782 research outputs found
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
A 3-dimensional singular kernel problem in viscoelasticity: an existence result
Materials with memory, namely those materials whose mechanical and/or
thermodynamical behaviour depends on time not only via the present time, but
also through its past history, are considered. Specifically, a three
dimensional viscoelastic body is studied. Its mechanical behaviour is described
via an integro-differential equation, whose kernel represents the relaxation
modulus, characteristic of the viscoelastic material under investigation.
According to the classical model, to guarantee the thermodynamical
compatibility of the model itself, such a kernel satisfies regularity
conditions which include the integrability of its time derivative. To adapt the
model to a wider class of materials, this condition is relaxed; that is,
conversely to what is generally assumed, no integrability condition is imposed
on the time derivative of the relaxation modulus. Hence, the case of a
relaxation modulus which is unbounded at the initial time t = 0, is considered,
so that a singular kernel integro-differential equation, is studied. In this
framework, the existence of a weak solution is proved in the case of a three
dimensional singular kernel initial boundary value problem.Comment: 15 page
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
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
The solution of the Volterra integral equation, where , and are smooth functions, can be represented as ,, where , are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate via , in a neighborhood of the origin and use (*) on the rest of the interval . 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 . Asymptotic error estimates are derived in order to examine the numerical stability of the methods
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