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Initial-boundary value problems for linear PDEs with variable coefficients

Published online by Cambridge University Press:  01 July 2007

P. A. TREHARNE  and
A. S. FOKAS
P. A. TREHARNE
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney 2006, Australia. e-mail: [email protected]
A. S. FOKAS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA. e-mail: [email protected]
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Abstract

A new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 221 - 242
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Fokas, A. S.. A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. A 453 (1997), 1411–1443.CrossRefGoogle Scholar
[2]Fokas, A. S.. Two-dimensional linear PDEs in a convex polygon. Proc. R. Soc. Lond. A. 457 (2001), 371–393.CrossRefGoogle Scholar
[3]Fokas, A. S.. A new transform method for evolution equations on the half-line. IMA J. Math. 67 (2002), 559–590.CrossRefGoogle Scholar
[4]Fokas, A. S. and Pelloni, B.. A transform method for linear evolution PDEs on a finite interval. IMA J. Appl. Math. 70 (2005), 564–587.CrossRefGoogle Scholar
[5]Dassios, G. and Fokas, A. S.. The basic elliptic equations in an equilateral triangle. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 2721–2748.Google Scholar
[6]Treharne, P. A. and Fokas, A. S.. Boundary value problems for systems of linear evolution equations. IMA J. Appl. Math. 69 (2004), 539–555.CrossRefGoogle Scholar
[7]Fokas, A. S., Its, A. R. and Sung, L-Y.. The nonlinear Schrödinger equation on the half-line. Nonlinearity 18 (2005), 1771–1822.CrossRefGoogle Scholar
[8]Fokas, A. S.. Integrable nonlinear evolution equations on the half-line. Comm. Math. Phys. 230 (2002), 1–39.CrossRefGoogle Scholar
[9]Fokas, A. S.. The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Comm. Pure Appl. Math. 58 (2005), 639–670.CrossRefGoogle Scholar
[10]Fokas, A. S.. Boundary Value Problems for Linear PDEs with Variable Coefficients. Proc. R. Soc. Lond. A 460 (2004), 1131–1151.CrossRefGoogle Scholar
[11]Beals, R., Deift, P. and Zhou, X.. The inverse scattering transform on the line, in Important Developments in Soliton Theory, Fokas, A. S., Zakharov, V. E., eds, (Springer, Berlin, 1993), 7–37.CrossRefGoogle Scholar
[12]Beals, R., Deift, P. and Tomei, C.. Direct and inverse scattering on the line. Mathematical Surveys and Monographs 28 (American Mathematical Society, Providence, RI, 1988).Google Scholar
[13]Aktosun, T. and Klaus, M.. Inverse theory: problem on the line, in Scattering, Pike, E. R., Sabatier, P. C., eds, (Academic Press, 2001), 770–785.Google Scholar
[14]Boutet de Monvel, A., Fokas, A. S. and Shepelsky, D.. The mKdV equation on the half-line. J. Inst. Math. Jussieu 3 (2004), 139–164.CrossRefGoogle Scholar
[15]Ablowitz, M. J. and Fokas, A. S.. Complex Variables and Applications (Cambridge University Press, 2nd ed., 2003).CrossRefGoogle Scholar
[16]Deift, P. and Trubowitz, E.. Inverse scattering on the line. Comm. Pure Appl. Math. 23 (1979), 121–151.CrossRefGoogle Scholar
[17]Ablowitz, M. J. and de Lillo, S.. Parametric forcing, bound states and solutions of a nonlinear Schrödinger type equation. Nonlinearity 7 (1994), 1143–1153.CrossRefGoogle Scholar
[18]Fokas, A. S. and Sung, L-Y.. Generalised Fourier Transforms, their nonlinearisation and the imaging of the brain. Notices of the AMS Feature Article 52 (2005), 1176–1190.Google Scholar