Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

2016 ◽  
Vol 40 (8) ◽  
pp. 3100-3110 ◽  
Author(s):  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Comparison Theory ◽  
Linear And Nonlinear ◽  
Picone Identity

Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1185-1194
Author(s):  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Comparison Theory ◽  
Second Order Differential Equations ◽  
Nonlinear Version ◽  
Linear And Nonlinear ◽  
Linear Differential ◽  
Picone Identity

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)??(y'))' + Xn,i=1 qi(t)??i(y)=0(1) and the second is the half-linear differential equations (k(t)??(x'))'+ p(t)??(x) = 0 (2) where ?*(s) = |s|*-1s and ?1 >...> ?m > ? > ?m+1 > ... > ?n > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone?s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

Linear and Nonlinear Differential Equations

1984 ◽  
Vol 52 (7) ◽  
pp. 670-670 ◽  
Author(s):  
I. D. Huntley ◽  
R. M. Johnson ◽  
Fred Brauer
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear And Nonlinear

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

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