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Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels

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Rekha P. Kulkarni, Gobinda Rakshit

Functional Analysis · Submitted 16 Apr 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods.

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arXiv ID1904.07895 (version 2)
Pith IDBDAXHSRM, Pith's permanent record of this paper and its review
CitationGobinda Rakshit, Rekha P. Kulkarni. "Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels." arXiv:1904.07895 (2019). https://pith.science/paper/BDAXHSRM
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,257 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.22 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green's function. For $r \geq 0,$ a space of piecewise polynomials of degree $\leq r $ with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr\"{o}m approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green's function. contribution 0.92 0.75 0.08 asserted 0.90
s1 For $r \geq 0,$ a space of piecewise polynomials of degree $\leq r $ with respect to an uniform partition is chosen to be the approximating space. method 0.99 0.80 0.03 asserted 0.99
s2 We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr\"{o}m approximation. method 0.98 0.93 0.06 asserted 0.99
s3 The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. interpretation 0.59 0.90 0.46 hedged possibility 0.87
s4 Numerical results are given for a specific example. result 0.66 0.93 0.05 asserted 0.97

What kind of paper the classifier reports this is: method or tool 0.96 · evidence from proof or derivation 0.85.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green's function. For $r \geq 0,$ a space of piecewise polynomials of degree $\leq r $ with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr\"{o}m approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.

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