scholarly journals Optimized implicit finite‐difference migration for VTI media

2006 ◽  
Author(s):  
Guojian Shan
Keyword(s):  
Finite Difference ◽  
Implicit Finite Difference ◽  
Vti Media

Optimized implicit finite-difference and Fourier finite-difference migration for VTI media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA189-WCA197 ◽  
Author(s):  
Guojian Shan
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Weighted Least Squares ◽  
Taylor Series Expansion ◽  
Relative Dispersion ◽  
Data Set ◽  
Isotropic Media ◽  
Implicit Finite Difference ◽  
Wavefield Extrapolation ◽  
Vti Media

Propagation velocity of seismic waves in heterogeneous VTI media depends not only on spatial location but also on their propagation direction, which leads to a much more complex dispersion relation than in isotropic media. As a result, designing implicit finite-difference (FD) schemes for wavefield extrapolation in anisotropic media through analytic Taylor-series expansion is more difficult. Implicit FD and Fourier finite-difference (FFD) schemes are developed for vertical transversely isotropic (VTI) media based on function fitting. The dispersion relation of VTI media is approximated with a rational function and its coefficients are estimated by weighted least-squares optimization. Because these coefficients are functions of Thomsen anisotropy parameters ([Formula: see text] and [Formula: see text]) and vary laterally in heterogeneous VTI media, they are calculated before wavefield extrapolation and stored in a table. Implicit FD and FFD schemes for VTI media are almost the same as for isotropic media, except that coefficients are looked up in a precalculated table. Impulse responses and relative dispersion-relation error show that accuracy of the FD scheme for VTI media is similar to its counterpart in isotropic media. Application to a synthetic data set showed that implicit FD and FFD can handle laterally varying VTI media.

scholarly journals A robust numerical solution to a time-fractional Black–Scholes equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. M. Nuugulu ◽  
F. Gideon ◽  
K. C. Patidar
Keyword(s):  
Finite Difference ◽  
Stock Options ◽  
Stochastic Dynamics ◽  
European Options ◽  
Numerical Convergence ◽  
Put Option ◽  
Implicit Finite Difference Scheme ◽  
Black Scholes ◽  
Implicit Finite Difference ◽  
Rigorous Mathematical Analysis

AbstractDividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.

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