scholarly journals Linear Barycentric Rational Method for Solving Schrodinger Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Peichen Zhao ◽  
Yongling Cheng
Keyword(s):  
Schrödinger Equation ◽  
Convergence Rate ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Interpolation Method ◽  
Rational Interpolation ◽  
Barycentric Interpolation ◽  
Rational Polynomial ◽  
The Matrix ◽  
Barycentric Rational Interpolation

A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.

scholarly journals Linear Barycentric Rational Collocation Method for Beam Force Vibration Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jin Li ◽  
Yu Sang
Keyword(s):  
Convergence Rate ◽  
Collocation Method ◽  
Convergence Rates ◽  
Rational Interpolation ◽  
Vibration Equation ◽  
Numerical Examples ◽  
Space And Time ◽  
Force Vibration ◽  
The Matrix ◽  
Barycentric Rational Interpolation

The linear barycentric rational collocation method for beam force vibration equation is considered. The discrete beam force vibration equation is changed into the matrix forms. With the help of convergence rate of barycentric rational interpolation, both the convergence rates of space and time can be obtained at the same time. At last, some numerical examples are given to validate our theorem.

scholarly journals Numerical Solution for Third-Order Two-Point Boundary Value Problems with the Barycentric Rational Interpolation Collocation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Qian Ge ◽  
Xiaoping Zhang
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Point Boundary ◽  
Third Order ◽  
The Third ◽  
The Matrix ◽  
Barycentric Rational Interpolation

The numerical solution for a kind of third-order boundary value problems is discussed. With the barycentric rational interpolation collocation method, the matrix form of the third-order two-point boundary value problem is obtained, and the convergence and error analysis are obtained. In addition, some numerical examples are reported to confirm the theoretical analysis.

scholarly journals The barycentric rational interpolation collocation method for boundary value problems

2018 ◽  
Vol 22 (4) ◽  
pp. 1773-1779 ◽  
Author(s):  
Dan Tian ◽  
Ji-Huan He
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Analytical Methods ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Higher Order ◽  
Barycentric Rational Interpolation

Higher-order boundary value problems have been widely studied in thermal science, though there are some analytical methods available for such problems, the barycentric rational interpolation collocation method is proved in this paper to be the most effective as shown in three examples.

scholarly journals Super Exponentially Convergent Approximation to the Solution of the Schrödinger Equation in Abstract Setting

2010 ◽  
Vol 10 (4) ◽  
pp. 345-358 ◽  
Author(s):  
I.P. Gavrilyuk
Keyword(s):  
Schrödinger Equation ◽  
Convergence Rate ◽  
Schrodinger Equation ◽  
Exponential Convergence ◽  
Exponential Convergence Rate ◽  
Abstract Setting

AbstractWe have developed an approximation to the solution of the Schrödinger equation in abstract setting. The accuracy of our approximation depends on the smoothness of this solution. We show that for the analytical initial vectors our approximation possesses a super exponential convergence rate.

scholarly journals Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 253 ◽  
Author(s):  
Aditya Kamath ◽  
Sergei Manzhos
Keyword(s):  
Schrödinger Equation ◽  
Vibrational Spectrum ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Collocation Points ◽  
Basis Size ◽  
Six Dimensions ◽  
Gaussian Basis Functions

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).

The Inviscid Limit of the Fractional Complex Ginzburg–Landau Equation

2016 ◽  
Vol 17 (6) ◽  
pp. 333-341
Author(s):  
Lijun Wang ◽  
Jingna Li ◽  
Li Xia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Convergence Rate ◽  
Schrodinger Equation ◽  
Landau Equation ◽  
Limit Behavior ◽  
Inviscid Limit ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

AbstractIn this paper, the inviscid limit behavior of solution of the fractional complex Ginzburg–Landau (FCGL) equation$${\partial _t}u + (a + i\nu){\Lambda ^{2\alpha}}u + (b + i\mu){\left| u \right|^{2\sigma}}u = 0, \quad (x, t) \in {{\Cal T}^n} \times (0, \infty)$$is considered. It is shown that the solution of the FCGL equation converges to the solution of nonlinear fractional complex Schrödinger equation, while the initial data${u_0}$is taken in${L^2}, $${H^\alpha}$, and${L^{2\sigma + 2}}$as$a,\, b$tends to zero, and the convergence rate is also obtained.

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