• East Asian mathematical journal
• /
• v.40 no.5
• /
• pp.515-526
• /
• 2024

We consider the initial value problem of the Schrödinger equation for an Euler operator 𝓡 on ℂⁿ that is an analogue of the harmonic oscillator in ℝⁿ. We get some regularity results of the Schrödinger equation on Fock spaces.

• Journal of Environmental Science International
• /
• v.11 no.9
• /
• pp.907-914
• /
• 2002

Various Eulerian-Lagrangian models for the one-dimensional longitudinal dispersion equation in nonuniform flow were studied comparatively. In the models studied, the transport equation was decoupled into two component parts by the operator-splitting approach; one part is governing advection and the other is governing dispersion. The advection equation has been solved by using the method of characteristics following fluid particles along the characteristic line and the results were interpolated onto an Eulerian grid on which the dispersion equation was solved by Crank-Nicholson type finite difference method. In the solution of the advection equation, Lagrange fifth, cubic spline, Hermite third and fifth interpolating polynomials were tested by numerical experiment and theoretical error analysis. Among these, Hermite interpolating polynomials are generally superior to Lagrange and cubic spline interpolating polynomials in reducing both dissipation and dispersion errors.

• The Korean Journal of Applied Statistics
• /
• v.23 no.2
• /
• pp.383-392
• /
• 2010

Modelling financial phenomena with diffusion processes is frequently used technique. This study reviews the earlier researches on the approximation problem of transition densities of diffusion processes, which takes important roles in estimating diffusion processes, and consider the method to obtain the coefficients of series efficiently, in series approximation method of transition densities. We developed a new efficient algorithm to compute the coefficients which are represented by repeated Dynkin operator on Hermite polynomial.

• Water for future
• /
• v.27 no.2
• /
• pp.155-166
• /
• 1994

Various Eulerian-Lagrangian numerical models for the one-dimensional longitudinal dispersion equation are studied comparatively. In the model studied, the transport equation is decoupled into two component parts by the operator-splitting approach ; one part governing adveciton and the other dispersion. The advection equation has been solved using the method of characteristics following fluid particles along the characteristic line and the results are interpolated onto an Eulerian grid on which the dispersion equation is solved by Crank-Nicholson type finite difference method. In solving the advection equation, various interpolation schemes are tested. Among those, Hermite interpolation polynomials are superior to Lagrange interpolation polynomials in reducing dissipation and dispersion errors in the simulation.

• Korean Journal of Hydrosciences
• /
• v.6
• /
• pp.51-66
• /
• 1995

Various Eulerian-Lagerangian numerical models for the one-dimensional longtudinal dispersion equation are studied comparatively. In the models studied, the transport equation is decoupled into two component parts by the operator-splitting approach ; one part governing advection and the other dispersion. The advection equation has been solved using the method of characteristics following flud particles along the characteristic line and the result are interpolated onto an Eulerian grid on which the dispersion equation is solved by Crank-Nicholson type finite difference method. In solving the advection equation, various interpolation schemes are tested. Among those, Hermite interpo;ation po;ynomials are superor to Lagrange interpolation polynomials in reducing both dissipation and dispersion errors.

• Water for future
• /
• v.26 no.3
• /
• pp.137-148
• /
• 1993

A hybrid finite difference method for the longitudinal dispersion equation was developed. The method is based on combining the Holly-Preissmann scheme with the fifth-degree Hermite interpolating polynomial and the generalized Crank-Nicholson scheme. Longitudinal dispersion of an instantaneously-loaded pollutant source was simulated by the model and other characteristics-based numerical methods. Computational results were compared with the exact solution. The present method was free from wiggles regardless of the Courant number, and exactly reproduced the location of the peak concentration. Overall accuracy of the computation increased for smaller value of the weighting factor, $\theta$ of the model. Larger values of $\theta$ overestimated the peak concentration. Smaller Courant number gave better accuracy, in general, but the sensitivity was very low, especially when the value of $\theta$ was small. From comparisons with the hybrid method using the third-degree interpolating polynomial and with split-operator methods, the present method showed the best performance in reproducing the exact solution as the advection becomes more dominant.