On graded meshes for weakly singular Volterra integral equations with oscillatory trigonometric kernels

Wu, Qinghua

doi:10.1016/j.cam.2013.12.039

Cited by 20 publications

(14 citation statements)

References 17 publications

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“…They possess the property that the higher oscillation, the higher accuracy. In this section, based on the Formulas (11), (13) and 22, we present some preliminary numerical experiments to verify the result of theoretical analysis. The experiments are performed on a 1.86 GHz PC with 2 GB of RAM.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In recent years, there has been tremendous interest in developing methods for solving highly oscillatory Volterra integral equation, such as discontinuous Galerkin method [5], Filon-type method [6,7], collocation method [4,8,9], collocation boundary value method [10,11], collocation method on uniform mesh [12], collocation method on graded mesh [13].…”

Section: Introductionmentioning

confidence: 99%

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Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Fang

Xiang

2019

Symmetry

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In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Fang

Xiang

2019

Symmetry

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“…where E m ðyÞ ¼ inf P m 2P m ky À P m k 1 and P m denotes the space of the algebraic polynomials of degree at most m. Theorem 1 Denote by y the unique solution of Eq (7) and by y n m the unique solution of (14). Then, if f(x) 2 C r ([−1, 1]) and the kernel K satisfies the conditions (15) and (16), the following error estimate holds true…”

Section: Plos Onementioning

confidence: 99%

Efficient Nyström-type method for the solution of highly oscillatory Volterra integral equations of the second kind

Wu,

Sun

2023

PLoS ONE

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Highly oscillatory Volterra integral equations are frequently encountered in engineering applications. The Nyström-type method is an important numerical approach for solving such problems. However, there remains scope to further optimize and accelerate the Nyström method. This paper presents a novel Nyström-type method to efficiently approximate solutions to second-kind Volterra integral equations with highly oscillatory kernels. First, the unknown function is interpolated at Chebyshev points. Then the integral equation is solved using the Nyström-type method, which leads to a problem of solving a system of linear equations. A key contribution is the technique to express the fundamental Lagrange polynomial in matrix form. The elements of the matrix, which involves highly oscillatory integrals, are calculated by using the classical Fejér quadrature formula with a dilation technique. The proposed method is more efficient than the one proposed in the recent literature. Numerical examples verify the efficiency and accuracy of the proposed method.

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“…in which > 0, > 0; for details see ( [20], pp 215). Once the integral [ , , , ] is obtained, by substituting into (10), the coefficient matrix is derived; then, we can compute the solution of integral by solving these linear algebraic equations. 0.0141286409694892 0.000146310591227839 −0.00107727880701281 8 2 0.0141193392298137 0.000145019893750479 −0.00107593754108643 16 2 0.0140583682955762 0.000144178555797097 −0.00106881635211410 0.00323202027752064 −0.0995508315886191 8 2 0.00321634589900894 −0.0994917961267987 16 2 0.00319500282444326 −0.0993143176430846…”

Section: [ ] = ∫mentioning

confidence: 99%

“…Recently, for weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels, it was found that the collocation methods are much more easily implemented and can get higher accuracy than discontinuous Galerkin methods under the same piecewise polynomials space; for details see [7][8][9][10]. In addition, collocation method only involves single integrals which are a little easier to evaluate.…”

Section: Introductionmentioning

confidence: 99%

The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels

2014

Journal of Applied Mathematics

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A method for approximating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatory trigonometric kernel is presented. The unknown function is approximated by expansion of Chebychev polynomial and the coefficients are determinated by classical collocation method. Due to the highly oscillatory kernels of integral equation, the discretised collocation equation will give rise to the computation of oscillatory integrals. These integrals are calculated by using recursion formula derived from the fundamental recurrence relation of Chebyshev polynomial. The effectiveness and accuracy of the proposed method are tested by numerical examples.

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On graded meshes for weakly singular Volterra integral equations with oscillatory trigonometric kernels

Cited by 20 publications

References 17 publications

Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Efficient Nyström-type method for the solution of highly oscillatory Volterra integral equations of the second kind

The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels

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