We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ∞ L2 error estimates of discontinuous Galerk...

Abstract. We consider a partial differential equation of Schrödinger type, known as the ‘parabolic’ approximation to the Helmholtz equation in the theory of sound propagation in an underwater, rangeand depth-dependent environment with a variable bottom. We solve an associated initialand boundary-value problem by a finite difference scheme of Crank-Nicolson type on a variable mesh. We prove that...

In the present investigation, the fully implicit and Crank–Nicolson difference schemes for solving option prices are analyzed. It is proved that the error expansions for the difference methods have the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. The difference methods are applied to European, American, and down-and-out knock-out ca...

We consider a model initialand boundary-value problem for a third-order p.d.e., a wide-angle ‘parabolic’ equation frequently used in underwater acoustics, with depthand rangedependent coefficients in the presence of horizontal interfaces and dissipation. After commenting on the existence–uniqueness theory of solution of the equation, we discretize the problem by a secondorder finite difference ...

In 1944, Freeman Dyson defined the concept of rank an integer partition and introduced without definition term crank partition. A for satisfying properties hypothesized it by was discovered in 1988 G.E. Andrews F.G. Garvan. this paper, we introduce truncated forms two theta identities involving generating functions partitions with non-negative crank. As corollaries derive new infinite families ...

A new numerical method is presented for the heat equation with discontinuous coefficients based on a Crank–Nicolson scheme and a conforming finite element space discretization. In the proposed method each node of the spatial discretization may have the global timestep split into an arbitrary number of local substeps in order to pursue a local improvement of the time discretization in the region...

This project work report provides a full solution of simplified Navier Stokes equations for The Incompressible Couette Problem. The well known analytical solution to the problem of incompressible couette is compared with a numerical solution. In that paper, I will provide a full solution with simple C code instead of MatLab or Fortran codes, which are known. For discrete problem formulation, im...

In this paper, we suggest a jump diffusion model in markets during financial crisis. Using risk-neutral pricing, we derive a partial differential equation (P.D.E.) for the prices of European options. We find a closed form solution of the P.D.E. in the particular case where the stock price is too large. Then, we use such a solution as a boundary condition in the numerical treatment of the P.D.E....

Numerical methods for the primitive equations (PEs) of oceanic flow are presented in this paper. First, a two-dimensional Poisson equation with a suitable boundary condition is derived to solve the surface pressure. Consequently, we derive a new formulation of the PEs in which the surface pressure Poisson equation replaces the nonlocal incompressibility constraint, which is known to be inconven...