American Institute of Mathematical Sciences

Advanced
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 - A, 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, (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.   Google Scholar

[3]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations,", Cambridge University Press, (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.  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.  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.  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.  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 (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.   Google Scholar

[10]

R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications,", Lecture Notes in Math., (1461).   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.  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.  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.  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.  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.  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.  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.  doi: 10.4171/ZAA/1304.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

G. Vainikko, First-kind cordial Volterra integral equations 1,, Numer. Funct. Anal. Optim., 33 (2012), 680.  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, (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.   Google Scholar

[3]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Equations,", Cambridge University Press, (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.  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.  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.  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.  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 (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.   Google Scholar

[10]

R. Gorenflo and S. Vessella, "Abel Integral Equations: Analysis and Applications,", Lecture Notes in Math., (1461).   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.  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.  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.  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.  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.  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.  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.  doi: 10.4171/ZAA/1304.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[1]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
[2]

Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293
[3]

Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[4]

Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169
[5]

István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089
[6]

M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42
[7]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079
[8]

Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089
[9]

Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008
[10]

Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251
[11]

Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493
[12]

Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53
[13]

Robert Lauter and Victor Nistor. On spectra of geometric operators on open manifolds and differentiable groupoids. Electronic Research Announcements, 2001, 7: 45-53.

[14]

Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053
[15]

Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control & Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191
[16]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[17]

Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826
[18]

Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28
[19]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[20]

Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences