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In this paper, we present high accuracy quadrature formulas for hyper-singular integrals α ≤ -1 (or α < -1 and α being non-integer), and the error power is O(h η ) with q(x, t) = x -t for α being integers ... Since the derivatives of the density function g(x) in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can easily be applied to solving corresponding hyper-singular ... Acknowledgements The authors would like to thank the editor and referees for useful comments and suggestions. ...doi:10.1186/s13660-015-0950-7 fatcat:yro46sg7pjgezdn6ypckzp3r5u
When an integrable singularity (x° with a >—1) is present, Navot’s generalization is valid. This introduces negative fractional powers of m into the expansion. ... They are applied to the lower-order formulae, basi- cally to the trapezoidal rule. ...
algorithms for solving first kind Abel integral equations are proposed, which have the high accuracy O(h 2 ) and asymptotic expansion of the errors. ... Then by means of Richardson extrapolation, an approximation with higher accuracy order O(h 3 ) is obtained. Moreover, an a posteriori error estimate for the algorithms is derived. ... Acknowledgement The authors would like to thank Professor Rudolf Gorenflo for his insightful suggestions. ...doi:10.1016/j.cam.2006.02.021 fatcat:c4nk6vkmgbg7lnyqmowbomdtpa
A numerical test of the "functional stratification" variance reduction algorithm is performed with the fractional Brownian motion. ... In particular, we derive the optimal quantization of the fractional Brownian motion by approximating the first terms of its Karhunen-Lo\'eve decomposition. ... However as pointed out in  , the Nyström method associated with the trapezoidal integration rule admits an asymptotic error expansion in even powers of the step sizes as soon as the covariance function ...arXiv:1009.1241v1 fatcat:7bnfd4memraqneg6trhou6b3km
Modified forms of the trapezoidal and mid- ordinate rules, taking account of the singularities, are obtained; it is then shown that the resulting approxima- tions can be extrapolated by Romberg’s method ... Further modifications to deal with the case when the integrand has an integrable branch singularity at one or both ends of the interval of integration are also briefly discussed.” J. ...
As is known from the theory of Abel integral equations, the rate of convergence of any finite-different method depends on the severity of kernel's singularity. ... We also determine the exact form of error constants. Next, we construct a finite-difference scheme based on a trapezoidal rule to solve a general first order integro-differential equation. ... Finding the whole asymptotic expansion of these quantities is one of the objectives of our future work. Proof Theorem 1 Let us start with the simplest case of the rectangular rule. ...doi:10.1007/s11075-016-0247-z fatcat:vns7breikzfudg2t2uelykivvm
We present a generic scheme to construct corrected trapezoidal rules with spectral accuracy for integral operators with weakly singular kernels in arbitrary dimensions. ... We provide a complete set of formulas for singularities of the type: (r) and r^-ν. ... We have presented the construction scheme for corrected trapezoidal rules for integral operators with weakly singular kernels. ...arXiv:1211.6074v1 fatcat:potmmrrmpfah7i5knd35yqyiuy
We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. ... The conformal map is a polynomial adjustment to the map, and allows the treatment of a finite number of singularities in the complex plane. ... Since the trapezoidal rule is equivalent to the integration of the Sinc expansion of a function  , the entire process of analyzing the convergence rates with different endpoint decay can also be useful ...arXiv:1406.3320v1 fatcat:lng7mzkrvjdyphqzuozaxx46hm
Acknowledgements I would like to thank Jet Wimp for his careful reading of the paper. ... He corrected my English in many places, he asked me to provide more explanations when needed, and suggested many improvements in the presentation. ... the approximation obtained by the trapezoidal rule with step size h to the deÿnite integral S and the g i are the known functions (forming an asymptotic sequence when h tends to zero) appearing in the ...doi:10.1016/s0377-0427(00)00360-5 fatcat:ohykewmvqndipdnyew32xrtglm
Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. ... We introduce a quadrature scheme-QBKIX-for the ubiquitous highorder accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all ... A popular method for on-surface quadrature is the product integration (in 2D, for the global trapezoid rule see [4, Section 4.2] or [40, Section 12.3] , and for panelbased rules see  ). ...doi:10.1007/s10543-017-0689-2 fatcat:loxwiozy3bhjtb5vjr5despjau
These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. ... Hence, by adjusting the weights locally, high-order-accurate versions of the trapezoidal rule can be constructed also in the presence of singularities and boundaries. ... Acknowledgements The support and enthusiasm with which Hans Riesel has met all our questions on Bernoulli numbers and the Riemann zeta function is gratefully acknowledged. Funding ...doi:10.1093/imanum/drt046 fatcat:4nqaphexzvdr3gtaljbcb2jhzq
We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. ... We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. ... Part of this research was performed while the third author was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation (Grant No. ...arXiv:2107.01438v3 fatcat:4oqghzmrebc7xgqb6fvgjdhikq
We present a simple discretization scheme for the hypersingular integral representation of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. ... Through singularity subtraction, we obtain a regularized integrand that is amenable to the trapezoidal rule with equispaced nodes, assuming a high degree of regularity in the underlying function (i.e., ... This gives an integral that is nice enough to admit discretization with the trapezoidal rule or related schemes. ...arXiv:1802.03770v3 fatcat:zgp75nly4nckli7ji5zvvdbvpq
While classically these transformations are applied with a product trapezoidal rule, we use adaptive methods in the PARINT software package, based on rules of higher polynomial degree for the integration ... We apply and compare results of transformations used to annihilate boundary singularities for multivariate integration over hyper-rectangular and simplicial domains. ... This work is further supported by Grant-in-Aid for Scientific Research (24540292) of JSPS, and the Large Scale Simulation Program No. 14/15-06, No. 15/16-06 of KEK. ...doi:10.1016/j.procs.2016.05.462 fatcat:nuh7bi2rwfconm4ygk7buzf62e
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. ... This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original ... We employ the convention of Heading, Dingle, Olver, and Berry, but other authors such as Bender and Orszag reverse the meaning of 'Stokes' and 'anti-Stokes'. ...doi:10.1023/a:1006145903624 fatcat:tkfmgemlend6xika2ss6comsvy
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