In this paper, the meshless local Petrov-Galerkin (MLPG) method is applied for solving a generalized Black-Scholes equation in financial problems. This equation is a PDE governing the price evolution of a European call or a European put under the Black-Scholes model. The θ-weighted method and MLPG are used for discretizing the governing equation in time variable and option pricing, respectively...

The truly Meshless Local Petrov-Galerkin (MLPG) method is extended to solve the incompressible Navier-Stokes equations. The local weak form is modified in a very careful way so as to ovecome the so-called Babus̃ka-Brezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (2000a) and Lin and Atluri (2000b) is used to stabilize the convection operator in the streamline di...

Abstract. We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in [29], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1 + d)-dimensional space-time hypercube, d = 1, 2, 3, · · · , subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence a...

This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt ...

Abstract In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide residual-based posteriori error estimate space-time formulation and the corresponding efficiently computable bound certification of method. introduce Petrov-Galerkin finite element discretization continuous problem use it as...

The standard in rod finite element formulations is the Bubnov–Galerkin projection method, where test functions arise from a consistent variation of ansatz functions. This approach becomes increasingly complex when highly nonlinear are chosen to approximate rod's centerline and cross-section orientations. Using Petrov–Galerkin we propose whole family nodal generalized virtual displacements veloc...

This paper presents the solution of coupled radiative transfer equation with heat conduction equation in complex three-dimensional geometries. Due to very different time scales for both physics, the radiative problem is considered steady-state but solved at each time iteration of the transient conduction problem. The discrete ordinate method along with the decentered streamline-upwind Petrov-Ga...

We analyze plane strain static thermoelastic deformations of a simply supported functionally graded (FG) plate by a meshless local Petrov–Galerkin (MLPG) method. Material moduli are assumed to vary only in the thickness direction. The plate material is made of two isotropic randomly distributed constituents and the macroscopic response is also modeled as isotropic. Displacements and stresses co...

We describe a consistent splitting approach to the pressure stabilized Petrov-Galerkin finite element method for incompressible flow. The splitting leads to (almost) explicit predictor and corrector steps linked by an implicit pressure equation which can be solved very efficiently. The overall secondorder convergence is proved in numerical experiments. Furthermore, the parallel implementation o...

Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial-boundary value problem defined only on a finite interval. A dual-Petrov-Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbin...