882 research outputs found
A M\"untz-Collocation spectral method for weakly singular volterra integral equations
In this paper we propose and analyze a fractional Jacobi-collocation spectral
method for the second kind Volterra integral equations (VIEs) with weakly
singular kernel . First we develop a family of fractional
Jacobi polynomials, along with basic approximation results for some weighted
projection and interpolation operators defined in suitable weighted Sobolev
spaces. Then we construct an efficient fractional Jacobi-collocation spectral
method for the VIEs using the zeros of the new developed fractional Jacobi
polynomial. A detailed convergence analysis is carried out to derive error
estimates of the numerical solution in both - and weighted
-norms. The main novelty of the paper is that the proposed method is
highly efficient for typical solutions that VIEs usually possess. Precisely, it
is proved that the exponential convergence rate can be achieved for solutions
which are smooth after the variable change for a
suitable real number . Finally a series of numerical examples are
presented to demonstrate the efficiency of the method
Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel
The Jacobi pseudo-spectral Galerkin method for the Volterra integro-differential equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the Lωα,β2-norm and the L∞-norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results
Spectral collocation method for compact integral operators
We propose and analyze a spectral collocation method for integral
equations with compact kernels, e.g. piecewise smooth kernels and
weakly singular kernels of the form We prove that 1) for integral equations, the convergence
rate depends on the smoothness of true solutions . If
satisfies condition (R): }, we obtain a geometric rate of convergence; if
satisfies condition (M): ,
we obtain supergeometric rate of convergence for both Volterra
equations and Fredholm equations and related integro differential
equations; 2) for eigenvalue problems, the convergence rate depends
on the smoothness of eigenfunctions. The same convergence rate for
the largest modulus eigenvalue approximation can be obtained.
Moreover, the convergence rate doubles for positive compact
operators. Our numerical experiments confirm our theoretical
results
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Generalised Dirichelt-to-Neumann map in time dependent domains
We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with
l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a
linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution.
For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit
weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane.
The above Volterra equations are shown to admit a unique solution
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