We construct a nonstandard finite difference (NSFD) scheme for a Burgers type partial differential equation (PDE) for which the diffusion coefficient has a linear dependence on the dependent variable. After a study of this PDE's traveling-wave solutions, we examine the corresponding properties...

We examine and discuss the general problem of how “denominator functions” are to be calculated when using nonstandard finite difference (NSFD) methods for constructing discrete models of differential equations. Using the example of the Fisher PDE with linear advection, we show that the...

We construct a discrete model for a particular PDE having nonlinear advection, diffusion, and reaction. The use of the author’s nonstandard finite difference methods form the basis for this construction. We demonstrate that the solutions to the scheme satisfy positivity and boundedness...

This study presents numerical schemes for solving two three-dimensional parabolic inverse problems. These schemes are developed for indentifying the parameter p(t) which satisfy ut=uxx+uyy+uzz+p(t)u+φ, in R×(0,T], u(x,y,z,0)=f(x,y,z),(x,y,z)∈R=[0,1]3. It is assumed that u is known on the...

A nonstandard finite difference scheme is constructed for the Burgers partial differential equation having no diffusion and a nonlinear logistic reaction term. This scheme preserves the positivity and boundedness properties of the original differential equation and includes the a priori...

Nonlinear diffusion processes can give rise to shock wave type solutions. These solutions are usually derived from the application of similarity methods since general solutions to the relevant partial differential equations are not known. We consider the Boltzmann problem and construct an exact...

We study the linearized stability of a discontinuous solution of a multidimensional hyperbolic system of conservation laws by linearizing the system around the basic solution; the resulting linearized system has discontinuous coefficients and involves nonconservative products. We propose a...

An explicit finite difference scheme for the numerical solution of the diffusion equation on a rectangular region, subject to a non-local boundary condition in the form of a double integral which simulates the specificaion of mass in a curved region is developed. The scheme, is based on the...