biswas–Arshed model which is a new model for soliton transmission through optical fibers is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

The Biswas-Arshed model in birefringent fibers which is first written in two component form for vector solitons is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

This paper examines soliton dynamics of sub pico-second pulses modeled by Kaup–Newell equation. The modified simple equation scheme and trial equation approach retrieves dark, bright as well as singular solitons to the model. The constraint relations guarantee the existence of such solitons.

This paper studies differential group delay in optical fibers with coupled Fokas-Lenells equation. Two integration schemes are adopted. They are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are retrieved together with their respective existence criteria that are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains bright, dark and singular dispersive optical soliton solutions to Schrodinger-Hirota equation by the aid of trial equation method. Both Kerr and power laws of nonlinearity are studied. Singular periodic solutions are also obtained as a byproduct of this scheme. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

The modified simple equation scheme is employed to secure dark and singular optical soliton solution to the perturbed Gerdjikov-Ivanov equation that models pulse dynamics in optical fibers and PCF. The solvability conditions for this algorithm is also presented. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solution to perturbed Gerdjikov-Ivanov equation by trial equation approach. Bright, dark and singular soliton solutions are derived. Additional solutions such as singular periodic solutions also emerge of this integration scheme. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solutions to the Kundu Eckhaus equation with general coefficients. The Riccati Bernoulli's sub-ODE method as well as Kudryashov's scheme are employed to obtain soliton solutions to the model. This generalized earlier reported result of the model with specific coefficients. Subsequently, the multiplier approach was utilized to secure a conserved quantity with the bright soliton solution that was reported earlier. Young Foundation of Wuhan Donghu University (2017DHKZ001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

Optical soliton pulses or molecules with the Sasa–Satsuma equation have been worked by trial equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model is examined to uncover quite important optical solitons in (2 + 1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Chen–Lee–Liu equation is examined to uncover quite important optical solitons of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by trial equation technique with parameter restrictions.

The Biswas-Arshed equation in birefringent fibers is considered to get optical solitos of its in this paper. Optical dark, bright and singular solitons are given by modified simple equation approach. Also, singular periodic solutions are located.

The Biswas-Arshed model which is a new model for soliton transmission through optical fibers is taken into account for the purpose of acquiring very valuable optical solitons of its in this paper. The modified simple equation methodology is utilized for obtaining both highly important solitons and singular periodic solutions.

This paper implements two efficient integration algorithms to retrieve soliton solutions that appear with weak nonlocal nonlinearity. These are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are recovered and their existence criteria are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

In this work, we consider the perturbed Radhakrishnan-Kundu-Lakshmanan equation (RKL) which is the higher order nonlinear Schrodinger equation with cubic nonlinear terms in Kerr law. This equation is used to model propagation of solitons through an optical fiber. The multi-waves, interactions and Peregrine-like rational solutions of this equation are investigated with the aid of logarithmic transformation and symbolic computation without using Hirota bilinear forms By selecting appropriate values of the parameter, three dimensional plots are drawn to obtained solitons. Moreover, physical meanings of the solutions obtained were also presented.

In this work, we derive the complexiton solutions for Date-Jimbo-Kashiwara-Miwa (DJKM) equation using the extended transformed rational function algorithm that relies on the Hirota bilinear form of the considered equation. Additional solutions such as complex-valued solutions also fall out of this integration scheme. Multisoliton-type solutions, in other words one-soliton, two-soliton and three-soliton solutions, which comprise both wave frequencies and generic phase shifts are presented through the medium of the multiple exp-function methodology which falls out as a result of generalisation of Hirota's perturbation technique.

The trial equation method is applied to obtain soliton solutions to Lakshmanan–Porsezian–Daniel model in optical fibers, PCF and metamaterials. This integration procedure is implemented into the model with three forms of nonlinearity. Bright, dark and singular soliton solutions are recovered with conditions that guarantee their existence.

This paper retrieves soliton solutions to the perturbed resonant nonlinear Schrodinger's equation with time-dependent coefficients using the modified simple equation method. The four types of exotic non-Kerr laws studied in this paper are quadratic-cubic law, anti cubic law, cubic-quintic-septic law and triple-power law. These soliton solutions appear with constraint conditions that guarantee their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper retrieves optical soliton solution to the perturbed complex Ginzburg-Landau equation that is studied with nine different forms of nonlinearity. The trial solutions approach is the integration algorithm adopted in this paper. The perturbation terms appear with full nonlinearity to get a taste of generalized setting. Bright, dark and singular soliton solutions are obtained. The existence criteria of such solitons are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper secures bright, dark and singualr soliton solutions in birefringent fibers. Both Kerr law and parabolic law nonlinearity are studied. The trial equation method successfully retrieves these solitons with constraint relations that assure their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

The modified simple equation method applied to perturbed complex Ginzburg-Landau equation gives dark and singular solutions to the model. The perturbation terms appear with full nonlinearity. There are eight nonlinear forms studied in this paper. These solitons appear with constraint conditions that gurantee its existence and these are also presented. Chutian Scholar Program of Hubei Government in China Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this study, we perform the extended Kudryashov method to nonlinear Schrdinger equation (NLSE) with spatio-temporal dispersion that arises in a propagation of light in nonlinear optical fibers, planar waveguides, Bose-Einstein condensate theory. Four types of nonlinearity - Kerr law, power law, parabolic law and dual-power law - are being considered for the model. By using this scheme, the topological, singular soliton and rational solutions are obtained. In addition, some graphical simulations of solutions are provided. It is demonstrated that the proposed algorithm is effective and can be handled for many other nonlinear complex differential equations.

Bu çalışma, 19-25 Ekim 2016 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics (ICNAAM)’da bildiri olarak sunulmuştur. In this paper, we study with some graph descriptors also called topological indices. These descriptors are useful in determination of some properties of chemical structures and preferred to some earlier descriptors as they are more practical. Especially the first and second Zagreb indices together with the first and second multiplicative Zagreb indices are considered and they are calculated in terms of the smallest and largest vertex degrees and vertex number for some well-known classes of graphs. Selçuk Üniversitesi

In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI) equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method. Keywords: Fractional Sawada–Kotera–Ito equation, Lie symmetry, Riemann–Liouville fractional derivative, Conservation laws, Exact solutions

In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

In this work, we present (G'/G,1/G)-expansion method for solving fractional differential equations based on a fractional complex transform. We apply this method for solving space-time fractional Cahn-Allen equation and space-time fractional Klein-Gordon equation. The fractional derivatives are described in the sense of modified Riemann-Lioville. As a result of some exact solution in the form of hyperbolic, trigonometric and rational solutions are deduced. The obtained solutions may be used for explaining of some physical problems. The (G'/G,1/G)-expansion method has a wider applicability for nonlinear equations. We have verified all the obtained solutions with the aid of Maple.

We applied the multiple exp-function scheme to the (2+1)-dimensional Sawada-Kotera (SK) equation and (3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

Bu çalışma, 19-25 Eylül 2012 tarihleri arasında Kos[Yunanistan]’da düzenlenen International Conference of Numerical Analysis and Applied Mathematics (ICNAAM)’da bildiri olarak sunulmuştur. The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number and in some of these methods, the minimal polynomials of several algebraic numbers are used. Here we obtain the minimal polynomial of one of those numbers, cos(2 pi/n), over the field of rationals by means of the better known Chebycheff polynomials for odd q and give some of their properties. We calculated this minimal polynomial for n is an element of N by using the Maple language and classifying the numbers n is an element of N into different classes. European Soc Computat Methods Sci, Engn & Technol (ESCMSET) R M Santilli Fdn

Bu çalışma, 19-25 Mayıs 2012 tarihleri arasında Kos[Yunanistan]’da düzenlenen International Conference of Numerical Analysis and Applied Mathematics (ICNAAM)’da bildiri olarak sunulmuştur. The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of these numbers over the field of rationals by means of the better known Chebycheff polynomials and the Maple language. European Soc Computat Methods Sci, Engn & Technol (ESCMSET) R M Santilli Fdn

In this work, we consider the motion of chain ball drawing with constant force in the frictionless surface which is a class of the Painleve–Gambier equations. We apply Jacobi's method which enables us to obtain Lagrangians of any second-order differential equation. It is comprised that the Lagrangian obtained by Musielak's method is the particular case of the many Lagrangians that can be obtained by Jacobi's method. In addition, we obtain integrating factors and first integrals for the equation in question by Ibragimov's variational derivative approach.

Bu çalışma, 19-15 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. The aim of this work is to derive the connection among cycle, proper cycle, right and left neighbour of indefinite reduced binary quadratic form F(x,y) = ax(2) + bxy + cy(2) of discriminant Delta = b(2) - 4ac. European Soc Comp Methods Sci & Engn

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. In this paper, we study the primes in the ring Z[w], where w = exp(2 pi i/3) is a cubic root of unity. We gave a classification of them and some results related to the use of them in the calculation of cubic residues are obtained. European Soc Comp Methods Sci & Engn

Bu çalışma, 21-26 Haziran 2010 tarihleri arasında Sozopol[Bulgaristan]'da düzenlenen 2. International Conference on Application of Mathematics in Technical and Natural Sciences'da bildiri olarak sunulmuştur. The main purpose of this paper is to construct partial twisted zeta-type function, which interpolates q-generalized Frobenius-Euler numbers and twisted q-generalized Frobenius-Euler numbers. We give some applications related to this function. Akdeniz Üniversitesi

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. The main purpose of this paper is to present a new example of deficiency one groups by considering the split extension of a finite cyclic group by a free abelian group having rank two. European Soc Comp Methods Sci & Engn

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. Let us define A = [a(ij)](i,j=0)(n-1) and B = [b(ij)](i,j=0)(n-1) as n x n Toeplitz and Hankel matrices, respectively, such that a(ij) = Pi-j and b(ij) = Pi+j, where P denotes the Pell number. We present upper and lower bounds for the spectral norms of these matrices. European Soc Comp Methods Sci & Engn

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. In this work, subgroups of a special class of discrete subgroups of PLS(2, R), namely the ones of the first kind with genus 0, have been studied. We establish a technique to compute the genus of these subgroups in terms of the genus of easier groups. The method established here can be used for triangle groups, surface groups and Hecke groups (including the well-known modular group). European Soc Comp Methods Sci & Engn

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rhodes[Yunanistan]’da düzenlenen International Conference on Numerical Analysis and Applied Mathematics’da bildiri olarak sunulmuştur. In this paper the Estrada index of Hermite matrix is firstly defined and investigated. In fact this is a natural generalization of Estrada, distance Estrada and Laplacian Estrada indices. Thus all properties about them can be handled by this new index. European Soc Comp Methods Sci & Engn

Bu çalışma, 19-25 Eylül 2010 tarihleri arasında Rodos[Yunanistan]'da düzenlenen International Conference on Numerical Analysis and Applied Mathematics'de bildiri olarak sunulmuştur. The aim of this paper is to unify generating functions of the some special numbers and polynomials. By using these generating functions, we modify Apostol-Bernoulli and Apostol-Euler polynomials. We also give some new relations on these numbers and polynomials.

Bu çalışma, 18-22 Eylül 2009 tarihleri arasında Rethymnon[Yunanistan]’da düzenlenen Exponential diophantine equationsprimitive divisors’da bildiri olarak sunulmuştur. We find all solutions of Diophantine equation x(2) + 11(2k) = y(n), x >= 1, y >= 1, k is an element of N, n >= 3. We give p-adic interpretation of this equation. Akdeniz Üniversitesi Greek Minist Educ & Religious Affairs European Soc Computat Methods Sci & Engn

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using multivariate p-adic q-integrals. These sums help us to study the relationships between p-adic q-integrals and non-archimedean combinatorial analysis. Akdeniz Üniversitesi

In this paper the authors investigate hypersurfaces M of a semi-Euclidean space E s n + 1 , n ≥ 4 , satisfying ( α C + β R ) ⋅ H = L k Q ( g , H k ) , k = 1 , 2 , 3 . Using obtained results they show additional curvature properties of investigated hypersurfaces.