Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

An analysis of the numerical solution of Fredholm integral equations of the first kind

Summary This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT _N x=y _N ,N=1,2, .... The operatorsT andT _N can be viewed as operators from eitherL ₂[a, b] toL ₂[c,d] or as operators fromL [a, b] toL [c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT _N x=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.     

A rapid solution of a kind of 1D Fredholm oscillatory integral equation

How to solve oscillatory integral equations rapidly and accurately is an issue that attracts special attention in many engineering fields and theoretical studies. In this paper, a rapid solution method is put forward to solve a kind of special oscillatory integral equation whose unknown function is much less oscillatory than the kernel function. In the method, an improved-Levin quadrature method is adopted to solve the oscillatory integrals. On the one hand, the employment of this quadrature method makes the proposed method very accurate; on the other hand, only a small number of small-scaled systems of linear equations are required to be solved, so the computational complexity is also very small. Numerical examples confirm the advantages of the method.     

A convergence rate for approximate solutions of Fredholm integral equations of the first kind

We study the convergence rate for solving Fredholm integral equations of the first kind by using the well known collocation method. By constructing an approximate interpolation neural network, we deduce the convergence rate of the approximate solution by only using continuous functions as basis functions for the Fredholm integral equations of the first kind. This convergence rate is bounded in terms of a modulus of smoothness.     

An algorithm for the solution of ill-posed linear systems arising from the discretization of Fredholm integral equation of the first kind

An algorithm for obtaining approximate solutions of ill-posed systems of linear equations arising from the discretization of Fredholm integral equation of the first kind is described. The ill-posed system is first replaced by an equivalent consistent system of linear equations. The method calculates the minimum length least squares solution of the consistent system. Starting from rank = 1 of the consistent system, the rank is increased by one in succession and a new solution is calculated. This is repeated until a certain simple criterion is satisfied. Linear programming techniques are used for which successive solutions are the basic solutions in the successive simplex tableaux. The algorithm is numerically stable. Numerical results show that this method compares favorably with other direct methods.     

Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

An adaptive numerical method for solving linear Fredholm integral equations of the first kind

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.Supported in part by the Office of Naval Research under Contract No. NR 044-457.Supported in part by the National Science Foundation under Grant No. GJ-31827.     

Numerical solution of a certain hypersingular integral equation of the first kind

In this paper, we first discuss the midpoint rule for evaluating hypersingular integrals with the kernel sin ⁻²[(x−s)/2] defined on a circle, and the key point is placed on its pointwise superconvergence phenomenon. We show that this phenomenon occurs when the singular point s is located at the midpoint of each subinterval and obtain the corresponding supercovergence analysis. Then we apply the rule to construct a collocation scheme for solving the relevant hypersingular integral equation, by choosing the midpoints as the collocation points. It’s interesting that the inverse of coefficient matrix for the resulting linear system has an explicit expression, by which an optimal error estimate is established. At last, some numerical experiments are presented to confirm the theoretical analysis.     

On the approximate solution of one-dimensional singular integral equations of first kind

We prove an estimate for the error in approximate solution of one-dimensional singular integral equations. The estimate is obtained by an approximation of the kernel. For a specific problem we give a comparison of numerical results. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

A numerical method for solving the Fredholm integral equation of the second kind

The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernels using the Toeplitz matrices. The solution has a computing time requirement ofO(N ²), where 2N + 1 is the number of discretization points used. Also, the error estimate is computed. Some numerical examples are computed using the MathCad package.     

Approximate solution of a complete singular integral equation of the first kind with a fixed hypersingularity by the overlapping method

We suggest a new method for the numerical solution of a singular integral equation of the first kind with a fixed hypersingularity, which arises in the problem on the flow past a profile with an ejector of the external flow. This method permits one to obtain a solution of the characteristic and complete integral equations with an interpolation degree of accuracy.     

On an approximate solution of a class of boundary integral equations of the first kind

We construct a method for computing an approximate solution of the boundary integral equation of the first kind corresponding to the Dirichlet boundary value problems for the Helmholtz equation.     

On asymptotic method in contact problems of Fredholm integral equation of the second kind

Besides asymptotic methods, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kernel which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite algebraic system is obtained.     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

The solution of mildly singular integral equation of the first kind on a disk

The integral equation $$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }}dy = G(x)} $$ x,y ∈ E², with 0 < λ < 2 is studied. Uniqueness for integrable solutions F is established under the assumption that G is integrable. Existence of an integrable solution F is then obtained under the further assumption that G ∈ C², with an explicit solution formula being given for F in terms of integral operators acting on derivatives of G.     

The solution of a mildly singular integral equation of the first kind on a ball

We consider the equations $$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }} dy = G(x)(\left| x \right| \leqslant 1)} $$ where x and y ∈ E^p, p ≥ 3 and λ < p. For λ ∈ (p-2,p) we show that this equation has at most one integrable solution which if G is twice differentiable actually exists and is, in fact, given by an explicit formula in terms of integral operators acting on G and its derivatives. When λ ≤ p-2 and λ ≠ ?2j (j=0,1,?) the equation also has at most one integrable solution for which, assuming it to exist and G to be sufficiently smooth, there also is an explicit formula; in this situation, though, the explicit formula does not usually provide an integrable solution, because, in general, such solutions do not exists when λ ≤ p-2 and λ ≠ ?2j (j=0,1,?), no matter how smooth G is. In the remaining case λ = ?2j (j=0,1,?), neither uniqueness nor existence holds for solutions of the equation.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

Convergence of approximate solution of system of Fredholm integral equations

In this paper numerical solution of system of linear Fredholm integral equations by means of the Sinc-collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The exponential convergence rate of the method is proved. The method is applied to a few test examples with continuous kernels to illustrate the accuracy and the implementation of the method.