In this paper, we develop two classes of finite difference schemes for the reaction– subdiffusion equations by using a mixed spline function in space direction, forward and backward difference in time direction, respectively. It has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters. By Fourier method, we pro...

In this paper, the Mickens non-standard discretization method which effectively preserves the dynamical behavior of linear differential equations is adapted to solve numerically the fractional order hyperbolic partial differential equations. The fractional derivative is described in the Riesz sense. Special attention is given to study the stability analysis and the convergence of the proposed m...

The applicability of the Dirichlet-to-Neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape. The original boundary value problem (BVP) for the multiple scattering problem is reformulated as an interface BVP. A heterogenous medium with variable physical properties in the vicinity of the obstacles is considere...

Wavelet analysis provides information on the energy present at various scales and locations throughout a computational domain. This information is precisely the information that is needed to define the appropriate gridpoint densities and the appropriate numerical order to resolve the physics at hand in the computationally most efficient manner. Here a two-dimensional version of the numerical me...

The motion of an incompressible, viscous rotating fluid contained in a spheroidal container is studied by a direct numerical simulation in an oblate spheroidal coordinate system. An appropriate formalism is first derived which allows us to expand any scalar field in spherical harmonics and to decompose any vector field into its sphero-poloidal and sphero-toroidal scalar parts. The spinover mode...

We present strongly stable semi-discrete finite difference approximations to the quarter space problem (x > 0, t > 0) for the first order in time, second order in space wave equation with a shift term. We consider space-like (pure outflow) and time-like boundaries, with either second or fourth order accuracy. These discrete boundary conditions suggest a general prescription for boundary conditi...

Two new finite difference schemes based on the method of characteristics are presented for convection-diffusion problems. Both of the schemes are of second order in time, and the matrices of the derived systems of linear equations are symmetric. No numerical integration is required for them. The one is of first order in space and the other is of second order. For the former scheme, an optimal e...

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and secondorder accurate in time for vanishing correlation. In our numerical study we obtain highorder numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all ...

This paper is concerned with the construction and developing of several nonstandard finite difference (NSFD) schemes in matrix form in order to obtain numerical solutions of epidemic models. In particular, we deal with a classical SIR epidemic model and a seasonal model associated with the evolution of the transmission of respiratory syncytial virus RSV in the human population. The first model ...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which ...