On The Use of Conformal Maps for the Acceleration of Convergence of the Trapezoidal Rule and Sinc Numerical Methods

Abstract The double exponential formula, or DE formula, is a high-precision integration formula using a change of variables called a DE transformation; it has the disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky & Olver (2015, On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. SIAM J. Sci. Comput., 37, A676–A700) attempted to improve the formula by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz–Christoffel transformation for which we can derive the explicit form. The new transformation formula can be regarded as a generalization of DE transformations. We confirm its effectiveness by numerical experiments.

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New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II

Applied System Innovation ◽

10.3390/asi1030030 ◽

2018 ◽

Vol 1 (3) ◽

pp. 30 ◽

Cited By ~ 2

Author(s):

Hussein ALKasasbeh ◽

Irina Perfilieva ◽

Muhammad Ahmad ◽

Zainor Yahya

Keyword(s):

Numerical Methods ◽

Differential Equations ◽

Trapezoidal Rule ◽

Approximation Methods ◽

System Of Differential Equations ◽

Fuzzy Transform ◽

Fuzzy Partition ◽

Fuzzy Approximation ◽

One Step ◽

Basic Functions

In this research, three approximation methods are used in the new generalized uniform fuzzy partition to solve the system of differential equations (SODEs) based on fuzzy transform (FzT). New representations of basic functions are proposed based on the new types of a uniform fuzzy partition and a subnormal generating function. The main properties of a new uniform fuzzy partition are examined. Further, the simpler form of the fuzzy transform is given alongside some of its fundamental results. New theorems and lemmas are proved. In accordance with the three conventional numerical methods: Trapezoidal rule (one step) and Adams Moulton method (two and three step modifications), new iterative methods (NIM) based on the fuzzy transform are proposed. These new fuzzy approximation methods yield more accurate results in comparison with the above-mentioned conventional methods.

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Handbook of Sinc Numerical Methods

10.1201/b10375 ◽

2016 ◽

Cited By ~ 16

Author(s):

Frank Stenger

Keyword(s):

Numerical Methods ◽

Sinc Numerical Methods

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Summary of sinc numerical methods

Journal of Computational and Applied Mathematics ◽

10.1016/s0377-0427(00)00348-4 ◽

2000 ◽

Vol 121 (1-2) ◽

pp. 379-420 ◽

Cited By ~ 75

Author(s):

Frank Stenger

Keyword(s):

Numerical Methods ◽

Sinc Numerical Methods

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On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations

Symmetry ◽

10.3390/sym11101200 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1200

Author(s):

Sanda Micula

Keyword(s):

Numerical Methods ◽

Integral Equations ◽

Trapezoidal Rule ◽

Picard Iteration ◽

Fredholm Integral Equations ◽

Convergence Conditions ◽

One Dimensional ◽

Cubature Formulas ◽

Uniqueness Of The Solution ◽

Iterative Numerical Methods

In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra–Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.

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