scholarly journals The Numerical Solution of Fractional Black-Scholes-Schrodinger Equation Using the RBFs Method

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Naravadee Nualsaard ◽  
Anirut Luadsong ◽  
Nitima Aschariyaphotha
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Option Prices ◽  
Time Step ◽  
Numerical Examples ◽  
Black Scholes ◽  
Semiclassical Solution ◽  
Time Linear ◽  
Simple Quadrature

In this paper, radial basis functions (RBFs) method was used to solve a fractional Black-Scholes-Schrodinger equation in an option pricing of financial problems. The RBFs method is applied in discretizing a spatial derivative process. The approximation of time fractional derivative is interpreted in the Caputo’s sense by a simple quadrature formula. This RBFs approach was theoretically proved with different problems of two numerical examples: time step arbitrage bubble case and time linear arbitrage bubble case. Then, the numerical results were compared with the semiclassical solution in case of fractional order close to 1. As a result, both numerical examples showed that the option prices from RBFs method satisfy the semiclassical solution.

Compact splitting symplectic scheme for the fourth-order dispersive Schrödinger equation with Cubic-Quintic nonlinear term

2019 ◽  
Vol 10 (02) ◽  
pp. 1950007 ◽  
Author(s):  
Lang-Yang Huang ◽  
Zhi-Feng Weng ◽  
Chao-Ying Lin
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Order Accuracy ◽  
Numerical Examples ◽  
Splitting Technique ◽  
Symplectic Scheme ◽  
Theoretical Results ◽  
Discrete Charge

Combining symplectic algorithm, splitting technique and compact method, a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schrödinger equation with cubic-quintic nonlinear term. The scheme has fourth-order accuracy in space and second-order accuracy in time. The discrete charge conservation law and stability of the scheme are analyzed. Numerical examples are given to confirm the theoretical results.

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).

scholarly journals Large Time-Stepping Spectral Methods for the Semiclassical Limit of the Defocusing Nonlinear Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-27
Author(s):  
Zongqi Liang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Spectral Methods ◽  
Schrodinger Equation ◽  
Large Time ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Time Stepping

We analyze a class of large time-stepping Fourier spectral methods for the semiclassical limit of the defocusing Nonlinear Schrödinger equation and provide highly stable methods which allow much larger time step than for a standard implicit-explicit approach. An extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Meanwhile, the first-order and second-order semi-implicit schemes are constructed and analyzed. Finally the numerical experiments are performed to demonstrate the effectiveness of the large time-stepping approaches.

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

2013 ◽  
Vol 12 (06) ◽  
pp. 1340001 ◽  
Author(s):  
ANDRÉ D. BANDRAUK ◽  
HUIZHONG LU
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Quantum Statistical Mechanics ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Parabolic Partial Differential Equation ◽  
Imaginary Time ◽  
Time Step ◽  
Order Accuracy

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

scholarly journals On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Bernstein Polynomials ◽  
Energy Levels ◽  
Basis Functions ◽  
Spline Collocation ◽  
Stationary Schrödinger Equation ◽  
B Splines ◽  
Domain Length

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

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

2018 ◽  
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.

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