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Novel numerical techniques based on Fokas transforms, for the solution of initial boundary value problems. (English) Zbl 1167.65446
Summary: The unified transform method of {\it A. S. Fokas} [Proc. R. Soc. Lond., Ser. A 453, No. 1962, 1411--1443 (1997; Zbl 0876.35102); A unified approach to boundary value problems. CBMS-NSF Regional Conference Series in Applied Mathematics 78. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xv (2008; Zbl 1181.35002)] has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work, we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the Fokas approach [loc. cit.], which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy $10^{ - 15}$, while the well-established Crank-Nicholson scheme requires 2048 time steps in order to reach a $10^{ - 8}$ accuracy.

MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
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References:
[1] Fokas, A. S.: A unified transform method for solving linear and certain nonlinear pdes, Proc. R. Soc. London A 453, 1411-1443 (1997) · Zbl 0876.35102 · doi:10.1098/rspa.1997.0077
[2] Fokas, A. S.: A unified approach to boundary value problems, (2007) · Zbl 1142.35061
[3] T.S. Papatheodorou, A.N. Kandili, New techniques for the numerical solution of elliptic and time dependent problems, in: SIAM Conference on Analysis of Partial Differential Equations, July 2006
[4] Flyer, N.; Fokas, A. S.: A hybrid analytical-numerical method for solving evolution partial differential equations. I. the half-line, Proc. roy. Soc. A 464, No. 2095, 1823-1849 (2008) · Zbl 1156.65089 · doi:10.1098/rspa.2008.0041
[5] Trefethen, L. N.; Weideman, J. A. C.; Schmelzer, T.: Talbot quadratures and rational approximations, BIT numer. Math. 46, 653-670 (2006) · Zbl 1103.65030 · doi:10.1007/s10543-006-0077-9