We consider real trigonometric polynomial Bernoulli equations of the form A(θ)y′=B1(θ)+Bn(θ)yn where n≥2, with A,B1,Bn being polynomials degree at most μ≥1 in variables θ and Bn(θ)≢0. also A(θ)yn−1y′=B0(θ)+Bn(θ)yn A,B0,Bn variable For first equation, we show that when n≥4, it has 3 solutions n is even 5 odd. second n≥3, odd even. provide two types mentioned above maximum number achieved. The pr...

A general class of convergent methods for the numerical solution of ordinary differential equations is employed to obtain a class of convergent generalized reducible quadrature methods for Volterra integral equations of the second kind and Volterra integro-differential equations.

The aim of this contribution is to show how the Ftransform technique can be generalized from the case of constant components to the case of polynomial components. For this purpose, we choose complete functional spaces with inner products. After a general presentation of Fm-transform where m ≥ 0, a detailed characterization of the F -transform is given. We applied a technique of numeric integrat...

We propose a numerical method (implemented in Matlab) for computing algebraic quadrature nodes and weights on compact multivariate domains. It relies on the search of maximum volume submatrices of Vandermonde matrices in suitable polynomial bases, by a greedy algorithm based on QR factorization with column pivoting. Such nodes are approximate Fekete points, and are good also for polynomial inte...

The differential quadrature method (DQM) is one of the most elegant and useful approximate methods for solving initial and/or boundary value problems. It is easy to use and also straightforward to implement. However, the conventional DQM is well-known to have some difficulty in implementing multiple initial and/or boundary conditions at a given discrete point. To overcome this difficulty, this ...

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...

In this article, we present an error estimate of the derivative approximation by the local multiquadricbased differential quadrature (LMQDQ) method. Radial basis function is different from the polynomial approximation, in which Taylor series expansion is not applicable. So, the present analysis is performed through the numerical solution of Poisson equation. It is known that the approximation e...