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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications," Lecture Notes in Math., 1461, Springer-Verlag, Berlin-Heidelberg, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory,, Quart. Appl. Math., 35 (): 165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. M. Srivastava and R. G. Buschmann, "Convolution Integral Equations," Wiley, New York, 1977.  Google Scholar

[20]

Ll. N. Trefethen and M. Embree, "Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators," Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[21]

F. Ursell, Integral equations with a rapidly oscillating kernel, J. London Math. Soc., 44 (1969), 449-459.  Google Scholar

[22]

G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 30 (2009), 1145-1172. doi: 10.1080/01630560903393188.  Google Scholar

[23]

G. Vainikko, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim., 31 (2010), 191-219. doi: 10.1080/01630561003666234.  Google Scholar

[24]

G. Vainikko, First-kind cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 33 (2012), 680-704. doi: 10.1080/01630563.2012.665260.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications," Lecture Notes in Math., 1461, Springer-Verlag, Berlin-Heidelberg, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory,, Quart. Appl. Math., 35 (): 165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. M. Srivastava and R. G. Buschmann, "Convolution Integral Equations," Wiley, New York, 1977.  Google Scholar

[20]

Ll. N. Trefethen and M. Embree, "Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators," Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[21]

F. Ursell, Integral equations with a rapidly oscillating kernel, J. London Math. Soc., 44 (1969), 449-459.  Google Scholar

[22]

G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 30 (2009), 1145-1172. doi: 10.1080/01630560903393188.  Google Scholar

[23]

G. Vainikko, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim., 31 (2010), 191-219. doi: 10.1080/01630561003666234.  Google Scholar

[24]

G. Vainikko, First-kind cordial Volterra integral equations 1, Numer. Funct. Anal. Optim., 33 (2012), 680-704. doi: 10.1080/01630563.2012.665260.  Google Scholar

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