11 research outputs found
Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels
In the present paper we consider discrete versions of the modified projection
methods for solving a Urysohn integral equation with a kernel of the type of
Green's function. For a space of piecewise polynomials of degree
with respect to an uniform partition is chosen to be the
approximating space. We define a discrete orthogonal projection onto this space
and replace the Urysohn integral operator by a Nystr\"{o}m approximation. The
order of convergence which we obtain for the discrete version indicates the
choice of numerical quadrature which preserves the orders of convergence in the
continuous modified projection methods. Numerical results are given for a
specific example.Comment: This is the the same paper with the arXiv identifier 1904.07895, but
the shortened version. A bit change in the title als
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
Superconvergence of the Iterated Galerkin Methods for Hammerstein Equations
In this paper, the well-known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numerical examples are presented to illustrate the superconvergence of the iterated Galerkin approximation for Hammerstein equations with weakly singular kernels. © 1996, Society for Industrial and Applied Mathematic
Approximate Analytical Methods For Solving Fredholm Integral Equations
Persamaan kamiran memainkan peranan penting dalam banyak bidang sains seperti
matematik, biologi, kimia, fizik, mekanik dan kejuruteraan. Oleh yang
demikian,pelbagai teknik berbeza telah digunakan untuk menyelesaikan persamaan
jenis ini. Kajian ini, memfokus kepada analisis secara matematik dan berangka bagi
beberapa kes persamaan kamiran Fredholm yang linear dan bukan linear. Kes-kes ini
termasuklah persamaan kamiran Fredholm satu dimensi jenis pertama dan kedua,
persamaan kamiran Fredholm dua dimensi jenis pertama dan kedua dan sistem
persamaan kamiran Fredholm satu dimensi dan dua dimensi.
Integral equations play an important role in many branches of sciences such as
mathematics, biology, chemistry, physics, mechanics and engineering. Therefore, many
different techniques are used to solve these types of equations. This study focuses on the
mathematical and numerical analysis of some cases of linear and nonlinear Fredholm
integral equations. These cases are one-dimensional Fredholm integral equations of the
first kind and second kind, two-dimensional Fredholm integral equations of the first kind
and second kind and systems of one and two-dimensional Fredholm integral equations