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The SEIR model for the transmission dynamics of measles is extended by the addition of second-order space derivatives to enable the geographic spread of the disease in a population which has not been vaccinated against it.The resulting system of three reaction-diffusion equations is solved by a convergent finite-difference technique which is second-order accurate in space and time. A parallel implementation procedure is studied and the method is tested using two initial distributions. [Copyright &y& Elsevier]

A first-order, unconditionally-stable, finite-difference scheme is developed for the numerical solution of the SIR model. It is seen that numerical simulations using the method reflect the long-term behaviour of the continuous-time system accurately. The introduction of seasonal variation into the SIR model leads to periodic and chaotic dynamics of epidemics which are present in the numerical simulations. [Copyright &y& Elsevier]

Abstract: In this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters, and it can be expressed by the symmetric bases. It is proved that the interpolation is stable. The concept of integral weights coefficients of the interpolation is given, which describes the “weight” of the interpolation points and the quantity as the interpolation data in the local interpolating region. [Copyright &y& Elsevier]

Abstract: A weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator. The degree of smoothness of this spline is C 2 in the interpolating interval when the parameters satisfy a continuous system. The sufficient and necessary conditions that constrain the interpolant curves to be convex in the interpolating interval or subinterval are derived. Also, the error estimate formulas of this interpolation are obtained. [Copyright &y& Elsevier]

In this paper a new method is developed to create a high-order smoothness interpolation using values of the function being interpolated. This is a kind of rational cubic interpolation with quadratic denominator. This rational spline not only belongs to C2 in the interpolating interval, but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region, all because of the selectable parameters in the rational spline itself. The more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters. [Copyright &y& Elsevier]

In this paper, our main purpose is to establish the existence of entire explosive positive radial solutions for quasi-linear elliptic equations. The main results of the paper extend previously known results both for the super-linear and sub-linear cases. [Copyright &y& Elsevier]

Abstract: In this paper a bivariate rational Hermite interpolation is developed to create a space surface using both function values and the first-order partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters. There are two schemes to construct the interpolation functions, an example shows that both of them approximate the function being interpolated very well. The basis of this interpolation is derived and the interpolating function can be C 1 in all the interpolation region when the parameters satisfy a simple condition. When a patch of the interpolating surface does not satisfy the need of design, say it is too high or too low at a point and its neighbourhood, the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed. For this, sufficient conditions are derived, and it is shown that this can be done just by selecting suitable parameters in the interpolation. [Copyright &y& Elsevier]

Abstract: We use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method. [Copyright &y& Elsevier]