International Journal of
Mathematics And its Applications

Abstract: The Kundu-Eckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Besides, we compare the behaviour of the solutions obtained with the only exact solutions given in the literature through fractional calculus. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

Keywords: Kundu-Eckhaus equation, Nonlinear Schr\"{o}dinger equation, Adomian decomposition method.

O.Gonz\'alez-Gaxiola
Departamento de Matem\'aticas Aplicadas y Sistemas, Universidad Aut\'onoma Metropolitana-Cuajimalpa, Cuajimalpa, M\'exico D.F.
E-mail: ogonzalez@correo.cua.uam.mx

Abstract: In this note we construct finite field of order $p$ (here $p$ is a positive prime) and $p^{2}$ for $p>2$ through even square elements of $Z_{2p} $. It has been already noticed that a finite field of order $p^{2} $, $p>2$ can be directly constructed without using the concept of quotient rings. We utilize the same technique to yield a finite field of order $p^{2} $ for $p>2$ however here we use the notion of even square elements of a ring $R$. It is noticed that for all the finite fields constructed in this article the reducing modulo $m$ is a composite integer.

Keywords: Finite field, matrix field, matrix, even square element.

S.K.Pandey
Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Daijar, Jodhpur, Rajasthan, India.
E-mail: skpandey12@gmail.com

Abstract: In this paper, authors investigate the summation and closed form relations to certain types of generalized higher order alpha difference equations for finding the values of alpha multi series to circular functions in the field of finite difference methods. We provide examples verified by MATLAB to illustrate the alpha multi series to circular functions.

Keywords: Alpha-difference equation, Alpha multi-series, Closed form solution, Circular functions.

G.Britto Antony Xavier
Department of Mathematics, Sacred Heart College, Tirupattur, Vellore, Tamil Nadu, India.
E-mail: brittoshc@gmail.com

Abstract: In this paper, the notion of intuitionistic fuzzy cone metric space is introduced. Some common fixed point theorems for occasionally weakly compatible mapping in intuitionistic fuzzy cone metric space are stated and proved.

Keywords: Intuitionistic Fuzzy Cone Metric, Occasionally Weakly Compatible, Fixed Point.

G.R.Kanna
Department of Mathematics, G.T.N. Arts College, Dindigul, Tamilnadu, India.
E-mail: ranjithkanna91@gmail.com

Abstract: In this paper, we introduce $\alpha, \ k$-Laplace Transform which is an extension of Generalized Laplace Transform (GLT) obtained by an inverse difference operator in the field of Digital Signal Processing (DSP). It provides a better platform to study the nature and functioning of relevant signal. Here we take $k$ as variable and $\alpha$ as parameter. Suitable examples with relevant diagrams which are generated and verified using MATLAB are inserted to validate our main results.

Keywords: Generalized Laplace Transform, Generalized difference operator, factorials and Polynomial-binomial.

G.Britto Antony Xavier
Department of Mathematics, Sacred Heart College, Tirupattur, Vellore, Taminadu, India.
E-mail: brittoshc@gmail.com

Abstract: Nowadays many real life problems are identified with Fuzzy set theory. The Fuzzy set theory is a useful tool to describe the situation in which data are imprecise or vague or uncertain. This set theory is completely described by its membership function. A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the interval $[0, 1]$ to indicate the degree of belongingness to the set under consideration. The degree of non belongingness is just automatically the complement to \ "1" of the membership degree. But many times, a human being does not express the degree of non membership as the complement to \ "1".There may be some hesitation about the belongingness and non-belongingness. This missing data or hesitation is accomplished by a set known as intuitionistic fuzzy set. In this paper, Milne's Predictor - Corrector method is used for finding numerical solution of an intuitionistic fuzzy differential equation (IFDE). The proposed method is based on the concept of generalized differentiability. IFDE is transformed into four ordinary differential systems and then Milne's Predictor - Corrector method is applied.Also,the convergence and stability of the proposed method is given and its applicability is illustrated by solving a first order IFDE.

Keywords: Intuitionistic fuzzy differential equation, Numerical methods, Milne's Predictor - Corrector, Generalised differentiability.

V.Nirmala
Department of Mathematics, University College of Engineering Tindivanam, Tamil Nadu, India.
E-mail: nirmalaucet@mail.com

Abstract: In this article we present a new proof of butterfly theorem.

Keywords: Cyclic Quadrilateral, Butterfly theorem, Ptolemy's theorem.

Dasari Naga Vijay Krishna
Department of Mathematics, Narayana Educational Instutions Machilipatnam, Bengalore, India.
E-mail: vijay9290009015@gmail.com

Abstract: In this paper, we introduce the new concept connected and independent edge detour domination number of a graph and obtain the connected and independent edge detour domination number for some well known graphs.

Keywords: Edge detour domination, efficient domination, efficient($\gamma$,ed)-number of graphs.

A.Mahalakshmi
Department of Mathematics, Sri Sarada College For Women, Tirunelveli, Tamil Nadu, India.
E-mail: lakshmi19maths@gmail.com

Abstract: In this paper we consider the quadratic fractional integral equation with generalized mittag leffler function and will find the positive solution of this integral equation. In this paper we will find the existence as well as approximations of the positive solutions for a nonlinear quadratic fractional integral equation with generalized mittag leffler function given on a closed and bounded interval of the real line.

Keywords: Quadratic fractional integral equation, Fractional derivatives and Integrals, approximate solution.

Mohammed Mazhar Ul Haque
Dr.B.A.M.University, Aurangabad, Maharashtra, India.
E-mail: mazhar-ul-haque@hotmail.com

Abstract: In this article we introduce the notion of even square semigroups and provide some examples. Even square semigroups have non-explicit appearances in the mathematical literatures and the idea to introduce them explicitly has emerged through author’s recent work on even square rings.

Keywords: Semigroup, even semigroup, even square semigroup, even square ring, nil element, nilpotent element.

S.K.Pandey
Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Daijar, Jodhpur, Rajasthan, India.
E-mail: skpandey12@gmail.com

Abstract: In this paper, We contribute some new results for Heronian Mean labeling of graphs. We prove that Double Triangular snake, Alternate Double Triangular snake, Double Quadrilateral Snake and Alternate Double Quadrilateral Snake graphs are Heronian mean graphs.

Keywords: Heronian Mean labeling, Triangular snake, Double Triangular snake, Quadrilateral snake, Double Quadrilateral Snake.

S.D.Deepa
Department of Mathematics, Nesamony Memorial Christian College, Marthandam, Tamilnadu, India.
E-mail: christodeepadit@gmail.com

Abstract: In this paper we develop Open-Type Quadrature Method. If the interval of definite integral can divided a number of equal subinterval then We are using the nodes of Quadrature Method as mid-point of each interval. We are comparing mid point method to Other Quadrature methods. Also we are developing the composite formula and estimated errors.

Keywords: Numerical Integration, Newton-Cotes method, Quadrature method, Mid points formula.

Mahesh Chalpuri
Department of Mathematics, Osmania University, Hyderabad-07, India.
E-mail: mahi99099@gmail.com

Abstract: In \cite{HR13} the notin of almost finitely copresented modules is introduced and studied. In this paper, we introduce and study a notion of $n$-almost finitely copresented $R$-modules.

Keywords: Finitely cogenerated modules, finitely copresented modules, almost finitely cogenerated modules, almost finitely copresented modules, $n$-copresented modules and $n$-almost finitely copresented modules.

Abdul-Qawe Kaed
Department of Mathematics, Faculty of applied Science, University of Thamar.
E-mail: Abdul-Qawe\_dabouan@yahoo.com

Abstract: In the present paper an unsteady receding flow of two immiscible viscous incompressible fluids through a porous medium bounded between two impermeable plates is investigated. The flow is generated by the motion of one of the plates with a constant velocity parallel to other while the other plate is kept at rest. After attaining the steady state flow, the moving plate is suddenly stopped and the partial differential equations governing subsequent receding flow are solved by variable separable method to study hydrodynamic properties of the fluids. Separate expressions for fluid velocity and shear stress in both regions are obtained. Variations in fluid velocities and shear stress with time, porosity parameter, viscosity ratio and width of channel are presented graphically.

Keywords: Two phase flow, Porous medium, Immiscible fluids, Variable separable method.

Deepak Kumar
Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India.
E-mail: deepakpatel0412@gmail.com

Abstract: In this paper, the derivation of exact solutions of time-fractional heat diffusion equations by employing the new integral transform, the Aboodh transform coupled to the Adomian decomposition method is proposed. The coupling is based on the Caputo fractional derivative definition solely chosen owing to its flexibility. The proposed method is in fact, very reliable as it gives exact solutions in little iteration without requiring either of discretization, perturbation or linearization among others. Some test examples are given with their graphical illustration.

Keywords: Fractional Diffusion Equation, Aboodh Transform, Decomposition Method.

Rahmatullah Ibrahim Nuruddeen
Department of Mathematics, Federal University Dutse, Jigawa State, Nigeria.
E-mail: rahmatullah.n@fud.edu.ng