The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example.
Ordinary Differential Equation (ODEs) problems can solve using Euler method. Euler method suitable for first-order numerical and it is an effective and easy method. The objective of this paper is to suggest a new scheme for better accuracy. The accuracy is determined by comparing with exact solution using average concept. This paper proposes Cube Polygon as modified Euler method to improved accuracy and complexity. A set of simulation were carried out to demonstrate the accuracy of the proposed method. Testing has been done into SCILAB 6.0 by recording the maximum error. Results indicate that Cube Polygon provide more accurate results and reducing complexity for both smaller and higher step size.
In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra Hn+1 from such basis of Hn is given.
In this article, we study the asymptotic behavior of even-order neutral delay differential equation ( a ⋅ ( u + ρ ⋅ u ∘ τ ) ( n − 1 ) ) ′ ( ℓ ) + h ( ℓ ) u ( g ( ℓ ) ) = 0 , ℓ ≥ ℓ 0 , {(a\cdot {(u+\rho \cdot u\circ \tau )}^{(n-1)})}^{^{\prime} }(\ell )+h(\ell )u(g(\ell ))=0,\hspace{1.0em}\ell \ge {\ell }_{0}, where n ≥ 4 n\ge 4 , and in noncanonical case, that is, ∫ ∞ a − 1 ( s ) d s < ∞ . \mathop{\int }\limits^{\infty }{a}^{-1}\left(s){\rm{d}}s\lt \infty . To the best of our knowledge, most of the previous studies were concerned only with the study of n n -order neutral equations in canonical case. By using comparison principle and Riccati transformation technique, we obtain new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Examples are presented to illustrate our new results.
This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria.
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