• Kyungpook Mathematical Journal
• /
• v.60 no.4
• /
• pp.869-881
• /
• 2020

Most first kind integral equations are ill-posed, and obtaining their numerical solution often requires solving a linear system of algebraic equations of large condition number, which may be difficult or impossible. This article proposes a regularization-direct method to numerically solve first kind Fredholm integral equations. The vector forms of block-pulse functions and related properties are applied to formulate the direct method and reduce the integral equation to a linear system of algebraic equations. We include a regularization scheme to overcome the ill-posedness of integral equation and obtain a stable numerical solution. Some test problems are solved using the proposed regularization-direct method to illustrate its efficiency for solving first kind Fredholm integral equations.

• Journal of applied mathematics & informatics
• /
• v.29 no.1_2
• /
• pp.23-37
• /
• 2011

In this paper, we use a modified Tikhonov regularization method to solve the Fredholm integral equation of the first kind. Under the assumption that measured data are contaminated with deterministic errors, we give two error estimates. The convergence rates can be obtained under the suitable choices of regularization parameters and the number of measured points. Some numerical experiments show that the proposed method is effective and stable.

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.223-236
• /
• 2000

In this paper, we solve the Fredholm integral equation of the first and second kind when the kernel takes a singular form. Also, some important relations for Chebyshev polynomial of integration are established.

• Journal of applied mathematics & informatics
• /
• v.17 no.1_2_3
• /
• pp.709-717
• /
• 2005

Integral equations occur naturally in many fields of mechanics and mathematical physics. We consider the Fredholm integral equation of the first kind.In this paper we are interested in integral equation of convolution type. We give approximate solution by Meyer wavelets

• The Transactions of the Korean Institute of Electrical Engineers D
• /
• v.53 no.4
• /
• pp.207-211
• /
• 2004

We consider a class of uncertain nonlinear systems containing the uncertainties without a priori information except that they are bounded. For such systems, we assume that the upper bound of the uncertainties is represented as a Fredholm integral equation of the first kind and we propose an adaptation law that is capable of estimating the upper bound. Using this adaptive upper bound, a continuous robust control which renders uncertain nonlinear systems uniformly ultimately bounded is designed.

• Journal of Institute of Control, Robotics and Systems
• /
• v.9 no.9
• /
• pp.658-663
• /
• 2003

In deterministic design of feedback controllers for uncertain dynamic systems, the upper bound of the uncertainty is very important to guarantee the stability of the closed loop system. In this paper, we assume that the upper bound of the uncertainty is formulated using a Fredholm integral equation of the first kind, that is, an integral of the product of a predefined kernel with an unknown influence function. We propose an adaptation law that is capable of estimating this upper bound. Using this adaptive upper bound, we design an adaptive variable structure control (AVSC), which guarantees asymptotic stability/ultimate boundedness of uncertain dynamic systems. The illustrative example shows the proposed AVSC is effective for uncertain dynamic systems.

• Structural Engineering and Mechanics
• /
• v.6 no.4
• /
• pp.457-466
• /
• 1998

The dynamic response of an elastic torsion shaft of revolution is analysed by the Line-Loaded Integral Equation Method (LLIEM). A "Dynamic Point Ring Couple" (DPRC) is used as a fictitious fundamental load and is distributed in an elastic space along the axis of the shaft outside the shaft occupation. According to the boundary condition, our problem is reduced to a 1-D Fredholm integral equation of the first kind, which is simpler for solving than that of a 2-D singular integral equation of the same kind obtanied by Boundary Element Method (BEM), for steady periodically varied loading. Numerical example of a shaft with quadratic generator under sinusoidal type of torque is given. Formulas for stresses and dangerous frequency are mentioned.

• Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
• /
• v.20 no.8
• /
• pp.42-47
• /
• 2006

In this paper, we propose a sliding mode control with the bound estimation for robot manipulators without requiring exact knowledge of the robot dynamics. For the bound estimation, the upper bound of the uncertain nonlinearities of robot dynamics is represented as a Fredholm integral equation of the first kind and we propose an adaptive scheme which is only dependent on the sliding surface function. Also, we prove the asymptotic stability for the robot systems using two important properties in the robot dynamics: skew-symmetry and positive-definiteness of robot parameters.

• The Transactions of the Korean Institute of Electrical Engineers D
• /
• v.52 no.7
• /
• pp.418-423
• /
• 2003

In order to describe the upper bound of the uncertainties without any information of the structure, we assume that the upper bound is represented as a Fredholm integral equation of the first kind, that is, an integral of the product of a predefined kernel with an unknown influence function. Based on the improved Lyapunov function, we propose an adaptation law that is capable of estimating the upper bound and we design a sliding mode control, which controls effectively for uncertain dynamic systems.

• Nuclear Engineering and Technology
• /
• v.16 no.2
• /
• pp.89-96
• /
• 1984

A toroidal-field (TF) coil with a pure tension D-shape curve is designed for the confinement of high-temperature plasmas in the SNUT-79, which is a tokamak being built at Seoul National University. A toroidal assembly of 16 D-shape TF coils is designed to produce the magnetic field of up to 3T, of which ripples appear to be below 4% of the average toroidal field in the plasma region. Exact positions and currents in six equilibrium coils distributed symmetrically in the z=0 plane are found by the solution of a set of linear equations which is transformed from a Fredholm integral equation of the first kind. The decay indices resulted from equilibrium field indicate that the stability condition for vertical and horizontal displacements is satisfied.