Error bounds for maximum entropy approximate solutions to Fredholm integral equations (original) (raw)
Properties of maximum entropy approximate solutions to Fredholm integral equations
Journal of Mathematical Physics, 1991
The properties of the maximum entropy approximate solutions to Fredholm integral equations of the second kind are discussed. It is proved that the approximate solutions exist and converge to the exact one for a wider class of problems than previously determined. The correspondence between the maximum entropy technique and the Galerkin method is established and on its basis an upper a posteriori bound for a solution is found.
Fredholm Integral Equations of the Second Kind (General Kernel
In Chap. 1, we conducted a thorough examination of the Fredholm integral equation of the second kind for an arbitrary complex parameter λ , assuming that the free term f (x) is complex-valued and continuous on the interval [a, b] and that the kernel K(x,t) is complex-valued, continuous, and separable on the square Q(a, b) = {(x,t) : [a, b] × [a, b]}. We stated the four Fredholm theorems and the Fredholm Alternative Theorem which provide for the construction of all possible solutions to the equation under these assumptions. A question naturally arises: What, if anything, can be proven if K(x,t) is a general kernel, i.e., an arbitrary kernel that is only assumed to be complex-valued and continuous? In this chapter, we will answer this question completely by proving that all of the Fredholm theorems continue to hold in this eventuality. In Sect. 2.1, we present several tools of the trade which are indispensible for the comprehension of the material in this chapter. In Sects. 2.2 and 2.3, we use these tools to show that the Fredholm integral equation of the second kind with a general kernel has a unique solution if the product of the parameter λ and the " size " of the kernel is small. In Sect. 2.4, we prove the validity of the Fredholm theorems for unrestricted λ and a general kernel. In Sect. 2.5, we show how to construct the resolvent kernel that appears in the solution to the integral equation recursively. In Sect. 2.6, we introduce numerical methods for producing an approximation to the solution of a Fredholm integral equation. These methods are necessary due to the inherent computational difficulties in constructing the resolvent kernel.
On the Regularization of Fredholm Integral Equations of the First Kind
SIAM Journal on Mathematical Analysis, 1998
In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the L 2 -norm, in the limit of noise vanishing.
A Fredholm-Type Theorem for Linear Integral Equations of Stieltjes Type
2008
We consider the linear integral equations of Fredholm and Volterra Z b a α(t, s) x(s) dg(s) = f (t), t 2 [a, b], and x(t) Z t a α(t, s) x(s) dg(s) = f (t), t 2 [a, b], in the frame of the Henstock-Kurzweil integral and we prove results on the existence and uniqueness of solutions. More precisely, we consider the above equations in the sense of Henstock-Kurzweil and we state a Fredholm Alternative theorem for the first equation and an existence and uniqueness result for the second equation for which the solution is given explicitly.
Malaysian Journal of Fundamental and Applied Sciences, 2023
In this note, the problems of solving by the aid of regularization method for a nonlinear Fredholm functional-integral equation of the first kind with degenerate kernel and integral maxima are considered. In using the regularization method, we denote the nonlinear function as a new unknown function. This method we combine with the method of the degenerate kernel. Fredholm functionalintegral equation of the first kind is ill-posed (non-correct) problem. We have used boundary conditions to ensure the uniqueness of the solution. We transform the implicit functional equation with integral maxima to another type of nonlinear functional-integral equation of the second kind by some integral transforms. Using the method of successive approximations and the method of compressing mapping, the theorem on single valued solvability of the problem is proved. It is obtained the necessary and sufficient conditions of existence and uniqueness of the solution of the problem with integral maxima. Two simple examples are analyzed with an exact and approximate solution.
Approximate solutions of Fredholm integral equations of the second kind
2009
This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail.