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On Volterra integral operators with highly oscillatory kernels
1. | Department of Mathematics, Hong Kong Baptist University, Hong Kong |
References:
[1] |
K. E. Atkinson, "The Numerical Solution of Integral Equations of the Second Kind," Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511626340. |
[2] |
A. Böttcher, H. Brunner, A. Iserles and S. P. Nørsett, On the singular values and eigenvalues of the Fox-Li and related operators, New York J. Math., 16 (2010), 539-561. |
[3] |
H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations," Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511543234. |
[4] |
H. Brunner, A. Iserles and S. P. Nørsett, "The Spectral Problem for a Class of Highly Oscillatory Fredholm Integral Operators," IMA J. Numer. Anal., 30 (2010), 108-130.
doi: 10.1093/imanum/drn060. |
[5] |
H. Brunner, A. Iserles and S. P. Nørsett, The computation of the spectra of highly oscillatory Fredholm integral operators, J. Integral Equations Appl., 23 (2010), 467-519.
doi: 10.1216/JIE-2011-23-4-467. |
[6] |
S. N. Curle, Solution of an integral equation of Lighthill, Proc. Roy. Soc. London Ser A, 364 (1978), 435-441.
doi: 10.1098/rspa.1978.0210. |
[7] |
T. Diogo and P. Lima, Superconvergence of collocation methods for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math., 218 (2008), 307-316.
doi: 10.1016/j.cam.2007.01.023. |
[8] |
B. Engquist, A. Fokas, E. Hairer and A. Iserles, "Highly Oscillatory Problems," London Math. Soc. Lecture Note Ser., 366, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9781139107136. |
[9] |
N. B. Franco, S. McKee and J. Dixon, A numerical solution of Lighthill's integral equation for the surface temperature distribution of a projectile, Mat. Apl. Comput., 2 (1983), 257-271. |
[10] |
R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications," Lecture Notes in Math., 1461, Springer-Verlag, Berlin-Heidelberg, 1991. |
[11] |
W. Han, Existence, uniqueness and smoothness results for second-kind Volterra equations with weakly singular kernels, J. Integral Equations Appl., 6 (1994), 365-384.
doi: 10.1216/jiea/1181075819. |
[12] |
A. Iserles, On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators, IMA J. Numer. Anal., 25 (2005), 25-44.
doi: 10.1093/imanum/drh022. |
[13] |
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772.
doi: 10.1007/s10543-004-5243-3. |
[14] |
A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillating integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399.
doi: 10.1098/rspa.2004.1401. |
[15] |
J.-P. Kauthen, A survey of singularly perturbed Volterra equations, Appl. Numer. Math., 24 (1997), 95-114.
doi: 10.1016/S0168-9274(97)00014-7. |
[16] |
H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory,, Quart. Appl. Math., 35 (): 165.
|
[17] |
M. J. Lighthill, Contributions to the theory of heat transfer through a laminar boundary layer, Proc. Roy. Soc. London Ser. A, 202 (1950), 359-377.
doi: 10.1098/rspa.1950.0106. |
[18] |
A. Pedas and G. Vainikko, Integral equations with diagonal and boundary singularities of the kernel, Z. Anal. Anwend., 25 (2006), 487-516.
doi: 10.4171/ZAA/1304. |
[19] |
H. M. Srivastava and R. G. Buschmann, "Convolution Integral Equations," Wiley, New York, 1977. |
[20] |
Ll. N. Trefethen and M. Embree, "Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators," Princeton University Press, Princeton, NJ, 2005. |
[21] |
F. Ursell, Integral equations with a rapidly oscillating kernel, J. London Math. Soc., 44 (1969), 449-459. |
[22] |
G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 30 (2009), 1145-1172.
doi: 10.1080/01630560903393188. |
[23] |
G. Vainikko, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim., 31 (2010), 191-219.
doi: 10.1080/01630561003666234. |
[24] |
G. Vainikko, First-kind cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 33 (2012), 680-704.
doi: 10.1080/01630563.2012.665260. |
show all references
References:
[1] |
K. E. Atkinson, "The Numerical Solution of Integral Equations of the Second Kind," Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511626340. |
[2] |
A. Böttcher, H. Brunner, A. Iserles and S. P. Nørsett, On the singular values and eigenvalues of the Fox-Li and related operators, New York J. Math., 16 (2010), 539-561. |
[3] |
H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations," Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511543234. |
[4] |
H. Brunner, A. Iserles and S. P. Nørsett, "The Spectral Problem for a Class of Highly Oscillatory Fredholm Integral Operators," IMA J. Numer. Anal., 30 (2010), 108-130.
doi: 10.1093/imanum/drn060. |
[5] |
H. Brunner, A. Iserles and S. P. Nørsett, The computation of the spectra of highly oscillatory Fredholm integral operators, J. Integral Equations Appl., 23 (2010), 467-519.
doi: 10.1216/JIE-2011-23-4-467. |
[6] |
S. N. Curle, Solution of an integral equation of Lighthill, Proc. Roy. Soc. London Ser A, 364 (1978), 435-441.
doi: 10.1098/rspa.1978.0210. |
[7] |
T. Diogo and P. Lima, Superconvergence of collocation methods for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math., 218 (2008), 307-316.
doi: 10.1016/j.cam.2007.01.023. |
[8] |
B. Engquist, A. Fokas, E. Hairer and A. Iserles, "Highly Oscillatory Problems," London Math. Soc. Lecture Note Ser., 366, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9781139107136. |
[9] |
N. B. Franco, S. McKee and J. Dixon, A numerical solution of Lighthill's integral equation for the surface temperature distribution of a projectile, Mat. Apl. Comput., 2 (1983), 257-271. |
[10] |
R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications," Lecture Notes in Math., 1461, Springer-Verlag, Berlin-Heidelberg, 1991. |
[11] |
W. Han, Existence, uniqueness and smoothness results for second-kind Volterra equations with weakly singular kernels, J. Integral Equations Appl., 6 (1994), 365-384.
doi: 10.1216/jiea/1181075819. |
[12] |
A. Iserles, On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators, IMA J. Numer. Anal., 25 (2005), 25-44.
doi: 10.1093/imanum/drh022. |
[13] |
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772.
doi: 10.1007/s10543-004-5243-3. |
[14] |
A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillating integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399.
doi: 10.1098/rspa.2004.1401. |
[15] |
J.-P. Kauthen, A survey of singularly perturbed Volterra equations, Appl. Numer. Math., 24 (1997), 95-114.
doi: 10.1016/S0168-9274(97)00014-7. |
[16] |
H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory,, Quart. Appl. Math., 35 (): 165.
|
[17] |
M. J. Lighthill, Contributions to the theory of heat transfer through a laminar boundary layer, Proc. Roy. Soc. London Ser. A, 202 (1950), 359-377.
doi: 10.1098/rspa.1950.0106. |
[18] |
A. Pedas and G. Vainikko, Integral equations with diagonal and boundary singularities of the kernel, Z. Anal. Anwend., 25 (2006), 487-516.
doi: 10.4171/ZAA/1304. |
[19] |
H. M. Srivastava and R. G. Buschmann, "Convolution Integral Equations," Wiley, New York, 1977. |
[20] |
Ll. N. Trefethen and M. Embree, "Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators," Princeton University Press, Princeton, NJ, 2005. |
[21] |
F. Ursell, Integral equations with a rapidly oscillating kernel, J. London Math. Soc., 44 (1969), 449-459. |
[22] |
G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 30 (2009), 1145-1172.
doi: 10.1080/01630560903393188. |
[23] |
G. Vainikko, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim., 31 (2010), 191-219.
doi: 10.1080/01630561003666234. |
[24] |
G. Vainikko, First-kind cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 33 (2012), 680-704.
doi: 10.1080/01630563.2012.665260. |
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