
Approximate solution of the integral equations involving kernel with additional singularity
The paper is devoted to the approximate solutions of the Fredholm integr...
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Numerical Solution of Nonlinear Abel Integral Equations: An hpVersion Collocation Approach
This paper is concerned with the numerical solution for a class of nonli...
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A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels
As is well known, using piecewise linear polynomial collocation (PLC) an...
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Discretetime Simulation of Stochastic Volterra Equations
We study discretetime simulation schemes for stochastic Volterra equati...
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A simple approach to proving the existence, uniqueness, and strong and weak convergence rates for a broad class of McKean–Vlasov equations
By employing a system of interacting stochastic particles as an approxim...
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Composite Quadrature Methods for Weakly Singular Convolution Integrals
The wellknown Caputo fractional derivative and the corresponding Caputo...
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Singular EulerMaclaurin expansion
We generalise the EulerMaclaurin expansion and make it applicable to th...
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Numerical methods for stochastic Volterra integral equations with weakly singular kernels
In this paper, we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a θEulerMaruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in L^p norm for any p≥ 1. For the θEulerMaruyama scheme the rate is min{1α,1/2β} for the Milstein scheme the rate ismin{1α,12β}whenα≠1/2, where(0<α<1, 0< β<1/2). These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of Itô formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies.
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