# zbMATH — the first resource for mathematics

Existence of solutions to nonlinear advection-diffusion equation applied to Burgers’ equation using Sinc methods. (English) Zbl 1340.35129
Summary: 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.
##### MSC:
 35K57 Reaction-diffusion equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] K. Al-Khaled: Theory and computation in hyperbolic model problems. Ph.D. Thesis. The University of Nebraska, Lincoln USA, 1996. [2] K. Al-Khaled: Sinc approximation of solution of Burgers’ equation with discontinuous initial condition. Recent Advances in Numerical Methods and Applications (O.P. Iliev, et al., ed.). Proceedings of the fourth international conference, NMA, 1998, Sofia, Bulgaria, World Scientific, Singapore, 1999, pp. 503-511. [3] Bateman, H., Some recent researches on the motion of fluids, Monthly Weather Review, 43, 163-170, (1915) [4] Cole, J.D., On a quasilinear parabolic equation occurring in aerodynamics, Q. Appl. Math., 9, 225-236, (1951) · Zbl 0043.09902 [5] Dafermos, C.M., Large time behavior of solutions of hyperbolic balance laws, Bull. Greek Math. Soc., 25, 15-29, (1984) · Zbl 0661.35059 [6] He, C.; Liu, C., Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball, Appl. Math. Lett., 24, 679-686, (2011) · 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) · Zbl 1212.35166 [8] E. Hopf: The partial differential equation $$u$$_{t}+uu_{x} = µ$$u$$_{xx}. Department of Mathematics, Indiana University, 1942. · Zbl 0039.10403 [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, 191-216, (1960) [10] J. Lund, K. L. Bowers: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, 1992. · Zbl 0753.65081 [11] J. Smoller: Shock Waves and Reaction-Diffusion Equation. Grundlehren der Mathematischen Wissenschaften 258, Springer, New York, 1983. · Zbl 0508.35002 [12] F. Stenger: Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics 20, Springer, New York, 1993. · Zbl 0803.65141 [13] T.D. Venttsel’: Quasilinear parabolic systems with increasing coefficients. Vestn. Mosk. Gos. Univ., Series VI (1963), 34-44. · Zbl 1213.35317
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.