Journal de mathématiques appliquées et computationnelles

In this paper, the numerical method to obtain the approximate solution of quadratic integral equations of Urysohn’s type is presented. This class of equations is often discussed in the different way focusing on the properties of involving functions in the integral equations. The use of radial basis functions for solving the Fredholm integral equations was offered by Makroglou [1] and we develop the method for solving the quadratic integral equations. In this study, the radial basis functions method with the collocation scheme to obtain the numerical solution of Fredholm and Volterra quadratic integral equations of Urysohn’s type is described. This technique plays an important role to reduce a quadratic integral equation to a system of equations. Some illustrative examples are demonstrated which confirm the efficiency, validity and applicability of the presented technique.

The aim of this work is to apply the homotopy perturbation method and homotopy analysis method to the problem of thermal explosion in a flammable gas mixture with the addition of volatile fuel droplets. The system of equations that describes the effects of heating, evaporation, and combustion of fuel in a polydisperse spray is simplified. Both convective and radiative heating of droplets is taken into account in the model. The model for the radiative heating of droplets takes into account the semitransparency of the droplets. The results of the analysis have been applied to the modeling of the thermal explosion in diesel engines. We applied the Homotopy Perturbation Method and the Homotopy Analysis Method to the new model and we found the region of the convergence of the considered solutions of the relevant physical parameters. The results demonstrate that these methods are very effective for solving nonlinear problems in science and engineering.

We study the stability character and periodic solutions of the following rational difference systems
1 1
1 1
1 1
= 1 , = 1 ,
1 1
n n
n n
n n n n
x x y y
y x x y
− −
+ +
− −
± ±
± ±
where the initial values 1 0 1 0 x , x , y , y − − are nonzero real numbers. Some numerical examples are given to illustrate our results.

In this article, an approximate analytical solution of fractional order nonlinear PDE’s with modified Riemann- Liouville derivative was obtained with the help of fractional variational iteration method (FVIM). It is showed that the solutions obtained by the FVIM are reliable and effective method for strongly nonlinear partial equations with modified Riemann-Liouville derivative. The solutions of our model equation can also be obtained from the known forms of the series solutions.

The symmetric smoothed particle hydrodynamics (SSPH) method is used to generate the basis functions to solve 2D homogeneous and non-homogeneous steady-state heat transfer problems. The SSPH basis functions together with the collocation method (i.e, the strong formulation of the problem) are used to solve sample problems. Comparisons are made with the results obtained by using different weight functions and particle numbers. The error norms for three sample problems are computed by the use of two different kernel functions such as the revised Gauss function and revised super Gauss function, among which the revised super Gauss function yields the smallest error norm. It is observed that the SSPH method yields large errors for non-homogenous problems, especially if the forcing term is not smooth.