In this paper, we studied the separability of the non-linear Schrodinger operator of the form S u x = − Δ u x + V x , u u x , where Δ u x = ∑ i = 1 n ∂ 2 u x ∂ x i 2 , with the non-linear matrix potential V x , u . We obtained the sufficient conditions for separability of this operator in the space L 2 R n ℓ and we established the suitable coercive inequalities. [ABSTRACT FROM AUTHOR]
NONLINEAR systems, NONLINEAR operators, NONLINEAR equations, BOUNDARY value problems, INTEGRAL equations, NEWTON-Raphson method
Abstract
We propose a class of Newton-like methods with increasing convergence order for approximating the solutions of systems of nonlinear equations. Novelty of the methods is that in each step the order of convergence is increased by an amount of three at the cost of only one additional function evaluation. Another important feature is the single use of an inverse operator in each iteration, which makes the schemes attractive and computationally more efficient. Theoretical results regarding convergence and computational efficiency are verified through numerical examples including those arising from boundary value problems and integral equations. By way of comparison, it is shown that the present methods are more efficient than their existing counterparts, particularly when applied to solve the large systems of equations. [ABSTRACT FROM AUTHOR]