Abstract
A new method for computing explicit formulae of exact solutions of the Cauchy problem for the time-dependent hyperbolic system of crystal optics with polynomials entries is given. This method is based on symbolic computations of all Taylor coefficients for a solution of the Cauchy problem using the initial data and inhomogeneous term which have polynomial presentation with respect to space variables. Computing explicit formulae of solutions is implemented by Maple 10. A computational example is presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2. International Science, New York (1962)
Haba, Z.: Green functions and propagation of waves in strongly inhomogeneous media. Journal of Physics A: Mathematical and General 37, 9295–9302 (2004)
Wijnands, F., et al.: Green’s functions for Maxwell’s equations: application to spontaneous emission. Optical and Quantum Electronics 29, 199–216 (1997)
Li, L.W., et al.: Circular cylindrical waveguide filled with uniaxial anisotropic media-electromagnetic fields and dyadic Green’s functions. IEEE transactions on microwave and techniques 49(7), 1361–1364 (2001)
Gottis, P.G., Kondylis, G.D.: Properties of the dyadic Green’s function for unbounded anisotropic medium. IEEE transactions on antennas and propagation 45, 154–161 (1995)
Ortner, N., Wagner, P.: Fundamental matrices of homogeneous hyperbolic system. Applications to crystal optics, elastodynamics, and piezoelectromagnetism. Z. Angew. Math. Mech. 84(5), 314–346 (2004)
Burridge, R., Qian, J.: The fundamental solution of the time-dependent system of crystal optics. European J. Appl. Math. 17(1), 63–94 (2006)
Yakhno, V.G.: Constructing Green’s function for the time-dependent Maxwell system in anisotropic dielectrics. Journal of Physics A: Mathematical and General 38(10), 2277–2287 (2005)
Yakhno, V.G., Yakhno, T.M., Kasap, M.: A novel approach for modelling and simulation of electromagnetic waves in anisotropic dielectrics. International Journal of Solids and Structures 43, 6261–6276 (2006)
Monk, P.: Finite element methods for Maxwell’s equations. Clarendon Press, Oxford (2003)
Cohen, G.C.: Higher-order numerical methods for transient wave equations. Springer, Berlin (2002)
Zienkiewicz, O.C., Taylor, R.L.: Finite elements method 1. Butterworth-Heinemann, Oxford (2000)
Cohen, G.C., Heikkola, E., Joly, P., Neittaan, M.P.: Mathematical and numerical aspects of waves propagation. Springer, Berlin (2003)
Werner, G.R., Cary, J.R.: A stable FDTD algorithm for non-diagonal, anisotropic dielectrics. Journal of Computational Physics 226, 1085–1101 (2007)
Ikawa, M.: Hyperbolic partial differential equations and wave phenomena. Translations of Mathematical Monographs, vol. 189. American Mathematical Society, Providence (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yakhno, V., Altunkaynak, M. (2009). Symbolic Computation of an Exact Solution of the Cauchy Problem for the System of Crystal Optics with Polynomial Data . In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_70
Download citation
DOI: https://doi.org/10.1007/978-3-642-00464-3_70
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00463-6
Online ISBN: 978-3-642-00464-3
eBook Packages: Computer ScienceComputer Science (R0)