in this work, we proposed an efective method based on cubic and pantic b-spline scaling functions to solve partial differential equations of frac- tional order. our method is based on dual functions of b-spline scaling func- tions. we derived the operational matrix of fractional integration of cubic and pantic b-spline scaling functions and used them to transform the mentioned equations to a ...

Let n ≥ 3, v1(x), . . . , vn−1(x) ∈ Z[x] and u ∈ {−1, 1}, then x(x− v1(a)y) · · · (x− vn−1(a)y) + uy = ±1 is called a parametrized familiy of Thue equations, if a ∈ Z and the solutions x, y are searched in Z; cf. Thomas (1993). There are several results concerning parametrized families of cubic and quartic families of Thue equations, see Mignotte et al. (1996) and the references therein. Thomas...

A new differential quadrature method based on quartic B-spline functions is introduced. The weighting coefficients are determined via a semi-explicit algorithm containing an algebraic equation system with four-band coefficient matrix. In order to validate the proposed method, the Burgers’ Equation is selected as test problem. The shock wave and the sinusoidal disturbance solutions of the Burger...

A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earli...

In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we give a proof of their reciprocity law which is not only considerably shorter but also she...

We develop a rational biquadratic G analogue of the non-uniform C B-spline paradigm. These G splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-pepresents them in spline form and uses the spline form to provide shape handles for locali...

For a smooth plane cubic B, we count curves C of degree d such that the normalizations of C\B are isomorphic to A, for d ≤ 7 (for d = 7 under some assumption). We also count plane rational quartic curves intersecting B at only one

In this paper simple quartic trigonometric polynomial blending functions, with a tensionparameter, are presented. These type of functions are useful for constructing trigonometricB´ezier curves and surfaces, they can be applied to construct continuous shape preservinginterpolation spline curves with shape parameters. To better visualize objects and graphics atension parameter is included. In th...

I. Introduction Rational spline is a commonly used spline function. In many cases the rational spline curves better approximating functions than the usual spline functions. It has been observed that many simple shapes including conic section and quadric surfaces can not be represented exactly by piecewise polynomials, whereas rational polynomials can exactly represent all conic sections and qua...

We consider the quartic center ẋ = −yf(x, y), ẏ = xf(x, y), with f(x, y) = (x+a)(y+b)(x+c) and abc 6= 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n− 1)/2] + 4 ≤ σ ≤ 5[(n− 1)/2 + 14, ...