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March  2014, 34(3): 915-929. doi: 10.3934/dcds.2014.34.915

On Volterra integral operators with highly oscillatory kernels

Hermann Brunner 1,

1. 

Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received  December 2012 Revised  January 2013 Published  August 2013

We study the high-oscillation properties of solutions to integral equations associated with two classes of Volterra integral operators: compact operators with highly oscillatory kernels that are either smooth or weakly singular, and noncompact cordial Volterra integral operators with highly oscillatory kernels. In the latter case the focus is on the dependence of the (uncountable) spectrum on the oscillation parameter. It is shown that the results derived in this paper merely open a window to a general theory of solutions of highly oscillatory Volterra integral equations, and many questions remain to be answered.
Keywords: Volterra integral operators, cordial Volterra operators, non-compact Volterra operators, highly oscillatory solutions., spectra, highly oscillatory kernels.
Mathematics Subject Classification: 45D05, 47G10, 65R2.
Citation: Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
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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