scholarly journals Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

Open Physics ◽  
2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Hakan Ciftci ◽  
Richard Hall ◽  
Nasser Saad
Keyword(s):  
Differential Equation ◽  
Coordinate Transformation ◽  
Iteration Method ◽  
Order Differential Equation ◽  
Approximate Solutions ◽  
Asymptotic Iteration Method ◽  
Perturbation Expansions ◽  
One Dimensional ◽  
Schrödinger’S Equation ◽  
Schrödinger's Equation

AbstractThe asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.

scholarly journals Exact and approximate solutions to Schrödinger’s equation with decatic potentials

Open Physics ◽  
2013 ◽  
Vol 11 (3) ◽  
Author(s):  
David Brandon ◽  
Nasser Saad
Keyword(s):  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Asymptotic Iteration Method ◽  
Polynomial Potential ◽  
Schrödinger’S Equation ◽  
Polynomial Solutions ◽  
Polynomial Coefficients ◽  
Linear Differential ◽  
Polynomial Potentials ◽  
Schrödinger's Equation

AbstractThe one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax 10 + bx 8 + cx 6 + dx 4 + ex 2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

scholarly journals On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Bulent Kilic ◽  
Mustafa Inc ◽  
Dumitru Baleanu
Keyword(s):  
Soliton Solutions ◽  
First Integral ◽  
Optical Solitons ◽  
Time Dependent ◽  
Ansatz Method ◽  
One Dimensional ◽  
Optical Metamaterials ◽  
Schrödinger’S Equation ◽  
The One ◽  
Schrödinger's Equation

AbstractThis paper integrates dispersive optical solitons in special optical metamaterials with a time dependent coefficient. We obtained some optical solitons of the aforementioned equation. It is shown that the examined dependent coefficients are affected by the velocity of the wave. The first integral method (FIM) and ansatz method are applied to reach the optical soliton solutions of the one-dimensional nonlinear Schrödinger’s equation (NLSE) with time dependent coefficients.

scholarly journals Wave Propagation Solution for Transverse Electric Mode in a Graded Interface between Left-Handed and Right-Handed Material

2016 ◽  
Vol 5 (3) ◽  
pp. 80
Author(s):  
B. N. Pratiwi ◽  
A. Suparmi ◽  
C. Cari
Keyword(s):  
Differential Equation ◽  
Wave Propagation ◽  
Iteration Method ◽  
Transverse Electric ◽  
Negative Permittivity ◽  
Asymptotic Iteration Method ◽  
Hyperbolic Function ◽  
Left Handed ◽  
Permittivity And Permeability ◽  
Electric Mode

Wave propagation for transverse electric (TE) mode in a graded interface between left-handed and right-handed material has been investigated by using asymptotic iteration method. By using hyperbolic functions for negative permittivity and negative permeability, we obtained the graded graphs of permittivity and permeability as a function of material thickness. Maxwell equation for the dielectric with the hyperbolic function in permittivity and permeability has been reduced to second orde differential equation. The second orde differential equation has been solved by using asymptotic iteration method with the eigen functions in complementary error functions. The eigen functions explained about the wave propagation in a graded interface of material. The distribution of the electric field and the wave vector were given in approximate solution.

Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system

2016 ◽  
Vol 13 (04) ◽  
pp. 1650045 ◽  
Author(s):  
Takahiro Yajima ◽  
Kazuhito Yamasaki
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Order Differential Equation ◽  
Geometric Theory ◽  
Transient State ◽  
Second Order ◽  
Two Dimensional ◽  
Overhead Crane ◽  
One Dimensional ◽  
Second Order Differential Equation

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

Numerically Efficient Computation of Eigensolution Spectrum in One-Dimensional Heterostructures

1998 ◽  
Vol 09 (07) ◽  
pp. 927-934 ◽  
Author(s):  
F. Farrelly ◽  
A. Petri
Keyword(s):  
Transfer Matrix ◽  
Density Of States ◽  
Spatial Behavior ◽  
Efficient Computation ◽  
Matrix Methods ◽  
One Dimensional ◽  
Schrödinger’S Equation ◽  
Schrödinger's Equation

We describe a method that allows an efficient determination of the density of states of one-dimensional heterostructures. We show that the propagation of an appropriate vector through the structure together with the use of the node theorem is much more effective than transfer matrix methods in those cases in which highly degenerate spectra are present. As a by-product, spatial behavior of solutions is also easily obtained. A case of elastic propagation is discussed in detail and application to Schrödinger's equation is presented.

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