Analysis of general quadrature methods for integral equations of the second kind (original) (raw)
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Quadrature methods for integral equations of the second kind over infinite intervals
Mathematics of Computation, 1981
Convergence results are proved for a class of quadrature methods for integral equations of the form y(t) = fit) + /ô° k(t, s)y(s) ds. An important special case is the Nystrom method, in which the integral term is approximated by an ordinary quadrature rule. For all of the methods considered here, the rate of convergence is the same, apart from a constant factor, as that of the quadrature approximation to the integral term.
An improved quadrature method for integral equations with logarithmic kernel
2003
In this paper we present a family of modied quadrature methods for the numer- ical approximation of integral equations of the rst kind with logarithmic kernel. We prove the stability and the existence, under some smoothness assumptions for the exact solution, of an expansion in powers of the discretization parameter of the error. Using this expansion we deduce that a particular method reaches order three. Some comments on the use of Richardson extrapolation are given.
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
SIAM Journal on Numerical Analysis, 1974
The solution of the Volterra integral equation f l K(t, s, x(s)) (*) x(t) g(t) + v-g2(t) + ,: _: ds, 0 <__ <= T, where gl(t), ga(t) and K(t, s,x) are smooth functions, can be represented as x(t)-u(t)+ x/v(t), 0 __< _< T, where u(t), v(t) are smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate x(t) via u(t), v(t) in a neighborhood of the origin and use (*) on the rest of the interval 0 __< _< T. In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order h7/2. Asymptotic error estimates are derived in order to examine the numerical stability of the methods.
Regularized Quadrature Methods for Fredholm Integral Equations of the First Kind
2018
Although quadrature methods for solving ill-posed integral equations of the first kind were introduced just after the publication of the classical papers on the regularization by A.N. Tikhonov and D.L. Phillips, there are still no known results on the convergence rate of such discretization. At the same time, some problems appearing in practice, such as Magnetic Particle Imaging (MPI), allow one only a discretization corresponding to a quadrature method. In the present paper we study the convergence rate of quadrature methods under general regularization scheme in the Reproducing Kernel Hilbert Space setting.
On fully discrete collocation methods for solving weakly singular integral equations
Mathematical Modelling and Analysis, 2009
In order to find approximate solutions of Volterra and Fredholm integrodifferential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non-smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.
Picard and Adomian methods for quadratic integral equation
Computational & Applied Mathematics, 2010
We are concerning with two analytical methods; the classical method of successive approximations (Picard method) [14] which consists the construction of a sequence of functions such that the limit of this sequence of functions in the sense of uniform convergence is the solution of a quadratic integral equation, and Adomian method which gives the solution as a series see
Journal of Advances in Mathematics and Computer Science, 2019
An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels. The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing n.