China
Africa
Explore chapters and articles related to this topic
Fredholm Integral Equation in Electrochemistry
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Mirela I. Iorga, Mihai V. Putz
Fredholm integral equation represents in mathematics an integral equation whose solutions have to give rise to Fredholm theory, Fredholm nuclei study, and Fredholm operators. Fredholm integral equation was the basis of Fredholm’s theorems, one of them being known as Fredholm alternative. The applicability of Fredholm integral equation was demonstrated in several domains, among them one of great importance is electrochemistry because the solutions of Fredholm integral equations facilitate the determination of process conditions, parameters, and so on.
Integral and Integro-Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
Despite the similarity between the Fredholm integral equation and the Volterra integral equation, there is a major diff erence between them. In particular, iffand K in (11.267) and (11.268) are continuous, there is a unique solution for the linear nonhomogeneous Volterra integral equation of the second kind. However, for a linear nonhomogeneous Fredholm integral equation of the second kind to have a unique solution, the Fredholm alternative theorem must be satisfied.
LP solutions to the parameterized Fredholm integral equations associated with Chandrasekhar kernels
Published in Applicable Analysis, 2022
As we all know, in mathematics, eigenvalue problems will lead to multiple or no solutions to linear integral equations of the first and second kinds. For example, when the second kind of homogeneous integral equation has a non-trivial solution, one may consider the corresponding non-homogeneous integral equation. In this case, if the function on the right side of the integral equation satisfies the compatibility condition of Fredholm alternative, then the integral equation has infinitely many solutions, otherwise, there is no solution. In physics, the well-posedness of partial differential equations is often a fundamental problem to be solved. For example, as one studies the Neumann problem of the Laplace equation in a simply connected region with a smooth boundary, an equivalent integral equation is usually derived, and a similar situation will also occur at this time [1]. The difficulty with the numerical solutions of the first kind of Fredholm equation and Volterra equation is that these solutions do not continuously depend on the data. This instability of the first kind of integral equation will continue to the solution of the algebraic equation system obtained by the discretization of the integral equation. Atkinson [2] proposes a numerical method to obtain the approximate solution of the second kind of non-uniquely solvable Fredholm integral equation. The main idea is to first transform the equation into a unique solvable equation, and then use numerical integration to approximate the integral operator.