Theory of Approximation and Applications
Volume:10 Issue: 2, Summer and Autumn 2016

In this paper, a numerical scheme for solving singular initial/boundary value problems presented. By applying the reproducing kernel Hilbert space method (RKHSM) for solving these problems, this method obtained to approximated solution. Numerical examples are given to demonstrate the accuracy of the present method. The result obtained by the method and the exact solution are found to be in good agreement with each other and it is noted that our method is of high signi cance. We compare our results with other paper. The comparison of the results with exact ones is made to con rm the validity and eciency.

In this paper the family of elliptic curves over Q given by the equation Ep : Y2 = (X - p)3 X3 (X p)3 where p is a prime number, is studied. It is shown that the maximal rank of the elliptic curves is at most 3 and some conditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 or rank(Ep(Q))≥2 are given.

The purpose of this paper is to obtain necessary and suffcient conditions for existence approximate fi xed point on Geraghty-contraction. In this paper, de nitions of approximate -pair fi xed point for two maps T α , Sαand their diameters are given in a metric space.

A variant of fixed point theorem is proved in the setting of S-metric spaces

In this paper, an effective technique is proposed to determine the numerical solution of nonlinear Volterra-Fredholm integral equations (VFIEs) which is based on interpolation by the hybrid of radial basis functions (RBFs) including both inverse multiquadrics (IMQs), hyperbolic secant (Sechs) and strictly positive definite functions. Zeros of the shifted Legendre polynomial are used as the collocation points to set up the nonlinear systems. The integrals involved in the formulation of the problems are approximated based on Legendre-Gauss-Lobatto integration rule. This technique is so convenience to implement and yields very accurate results compared with the other basis. In addition a convergence theorem is proved to show the stability of this technique. Illustrated examples are included to confirm the validity and applicability of the proposed method. The comparison of the errors is implemented by the other methods in references using both inverse multiquadrics (IMQs), hyperbolic secant (Sechs) and strictly positive definite functions.

In this paper, the reduced di erential transform method is investigated for a nonlinear partial di erential equation modeling nematic liquid crystals, it is called the Hunter-Saxton equation. The main advantage of this method is that it can be applied directly to nonlinear di erential equations without requiring linearization, discretization, or perturbation. It is a semi analytical- numerical method that formulizes Taylor series in a very di erent manner. The numerical results denote that reduced di erential transform method is ecient and accurate for Hunter-Saxton equation.