Abstract
We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.
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Agbolade, O.A., Anake, T.A.: Solutions of first-order Volterra type linear integrodifferential equations by collocation method. J. Appl. Math. 2017, 1–5 (2017). https://doi.org/10.1155/2017/1510267
Allaei, S.S., Yang, Z.W., Brunner, H.: Collocation methods for third-kind VIEs. IMA J. Numer. Anal. 37. https://doi.org/10.1093/imanum/drw033 (2017)
Apartsyn, A.S.: On some classes of linear Volterra integral equations abstract and applied analysis. https://doi.org/10.1155/2014/532409 (2014)
Atkinson, K.E.: A survey of numerical methods for solving nonlinear integral equations. Journal of Integral Equations and Applications 4(1). https://doi.org/10.1216/jiea/1181075664 (1992)
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5). https://doi.org/10.1137/S1064827503430126 (2004)
Beals, R., Wong, R.: Special Functions and Orthogonal Polynomials. No. 153 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)
Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017). https://doi.org/10.1137/141000671
Biazar, J., Ebrahimi, H.: Chebyshev wavelets approach for nonlinear systems of Volterra integral equations. Computers & Mathematics with Applications 63(3), 608–616 (2012). https://doi.org/10.1016/j.camwa.2011.09.059
van den Bosch, F., Metz, J.A.J., Zadoks, J.C.: Pandemics of focal plant disease, a model. Phytopathology 89(6), 495–505 (1999). https://doi.org/10.1094/PHYTO.1999.89.6.495
Böttcher, A., Silbermann, B., Karlovich, A.: Analysis of Toeplitz Operators, 2nd edn. Springer Monographs in Mathematics. Springer, Berlin (2006). OCLC: 181538992
Brunner, H.: On the numerical solution of nonlinear Volterra integro-differential equations. BIT Numer. Math. 13 (4), 381–390 (1973). https://doi.org/10.1007/BF01933399
Brunner, H.: High-order Methods for the numerical solution of Volterra integro-differential equations. J. Comput. Appl. Math. 15(3). https://doi.org/10.1016/0377-0427(86)90221-9 (1986)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)
Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D., Brown, B.P.: Dedalus: a flexible framework for numerical simulations with spectral methods. arXiv:1905.10388[astro-ph, physics:physics] (2019)
Driscoll, T.A.: Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations. J. Comput. Phys. 229(17). https://doi.org/10.1016/j.jcp.2010.04.029 (2010)
Driscoll, T.A., Bornemann, F., Trefethen, L.N.: The chebop system for automatic solution of differential equations. BIT Numerical Mathematics 48(4). https://doi.org/10.1007/s10543-008-0198-4 (2008)
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, Second Edn. No. 155 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2014)
Ezquerro, J.A., Hernández, M.A., Romero, N.: Solving nonlinear integral equations of Fredholm type with high order iterative methods. J. Comput. Appl. Math. 236(6). https://doi.org/10.1016/j.cam.2011.09.009 (2011)
Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2004)
Geiser, J.: An iterative splitting approach for linear integro-differential equations. Appl. Math. Lett. 26, 1048–1052 (2013). https://doi.org/10.1016/j.aml.2013.05.012
Ghasemi, M., Kajani, M.T., Babolian, E.: Numerical solutions of the nonlinear Volterra–Fredholm integral equations by using homotopy perturbation method. Appl. Math. Comput. 188(1), 446–449 (2007). https://doi.org/10.1016/j.amc.2006.10.015
Gordji, M.E., Baghani, H., Baghani, O.: On existence and uniqueness of solutions of a nonlinear integral equation. https://doi.org/10.1155/2011/743923 (2011)
Gutleb, T.S.: TSGut/SparseVolterraExamples.jl: v0.1.1. https://doi.org/10.5281/zenodo.4382253 (2020)
Gutleb, T.S., Olver, S.: A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle. SIAM J. Numer. Anal. 58(3), 1993–2018 (2020). https://doi.org/10.1137/19M1267441
Hackbusch, W.: Integral Equations: Theory and Numerical Treatment. No. 120 in International Series of Numerical Mathematics. Basel, Birkhäuser (1995)
Hale, N.: An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type. IMA J. Numer. Anal. 39(4), 1727–1746 (2019). https://doi.org/10.1093/imanum/dry042
Hale, N., Olver, S.: A fast and spectrally convergent algorithm for Rational-Order fractional integral and differential equations. SIAM J. Sci. Comput. 40. https://doi.org/10.1137/16M1104901 (2018)
Hethcote, H.W., Tudor, D.W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9(1), 37–47 (1980). https://doi.org/10.1007/BF00276034
Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F., Cattani, C.: Wavelets Method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simul. https://doi.org/10.1016/j.cnsns.2013.04.026 (2014)
Krimer, D.O., Putz, S., Majer, J., Rotter, S.: Non-markovian dynamics of a single-mode cavity strongly coupled to an inhomogeneously broadened spin ensemble. Phys. Rev. A 90(4). https://doi.org/10.1103/PhysRevA.90.043852 (2014)
Krimer, D.O., Zens, M., Putz, S., Rotter, S.: Sustained photon pulse revivals from inhomogeneously broadened spin ensembles. Laser Photonics Rev. (6): 1023–1030. https://doi.org/10.1002/lpor.201600189 (2016)
Lepik, U.: Haar wavelet method for nonlinear integro-differential equations. Appl. Math. Comput. 176. https://doi.org/10.1016/j.amc.2005.09.021 (2006)
Lintner, S.K., Bruno, O.P.: A generalized calderoń formula for open-arc diffraction problems: theoretical considerations. P. Roy. Soc. Edinb. A 145(2). https://doi.org/10.1017/S0308210512000807 (2015)
Meehan, M., O’Regan, D.: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear analysis: theory. Methods & Applications 31. https://doi.org/10.1016/S0362-546X(96)00313-6(1998)
Micke, A., Bülow, M.: Application of Volterra integral equations to the modelling of the sorption kinetics of multi-component mixtures in porous media. Gas Separation & Purification 4(3), 158–164 (1990). https://doi.org/10.1016/0950-4214(90)80018-G
Mogensen, P.K., Carlsson, K., Villemot, S., Lyon, S., Gomez, M., Rackauckas, C., Holy, T., Widmann, D., Kelman, T., Macedo, M.R.G., Benneti, Bojesen, T.A., Arakaki, T., Christ, S., Byrne, S., Lubin, M., Barton, D., Kwon, C., Lucibello, C., Riseth, A.N., Levitt, A.: JuliaNLSolvers/NLsolve.jl: v4.2.0. https://doi.org/10.5281/zenodo.3527404 (2019)
Nedaiasl, K., Bastani, A.F., Rafiee, A.: A product integration method for the approximation of the early exercise boundary in the american option pricing problem. Mathematical Methods in the Applied Sciences 42(8), 2825–2841 (2019). https://doi.org/10.1002/mma.5553
Olver, F., Daalhuis, A., Lozier, D., Schneider, B., Boisvert, R., Clark, C., Miller, B., Saunders (eds.), B.V.: NIST Digital Library of Mathematical Functions (2018). https://dlmf.nist.gov/
Olver, S.: JuliaApproximation/ApproxFun.jl (2019). https://github.com/JuliaApproximation/ApproxFun.jl
Olver, S., Townsend, A.: A fast and well-conditioned spectral method. SIAM Rev. 55(3). https://doi.org/10.1137/120865458 (2013)
Olver, S., Townsend, A.: A practical framework for infinite-dimensional linear algebra. In: 2014 First Workshop for High Performance Technical Computing in Dynamic Languages. IEEE, LA, USA. https://doi.org/10.1109/HPTCDL.2014.10(2014)
Olver, S., Townsend, A., Vasil, G.: A sparse spectral method on triangles. arXiv:1902.04863 (2019)
Pachon, R., Platte, R.B., Trefethen, L.N.: Piecewise-smooth chebfuns. IMA Journal of Numerical Analysis 30(4). https://doi.org/10.1093/imanum/drp008 (2010)
Prüss, J.: Evolutionary Integral Equations and Applications. Modern Birkhäuser Classics. Springer, Basel, New York (2012). OCLC: ocn796763028
Saeedi, H., Mohseni Moghadam, M.: Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1216–1226 (2011). https://doi.org/10.1016/j.cnsns.2010.07.017
Sahu, P.K., Ray, S.S.: Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Appl. Math. Comput. https://doi.org/10.1016/j.amc.2015.01.063 (2015)
Shayanfard, F., Laeli Dastjerdi, H., Maalek Ghaini, F.: collocation method for approximate solution of Volterra integro-differential equations of the third-kind. Appl. Numer. Math. https://doi.org/10.1016/j.apnum.2019.09.020 (2019)
Slevinsky, R.M.: Conquering the pre-computation in two-dimensional harmonic polynomial transforms. arXiv:1711.07866 (2017)
Slevinsky, R.M.: Fast and backward stable transforms between spherical harmonic expansions and bivariate fourier series. Appl. Comput. Harmon Anal. https://doi.org/10.1016/j.acha.2017.11.001 (2017)
Slevinsky, R.M.: FastTransforms v0.1.1. https://github.com/MikaelSlevinsky/FastTransforms. Original-date: 2018-03-15T23:11:52Z (2019)
Slevinsky, R.M., Olver, S.: A fast and well-conditioned spectral method for singular integral equations. J. Comput. Phys. 332, 290–315 (2017). https://doi.org/10.1016/j.jcp.2016.12.009
Snowball, B., Olver, S.: Sparse spectral and-finite element methods for partial differential equations on disk slices and trapeziums. Stud. Appl. Math. 145(1), 3–35 (2020)
Song, H., Yang, Z., Brunner, H.: Analysis of collocation methods for nonlinear Volterra integral equations of the third kind calcolo. https://doi.org/10.1007/s10092-019-0304-9 (2019)
Townsend, A., Olver, S.: The automatic solution of partial differential equations using a global spectral method. J. Comput. Phys. 299. 10.1016/j.jcp.2015.06.031 (2015)
Unterreiter, A.: Volterra integral equation models for semiconductor devices. Mathematical Methods in the Applied Sciences 19 (6), 425–450 (1996). https://doi.org/10.1002/(SICI)1099-1476(199604)19:6<425::AID-MMA744>3.0.CO;2-M
Wazwaz, A.M.: Linear an Nonlinear Integral Equations: Methods and Applications. Higher Education Press, Beijing (2011)
Wazwaz, A.M.: The regularization method for Fredholm integral equations of the first kind. Computers & Mathematics with Applications 61(10), 2981–2986 (2011)
Xiang, S., Brunner, H.: Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels. BIT Numer. Math. 53(1), 241–263 (2013). 10.1007/s10543-012-0399-8
Zakes, F., Sniady, P.: Application of Volterra Integral Equations in Dynamics of Multispan Uniform Continuous Beams Subjected to a Moving Load (2016). https://doi.org/10.1155/2016/4070627
Zhang, P., Hao, X.: Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces. Advances in Difference Equations 2018. https://doi.org/10.1186/s13662-018-1681-0 (2018)
Zhu, L., Fan, Q.: Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Commun. Nonlinear Sci. Numer. Simul. 17. https://doi.org/10.1016/j.cnsns.2011.10.014 (2012)
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The author would like to thank Sheehan Olver for reading a draft and providing helpful comments, and the anonymous reviewers for their useful comments and suggestions.
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Gutleb, T.S. A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels. Adv Comput Math 47, 42 (2021). https://doi.org/10.1007/s10444-021-09866-7
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DOI: https://doi.org/10.1007/s10444-021-09866-7
Keywords
- Volterra
- Integral equations
- Nonlinear
- Integro-differential
- General kernels
- Spectral methods
- Multivariate orthogonal polynomials