A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

The Fourier method for one-way wave propagation is efficient, but potentially inaccurate in complex media. The implicit finite-difference method can handle arbitrarily complex media, but can be inefficient in 3D and has limited dip bandwidth. We proposed a new Fourier method based on Chebyshev expansion of the second kind. Both theoretical analyses and numerical experiments show that the proposed method is comprehensively superior to a similar method based on Chebyshev expansion of the first kind in terms of balanced amplitude and error tolerance. Within the dip bandwidth from 0 to 65°, the fourth-order form of our method has an error tolerance of 2%, which is about one-third that of Chebyshev expansion of the first kind. Our method is also superior to the implicit finite-difference method in several important aspects: effective bandwidth, computational efficiency, numerical dispersion and two-way splitting error. It can be easily extended from 2D to 3D compared with the finite-difference method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. The proposed method shows better imaging results of the SEG/EAGE model by providing a well-focused salt dome, flank and bottom as well as the detailed structures beneath the salt body, compared with the implicit finite-difference method and Chebyshev expansion of the first kind; meanwhile, our method has less imaging artifacts since it can better position the reflectors.

Finite Difference Method in Fluid Potential Function and Velocity Calculation

There is no deterministic solution for many fluid problems but by applying analytical solutions many of them are approximated. In this study an implicit finite difference method presented which solves the potential function and further expanded to drive out the velocity components in 2D-space by applying a point-by-point swiping approach. The results showed the rotational behavior of both potential function as well as velocity components while encountering central obstacle.

Implicit Finite Difference Simulation of Prandtl-Eyring Nanofluid over a Flat Plate with Variable Thermal Conductivity: A Tiwari and Das Model

The current article presents the entropy formation and heat transfer of the steady Prandtl-Eyring nanofluids (P-ENF). Heat transfer and flow of P-ENF are analyzed when nanofluid is passed to the hot and slippery surface. The study also investigates the effects of radiative heat flux, variable thermal conductivity, the material’s porosity, and the morphologies of nano-solid particles. Flow equations are defined utilizing partial differential equations (PDEs). Necessary transformations are employed to convert the formulae into ordinary differential equations. The implicit finite difference method (I-FDM) is used to find approximate solutions to ordinary differential equations. Two types of nano-solid particles, aluminium oxide (Al2O3) and copper (Cu), are examined using engine oil (EO) as working fluid. Graphical plots are used to depict the crucial outcomes regarding drag force, entropy measurement, temperature, Nusselt number, and flow. According to the study, there is a solid and aggressive increase in the heat transfer rate of P-ENF Cu-EO than Al2O3-EO. An increment in the size of nanoparticles resulted in enhancing the entropy of the model. The Prandtl-Eyring parameter and modified radiative flow show the same impact on the radiative field.

Axisymmetric Turbulent Methane Jet Propagation in a Co-Current Air Flow Under Combustion at a Finite Velocity

The paper introduces a numerical method for solving the problem of the axisymmetric methane jet propagation in an infinite co-current air flow. For modeling, we used the dimensionless equations of the turbulent boundary layer of reacting gases in the Mises coordinates. To close the Reynolds equation, a modified k - ε turbulence model was used. The k - ε model is considered a low Rhine turbulence model. Assuming that the intensities of convective and turbulent transfers of components are the same and using the stoichiometric ratios of the concentrations of components during combustion, we reduced five equations for the transfer and conservation of the mass of components to two equations for the relative excess concentration of the combustible gas. The concentrations of the components were determined from the solutions of these equations. By using relatively excessive velocities and total enthalpy, we reduced the boundary conditions for the three equations to a general form. To solve the problem in the Mises coordinates, we used a two-layer, six-point implicit finite-difference scheme, which provides the second order of accuracy of approximation in coordinates. The equations for the conservation and transfer of substances being non-linear, an iterative process was implemented. The influence of the radius of the fuel nozzle on the indices of the turbulent jet and flame was investigated. Findings of research show that in an endless co-current flow of fuel with a decrease in the radius of the nozzle, the rate of the chemical reaction and the maximum temperature in the calculation area decrease, and the amount of unburned part of the combustible gas increases