JOURNAL OF ADVANCES IN MATHEMATICS Interested in submitting to this journal? We recommend that you review the About the Journal page for the journal's section policies, as well as the Author Guidelines. Authors need to register with the journal prior to submitting or, if already registered, can simply log in and begin the five-step process. KHALSA PUBLICATIONS en-US JOURNAL OF ADVANCES IN MATHEMATICS 2347-1921 <p>Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the&nbsp;<a href="">Creative Commons Attribution License</a>, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.</p> <p>&nbsp;</p> A Lie Symmetry Solutions of Sawada-Kotera Equation <p>In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth order Sawada-Kotera equation. The technique is among the most powerful approaches currently used to achieveprecise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generatorsand invariant transformations hence symmetry solutions of the fifth order Sawada-Kotera equation.</p> <p><strong><em>Key Words</em></strong>- Lie symmetry analysis. Sawada-Kotera equation. Symmetry groups. Prolongations. Invariant solutions. Power series solutions. Symmetry solutions.</p> Winny Chepngetich Bor Owino M. Oduor John K. Rotich Copyright (c) 2019 Winny Chepngetich Bor, Owino M. Oduor, John K. Rotich 2019-07-30 2019-07-30 17 1 11 10.24297/jam.v17i0.8364 Breaking Boundaries: Discovering the Impossible Counterproof of Beal’s Conjecture <p>This paper will attempt to logically differentiate between two types of fractions and discuss the idea of Zero as a neutral integer. This logic can then be followed to create a counterexample and a proof for Beal’s conjecture.</p> Halima Jibril Mohamed Adela Zyfi Ghedlawit Futzum Copyright (c) 2019 Halima Jibril 2019-07-23 2019-07-23 17 12 18 10.24297/jam.v17i0.8279 Convergence of the Collatz Sequence <p>For any natural number was created the supplement sequence, that is convergent together with the original Collatz sequence. The numerical parameter - index was defined, that is the same for both sequences. This new method provides the following results:</p> <ol> <li>All natural numbers were distributed into six different classes;</li> <li>The properties of index were found for the different classes;</li> <li>For any natural number was constructed the bounded sequence of increasing numbers,</li> </ol> <p>&nbsp;&nbsp;&nbsp; that is convergent together with the regular Collatz sequence.</p> Anatoliy Nikolaychuk Copyright (c) 2019 Anatoliy Nikolaychuk 2019-07-23 2019-07-23 17 19 33 10.24297/jam.v17i0.8336 Karp's Theorem in Inverse Obstacle Scattering Problems <p>In this work, we provide a proof of the so-called Karp's theorem in a different approach. We&nbsp;use the unique continuation principle together with the monotonicity of eigenvalues for the&nbsp;negative Laplace operator. This method is new and would be applicable to other types of&nbsp;inverse scattering problems.</p> Jaemin Shin Copyright (c) 2019 Jaemin Shin 2019-08-14 2019-08-14 17 34 38 10.24297/jam.v17i0.8399 Division And Combination In Linear Algebra <p>In this paper, the relationship between matrix operation, linear equations, linear representation of vector groups and linear correlation is discussed, and the idea of division and combination in linear algebra is discussed to help learners understand the connections between various knowledge points of linear algebra from multiple angles, deep levels, and high dimensions.</p> Liang Fang Rui Chena Copyright (c) 2019 Liang Fang, Rui Chena 2019-09-06 2019-09-06 17 39 146 10.24297/jam.v17i0.8413 Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method <p>We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded&nbsp;integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration&nbsp;step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different&nbsp;nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta&nbsp;(RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we&nbsp;show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval&nbsp;where the IVP is being considered.</p> Abhinandan Chowdhury Sammie Clayton Mulatu Lemma Copyright (c) 2019 Abhinandan Chowdhury 2019-09-16 2019-09-16 17 147 154 10.24297/jam.v17i0.8408 A Parametric Approach for Solving Interval–Valued fractional Continuous Static Games <p>The aim of this paper is to show that a parametric approach can be used to solve fractional continuous static games with interval-valued in the objective function and in the constraints. In this game, cooperation among all the players is possible, and each player helps the others up to the point of disadvantage to himself, so we use the Pareto-minimal solution concept to solve this type of game. The Dinkelbach method is used to transform fractional continuous static games into non- fractional continuous static games. Moreover, an algorithm with the corresponding flowchart to explain the suggested approach is introduced. Finally, a numerical example to illustrate the algorithm’s steps is given.</p> Mervat Elshafei Copyright (c) 2019 Mervat Elshafei 2019-09-16 2019-09-16 17 155 164 10.24297/jam.v17i0.8419 Angle Trisection <p>We seek to increase the development of science, but there are several fundamental questions about what is. Without solving the question is a false reflection of the history of science and the beginning of cognition. We know that their investigation and resolution, with the exception of rooting and knowledge of morphophonemic, do not come. Research on certain natural or pure mathematical phenomena is an example of my fundamental research that will lead to the definition of general principles and scientific theories.</p> Mehryar Husyan Pour Shad Copyright (c) 2019 Mehryar Husyan Pour Shad 2019-09-16 2019-09-16 17 165 231 10.24297/jam.v17i0.8412 Fuzzy Graphs <p>In this paper, neighbourly irregular fuzzy graphs, neighbourly total irregular fuzzy graphs, highly irregular fuzzy graphs and highly total irregular fuzzy graphs are introduced. A necessary and sufficient condition under which neighbourly irregular and highly irregular fuzzy graphs are equivalent is provided. We define d2 degree of a vertex in fuzzy graphs and total d2 -degree of a vertex in fuzzy graphs and (2, k)-regular fuzzy graphs, totally (2, k)- regular fuzzy graphs are introduced. (2, k)- regular fuzzy graphs and totally (2, k)-regular fuzzy graphs are compared through various examples.</p> Huda Mutab Al Mutab Copyright (c) 2019 Huda Mutab Al Mutab 2019-10-03 2019-10-03 17 232 247 10.24297/jam.v17i0.8443