NUMERICAL EXPERIMENTS FOR NONLINEAR BURGER’S PROBLEM

Authors:

Jawad Kadhim Tahir,

DOI NO:

https://doi.org/10.26782/jmcms.2021.11.00002

Keywords:

Burger's problem,numerical solution,Cole-Hopf transformation,non-classical variational.,

Abstract

The article contains the results of computational experiments for the non-homogeneous Burger's problem and finding its solution by using the non-classical variational-Cole-Hopf transformation approach. On using exact linearization via Cole-Hopf transformation, as well as the application of the non-classical variational approach, then the non-homogeneous Burger's problem has been solved. The solution which is obtained by this approach is in a compact form so that the original nonlinear solution is easy to be approximated. The accuracy of this method is dependent on the types of selected basis and its number.

Refference:

I. Ames, W. F., “Nonlinear Partial Differential Equations in engineering”, Academic pres, Inc., London, 1965.
II. Bateman, H., “Some recent Researches on the Motion of Fields”, Mon. Weather rev., 1915.
III. Chern, I. L., “Long-Time effect of Relaxation for Hyperbolic conservation Laws”, Communications in mathematical Physics, 1995.
IV. Chern, I. L., “Multiple-Mode Diffusion waves for Viscous Nonstrictly Hyperbolic Conservation Laws”, Communications in mathematical Physics, 1991.
V. Chern, I. L. and Tai-Ping Liu, “Convergence of Diffusion Waves of Solutions for Viscous Conservation Laws”, Communications in mathematical Physics, 1987.
VI. Cole, J. D., “On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics”, Quart. Appl. Math., 1951.
VII. DeLillo, S., “The Burger’s Equation Initial-Boundary Value Problems on the Semiline”, Springer-Verlag, Berlin, Heidelberg, 1990.
VIII. Eschedo, M. and Zua Zua, E., “Long-Time Behaviour for a Convection-Diffusion Equation in Higher Dimension”, SIAM J. Math. Anal., 1997.
IX. Flether, C. A. J., “Burger’s Equation; a Model for all reasons in Numerical Solutions of Partial Differential Equations”, J. Noyle, ed., North-Holland, Amsterdam, New York, 1982.
X. Jawad K., Tahir. : ‘DEVELOPING MATHEMATICAL MODEL OF CROWD BEHAVIOR IN EXTREME SITUATIONS’. J. Mech. Cont.& Math. Sci., Special Issue, No.- 8, April (2020) pp 155-164. DOI : 10.26782/jmcms.spl.8/2020.04.00012
XI. Hopf, E., “The Partial Differential Equation ut + uux  uxx”, Comm. Pure and Applied Math., 1950.
XII. Lighthill, M. J., “Viscosity Effects in Sound Waves of Finite Amplitude”, C.U.P., Cambridge, 1956.
XIII. Magri, F., “Variaional Formulation for Every Linear Equation”, Int. J. Engng. Sci., Vol.12, pp.537-549, 1974.
XIV. Miller, J. C., Bernoff, A. J.; Rate of Convergence to Self-Similar Solutions of Burger’s Equation, Stud. Appl. Math. 111, 29-40, 2003.
XV. Moran, J. P. and S. F. Shen, “On the Formulation of Weak Plane Shock Waves by Impulsive Motion of a Piston”, Journal of Fluid Mechanics, 1966.
XVI. Nguyen, V. Q., “A Numerical study of Burger’s Equation with Robin Boundary Conditions”, M.Sc. Thesis, Virginia Polytechnic Institute and state University, 2001.
XVII. Radhi, A. Z., “Non-Classical Variational Approach to Boundary Problem in Heat Flow and Diffusion”, M.Sc. Thesis, al-Nahrain University, 1993.
XVIII. Wang, W. and Roberts, A. J.; Diffusion Approximation for Self-Similarity of Stochastic Advection in Burger’s Equation. Communication in Mathematical Physics, Vol.333, pp.1287-1316, 2014.
XIX. Whithman, G. B., “Linear and Nonlinear Waves”, John Wiley and Sons, 1974.
XX. W. A. Shaikh, A. G. Shaikh, M. Memon, A. H. Sheikh, A. A. Shaikh. : ‘NUMERICAL HYBRID ITERATIVE TECHNIQUE FOR SOLVING NONLINEAR EQUATIONS IN ONE VARIABLE’. J. Mech. Cont. & Math. Sci., Vol.-16, No.-7, July (2021) pp 57-66. DOI : 10.26782/jmcms.2021.07.00005

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