Linear equations

Krasnosel’skii, M. A.; Vainikko, Gennadi; Zabreiko, P. P.; Rutitskii, Ya. B.; Stetsenko, V. Ya.

doi:10.1007/978-94-010-2715-1_2

Cited by 11 publications

(4 citation statements)

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“…Other root-finding methods for finding the solutions of nonlinear operators, like IDEs, exist and have been in use for many years (Rall 1969 ; Krasnosel’skii et al. 1972 ; Hutson and Pym 1980 ; Werner 1981 ; Kantorovich and Akilov 1982 ; Precup 2002 ). These methods are, however, chiefly numerical and require solving for the population density at every location in the spatial domain.…”

Section: Discussionmentioning

confidence: 99%

Block-pulse integrodifference equations

Gilbertson,

Kot

2023

J. Math. Biol.

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We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing growth, for populations that are confined to a finite habitat-patch. This method is based on approximating the growth function of the IDE with a piecewise-constant function, and we call the resulting model a block-pulse IDE. We explicitly write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. We characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including formal stability analyses, and we explore the bifurcation structure of these models. These simple models display rich dynamics, with numerous fold bifurcations. We then use three-, five-, and ten-step block-pulse IDEs, with a numerical root finder, to approximate models with compensatory Beverton–Holt growth and depensatory, or Allee-effect, growth. Our method provides a good approximation for the equilibrium distributions for compensatory and depensatory growth and offers numerical and analytical advantages over the original growth models.

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Section: Discussionmentioning

confidence: 99%

Block-pulse integrodifference equations

Gilbertson,

Kot

2023

J. Math. Biol.

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“…An approximate solution of the Fredholm integral equation (16) or (30) in the Hilbert space L true~ 2 can be found by one of the known methods [6, 7], in particular by highly efficient methods for equation with coercive self-adjoint operators [7, 8]. As it will be seen from the results of the next section, an approximate solution in L true~ 2 is sufficient for efficient calculations of mechanical characteristics and there is no necessity for an approximate solution in C ( 0 , η ) .…”

Section: Regularization Of the Dual Integral Equationsmentioning

confidence: 99%

Debonding of an elastic layer with a cavity from a rigid substrate caused by rotation of a bonded rigid cylinder

Malits

2024

Mathematics and Mechanics of Solids

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Debonding of an elastic layer with a circular cylindrical cavity [Formula: see text], [Formula: see text], from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone [Formula: see text] is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As [Formula: see text] and [Formula: see text], a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions.

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“…By [23, theorem 1.1] we know this for m = 1, and in particular we have injectivity W m, p → W 1, p . The inequalities of Young and Hölder together with (18) imply for any multi-indices β, β with |β|…”

Section: Theorem 33 Describes Equivalent Properties Of γ and T(k)mentioning

confidence: 99%

“…Here we will describe a projection method for solving µ as generally described in [18] and [19]. A detailed convergence proof is given in [24].…”

Section: Numerical Solution Of the Lippmann-schwinger Equationmentioning

confidence: 99%

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

Siltanen¹,

2000

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The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R 2 from knowledge of the Dirichletto-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C 2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman's proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast.

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Linear equations

Cited by 11 publications

References 0 publications

Block-pulse integrodifference equations

Block-pulse integrodifference equations

Debonding of an elastic layer with a cavity from a rigid substrate caused by rotation of a bonded rigid cylinder

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

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