Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

The differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.

The differential quadrature method has been widely used in scientific and engineering computation. However, for the basic characteristics of time domain differential quadrature method, such as numerical stability and calculation accuracy or order, it is still lack of systematic analysis conclusions. In this paper, according to the principle of differential quadrature method, it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature. Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method, it has been proved that the differential quadrature method is A-stable and s-stage s-order method. On this basis, in order to further improve the accuracy of the time domain differential quadrature method, a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Padé approximations. The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson’s ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.

A nondestructive detection method of surface cracks in two dimensional plate structures using modal strain energy method is investigated in this paper. Experimental modal analysis (EMA) is conducted on an aluminum alloy 6061 thin plate to obtain the mode shapes before and after damage under a completely free boundary condition. The measured mode shapes are used to compute the strain energy of the plate. Limited by the measured points, a differential quadrature method is employed to compute the partial differential terms in strain energy formula. A damage index is then defined based on strain energy ratio of the plate before and after damage. This damage index is employed to identify the location of surface crack in plate structure. A finite element analysis (FEA) is also performed to access this approach and demonstrate a feasible process for the experimental work. Good correlation between FEA and EMA results is obtained. The damage index obtained from both FEA and EMA successfully identify the location of surface crack in the aluminum plate. Only few measured mode shapes of the plate are required in this method, which provides a quick, flexible, inexpensive and nondestructive technique to identify the damagein local and global 2D plate structures.