In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including ?-method, mixed method, Richardson Du Fort Frankel MADE (modified alternating directional explicit) method. These produce drawbacks such as spurious oscillations negative values in solution when volatility is much smaller than interest rate. The method sacrifices accuracy obtain sta...

In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson’s internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Gödel’s system T can be extr...

In the past few decades, nonstandard methods, as a branch of mathematical logic, have been successfully applied to obtain new results in additive/combinatorial number theory (cf. [BJ, Ji1, Ji2, Ji3, Ji4, Ji5, Ji6, JK, Le1, Le2, Le3, Le4]). Although the nonstandard techniques in these applications are elementary from a nonstandard analyst’s point of view, it is extremely difficult for a reader w...

this paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. the method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions.the construction of the nonstandard method is based on the special treatment of the interface using suitable jump condition...