Search

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0

In this work, we have studied a traveling wave packet when it travels through a rectangular quantum barrier and well. For this purpose, we have calculated the reflection, trapping and transmission coefficient of the cited quantum systems. We show that trapped part of the wave packet increases and then decreases with time. Deeper quantum wells have smaller trapping coefficient while higher quantum barriers have larger trapping coefficient. Changing of the velocity of the wave packet does not change the trapping coefficient. For small barrier heights (well depths), the reflection increases and then approximately saturates when the nonlinear coefficient increases. Finally, there are abrupt changes in the reflection, trapping and transmission coefficients when the barrier (or well) widths increase.

Researchers usually neglect the electron–electron interaction effect when they study the optical properties of semiconducting nanostructures through the compact density matrix approach. In the existing papers, this work has also been done through self-consistent solution of the Schrodinger and Poisson equations. For the first time, we have investigated the electronic and optical properties of wurtzite GaN/AlN quantum wells in the presence of electron–electron interactions assuming weak electron–electron scattering through a numerical solution of a nonlinear Schrodinger equation. It is shown that the electron–electron interaction effect leads to the removal of the N-fold degeneracy in the energy levels of N-well multiple quantum well systems. Effects of different parameters such as electric field, the strength of the electron–electron interaction, a total length of the system and also two different numbers of wells (1 and 2) on the mentioned optical properties have been investigated. Also, we observed redshifts in the optical rectification coefficient peak positions when the nonlinear parameter increased. Finite difference self-consistent method was employed to solve the nonlinear Schrodinger equation and we demonstrated the method convergence. Finally, we illustrated the accuracy of the method by comparing our energy values with those of the Sinc method.

In this work, we have studied a traveling soliton through a triangular refractive index waveguide. Our calculations have been performed with a 4th order Runge Kutta method which we have presented the algorithm. Then we have checked the numerical accuracy and found it desirable. In our investigations, the light beam inside a waveguide have been oscillated which perfectly agree with the so-called “swing effect”. In comparison with a soliton with amplitude Q = 1, for soliton amplitude Q = 2 a shorter oscillation period have been observed. In the meantime, for lower waveguide widths, soliton have had shorter oscillation period. Finally, for η ≠ 0, the soliton oscillate with higher amplitude and it recede from center of waveguide which is not desirable.

In this work, we have studied the wave localization in a two dimensional parabolic periodic refractive index profile. Our calculations have been performed by developing a 4th order Runge–Kutta method. Effects of different parameters in refractive index profile and incident wave shape on the wave intensity and shape in the future are shown. Effects of the mentioned parameters on localization degree and total momentum of the system are also investigated. We find different parameters change intervals within which the excitation disperses. Thus no bound state is possible. Finally, we show when a bound state is present.