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Sinc-Galerkin method; advection-diffusion equation; numerical solution
This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution is shown to converge to the exact solution at an exponential rate. A numerical example is given to illustrate the accuracy of the method.
[1] Al-Khaled, K.: Theory and computation in hyperbolic model problems. Ph.D. Thesis. The University of Nebraska, Lincoln, USA (1996). MR 2694551
[2] Al-Khaled, K.: Sinc approximation of solution of Burgers' equation with discontinuous initial condition. O. P. Iliev, et al. Recent Advances in Numerical Methods and Applications Proceedings of the fourth international conference, NMA, 1998, Sofia, Bulgaria World Scientific, Singapore (1999), 503-511. MR 1786646 | Zbl 0980.65106
[3] Bateman, H.: Some recent researches on the motion of fluids. Monthly Weather Review 43 (1915), 163-170. DOI 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2
[4] Cole, J. D.: On a quasilinear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9 (1951), 225-236. DOI 10.1090/qam/42889 | MR 0042889 | Zbl 0043.09902
[5] Dafermos, C. M.: Large time behavior of solutions of hyperbolic balance laws. Bull. Greek Math. Soc. 25 (1984), 15-29. MR 0815565 | Zbl 0661.35059
[6] He, C., Liu, C.: Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball. Appl. Math. Lett. 24 (2011), 679-686. DOI 10.1016/j.aml.2010.12.005 | MR 2765117 | Zbl 1213.35317
[7] Holden, H., Karlsen, K. H., Mitrovic, D., Panov, E. Yu.: Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function. Acta Math. Sci., Ser. B, Engl. Ed. 29 1573-1612 (2009). DOI 10.1016/S0252-9602(10)60004-5 | MR 2589093 | Zbl 1212.35166
[8] Hopf, E.: The partial differential equation $u_{t}+uu_{x}=\mu u_{xx}$. Department of Mathematics, Indiana University (1942).
[9] Il'in, A. M., Oleĭnik, O. A.: Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time. Mat. Sb. 51 (1960), 191-216 Russian. MR 0120469
[10] Lund, J., Bowers, K. L.: Sinc Methods for Quadrature and Differential Equations. SIAM Philadelphia (1992). MR 1171217 | Zbl 0753.65081
[11] Smoller, J.: Shock Waves and Reaction-Diffusion Equation. Grundlehren der Mathematischen Wissenschaften 258 Springer, New York (1983). DOI 10.1007/978-1-4684-0152-3_14 | MR 0688146
[12] Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics 20 Springer, New York (1993). DOI 10.1007/978-1-4612-2706-9 | MR 1226236 | Zbl 0803.65141
[13] Venttsel', T. D.: Quasilinear parabolic systems with increasing coefficients. Vestn. Mosk. Gos. Univ., Series VI (1963), 34-44.
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