Error Analysis of Meshfree Approximation in Nonlinear Black-Scholes Model

The transaction cost model of Guy Barles and Halil Mete Soner is incorporated into the standard Black Scholes Equation. The resulting model is solved by a numerical method, called, the meshfree approximation using radial basis function. The errors produced by the scheme are discussed and presented in diagrams and tables.

Radial Basis Function in Nonlinear Black-Scholes Option Pricing Equation with Transaction Cost

Differential equations play significant role in the world of finance since most problems in these areas are modeled by differential equations. Majority of these problems are sometimes nonlinear and are normally solved by the use of numerical methods. This work takes a critical look at Nonlinear Black-Scholes model with special reference to the model by Guy Barles and Halil Mete Soner. The resulting model is a nonlinear Black-Scholes equation in which the variable volatility is a function of the second derivative of the option price. The nonlinear equation is solved by a special class of numerical technique, called, the meshfree approximation using radial basis function. The numerical results are presented in diagrams and tables.

Enhanced Discrete Element Method (EDEM) and its Application to Stability Analysis of the Slope with Non-Through Joints

A new enhanced Discrete Element Method (EDEM) for modeling the system composed of cracked solids is developed by coupling the traditional Discontinuous Deformation Analysis method (DDA, a kind of implicit version of DEM) with Moving Least-Squares (MLS) meshfree approximation functions. Tracing crack growth inside fracturing blocks and other related capabilities are available in the postprocessing procedure at each iteration step. Some numerical examples are provided to verify this method, and it is prospective to solve stability problems of the slope with non-through joints and other fracture mechanics problems in a new way.

Hermite Reproducing Kernel Meshfree Thermal Buckling Analysis of Euler-Bernoulli Beams with Elastic Foundation

The Hermite reproducing kernel meshfree method is employed for the stability analysis of Euler-Bernoulli beams with particular reference to the thermal buckling problem. This meshfree approximation employs both the nodal deflectional and rotational variables to construct the deflectional approximant according to the reproducing kernel conditions. In this paper, we apply this HRK meshfree method to the thermal buckling analysis of Euler-Bernoulli beam on elastic foundation. By comparison to the Gauss Integration method, HRK meshfree method shows much better solution accuracy.

A Kernel-Enriched Quadratic Convex Meshfree Approximation

Convex meshfree approximation with non-negative shape functions yields strict positive mass matrix and is particularly favorable for dynamic analysis. In this work, a kernel-enriched quadratic convex meshfree formulation with adjustable local approximation feature is presented. This formulation is built upon the generalized meshfree approximation with a relaxed quadratic reproducing condition. The resulting shape functions of the kernel-enriched quadratic convex meshfree formulation are presented in detail. The convergence behaviors for both static and vibration problems are discussed. Numerical results show that better accuracy can be achieved with the present formulation.