In this work, the pulse propagation in a nonlinear dispersive optical medium is numerically investigated. The finite difference time-domain scheme of third order and periodic boundary conditions are used to solve generalized nonlinear Schr¨odinger equation governing the propagation of the pulse. As a result a discrete system of ordinary differerential equations is obtained and solved numerically by fourth order Runge-Kutta algorithm. Varied input ultrashort laser pulses are used. Accurate results of the solutions are obtained and the comparison with other results is excellent.
In this paper, an isogeometric error estimate for transport equation is obtained in 2D to prove the convergence of isogeometric method. The result that we have obtained, generalizes Ern result, got in finite elements method. For the time discretization, the two stage Heun scheme is used to prove this result. For a polynomial of degree 1 k ≥ , the order of convergence in space is 2 and in time is 1 2 k + .
Newton's method is used to find the roots of a system of equations ( ) 0 f x = . It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton's method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1 2 + . On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method.Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.
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