Evolution partial differential equations with discontinuous data

Trogdon, Thomas; Biondini, Gino

doi:10.1090/qam/1526

Authors: Thomas Trogdon and Gino Biondini
Journal: Quart. Appl. Math. 77 (2019), 689-726
MSC (2010): Primary 35Q99; Secondary 65M99, 37L50, 42B20
DOI: https://doi.org/10.1090/qam/1526
Published electronically: November 28, 2018
MathSciNet review: 4009329
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using the unified transform method we characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the solutions in the presence of corner singularities. The characterization focuses on an expansion in terms of computable special functions.

References

Gino Biondini and Thomas Trogdon, Gibbs phenomenon for dispersive PDEs on the line, SIAM J. Appl. Math. 77 (2017), no. 3, 813–837. MR 3648078, DOI https://doi.org/10.1137/16M1090892
Norman Bleistein and Richard A. Handelsman, Asymptotic expansions of integrals, 2nd ed., Dover Publications, Inc., New York, 1986. MR 863284
John P. Boyd and Natasha Flyer, Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 281–309. MR 1702205, DOI https://doi.org/10.1016/S0045-7825%2898%2900358-2
Qingshan Chen, Zhen Qin, and Roger Temam, Treatment of incompatible initial and boundary data for parabolic equations in higher dimension, Math. Comp. 80 (2011), no. 276, 2071–2096. MR 2813349, DOI https://doi.org/10.1090/S0025-5718-2011-02469-5
C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), 197–205. MR 117885, DOI https://doi.org/10.1007/BF01386223

B. Deconinck, B. Pelloni, and N. E. Sheils, Non-steady state heat conduction in composite walls, Proc. Roy. Soc. A., 470, 20130605–20130605, (2014)

Bernard Deconinck, Natalie E. Sheils, and David A. Smith, The linear KdV equation with an interface, Comm. Math. Phys. 347 (2016), no. 2, 489–509. MR 3545514, DOI https://doi.org/10.1007/s00220-016-2690-z

Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI https://doi.org/10.1137/110821871

Jeffery C. DiFranco and Kenneth T.-R. McLaughlin, A nonlinear Gibbs-type phenomenon for the defocusing nonlinear Schrödinger equation, IMRP Int. Math. Res. Pap. 8 (2005), 403–459. MR 2204638

Natasha Flyer and Bengt Fornberg, Accurate numerical resolution of transients in initial-boundary value problems for the heat equation, J. Comput. Phys. 184 (2003), no. 2, 526–539. MR 1959406, DOI https://doi.org/10.1016/S0021-9991%2802%2900034-7

Natasha Flyer and Bengt Fornberg, On the nature of initial-boundary value solutions for dispersive equations, SIAM J. Appl. Math. 64 (2003/04), no. 2, 546–564. MR 2049663, DOI https://doi.org/10.1137/S0036139902415853

Natasha Flyer, Zhen Qin, and Roger Temam, A penalty method for numerically handling dispersive equations with incompatible initial and boundary data, Numer. Methods Partial Differential Equations 28 (2012), no. 6, 1996–2009. MR 2981879, DOI https://doi.org/10.1002/num.21693

Natasha Flyer and Paul N. Swarztrauber, The convergence of spectral and finite difference methods for initial-boundary value problems, SIAM J. Sci. Comput. 23 (2002), no. 5, 1731–1751. MR 1885081, DOI https://doi.org/10.1137/S1064827500374169

Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953

N. Flyer and A. S. Fokas, A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008), no. 2095, 1823–1849. MR 2403130, DOI https://doi.org/10.1098/rspa.2008.0041

A. S. Fokas and B. Pelloni (eds.), Unified transform for boundary value problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015. Applications and advances. MR 3364228

A. S. Fokas and L. Sung, Initial-boundary value problems for linear dispersive evoluttion equations on the half-line, preprint at http://imi.cas.sc.edu/papers/209/, unpublished (2000).

Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836

Justin Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations 18 (2005), no. 6, 647–668. MR 2136703

Justin Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1151–1190. MR 2254610, DOI https://doi.org/10.1080/03605300600718503

Lin Huang and Jonatan Lenells, Construction of solutions and asymptotics for the sine-Gordon equation in the quarter plane, J. Integrable Syst. 3 (2018), no. 1, xyy014, 92. MR 3842394, DOI https://doi.org/10.1093/integr/xyy014

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822

Sean D. Lawley, Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives, SIAM J. Appl. Dyn. Syst. 15 (2016), no. 3, 1410–1433. MR 3531725, DOI https://doi.org/10.1137/15M1038426

S. Olver, RHPackage: A Mathematica package for computing solutions to matrix-valued Riemann–Hilbert problems, http://www.maths.usyd.edu.au/u/olver/projects/RHPackage.html

Zhen Qin and Roger Temam, Penalty method for the KdV equation, Appl. Anal. 91 (2012), no. 2, 193–211. MR 2876749, DOI https://doi.org/10.1080/00036811.2011.579564

Jeffrey B. Rauch and Frank J. Massey III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. MR 340832, DOI https://doi.org/10.1090/S0002-9947-1974-0340832-0

Natalie E. Sheils and Bernard Deconinck, Interface problems for dispersive equations, Stud. Appl. Math. 134 (2015), no. 3, 253–275. MR 3322696, DOI https://doi.org/10.1111/sapm.12070

N. E. Sheils and D. A. Smith, Heat equation on a network using the Fokas method, J. Phys. A 48 (2015), no. 33, 335001, 21. MR 3376028, DOI https://doi.org/10.1088/1751-8113/48/33/335001

Stephen Smale, Smooth solutions of the heat and wave equations, Comment. Math. Helv. 55 (1980), no. 1, 1–12. MR 569242, DOI https://doi.org/10.1007/BF02566671

Michael Taylor, Short time behavior of solutions to nonlinear Schrödinger equations in one and two space dimensions, Comm. Partial Differential Equations 31 (2006), no. 4-6, 945–957. MR 2233047, DOI https://doi.org/10.1080/03605300500361537

R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations, J. Differential Equations 43 (1982), no. 1, 73–92. MR 645638, DOI https://doi.org/10.1016/0022-0396%2882%2990075-4

T. Trogdon, Numerical methods for evolution problems, Unified transform for boundary value problems, SIAM, Philadelphia, PA, 2015, pp. 259–292. MR 3364240

Thomas Trogdon and Sheehan Olver, Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. MR 3450072

Athanassios S. Fokas, A. Alexandrou Himonas, and Dionyssios Mantzavinos, The nonlinear Schrödinger equation on the half-line, Trans. Amer. Math. Soc. 369 (2017), no. 1, 681–709. MR 3557790, DOI https://doi.org/10.1090/S0002-9947-2016-06734-3

Athanassios S. Fokas, A. Alexandrou Himonas, and Dionyssios Mantzavinos, The Korteweg–de Vries equation on the half-line, Nonlinearity 29 (2016), no. 2, 489–527. MR 3461607, DOI https://doi.org/10.1088/0951-7715/29/2/489