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A generalised cole-hopf transformation for nonlinear parabolic and hyperbolic equations

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Summary

The Cole-Hopf transformation has been generalized to generate a large class of nonlinear parabolic and hyperbolic equations which are exactly linearizable. These include model equations of exchange processes and turbulence. The methods to solve the corresponding linear equations have also been indicated.

Sommaire

La transformation de Cole et de Hopf a été généralisée en vue d'engendrer une classe d'équations nonlinéaires paraboliques et hyperboliques qui peuvent être rendues linéaires de façon exacte. Elles comprennent des équations modèles de procédés d'échange et de turbulence. Les méthodes pour résoudre les équations linéaires correspondantes ont également été indiquées.

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References

  1. E. Hopf, The Partial Differential Equation970-1, Comm. Pure Appl. Maths.3, 201–230 (1950).

    Google Scholar 

  2. J. D. Cole,On a Quasilinear Parabolic Equation Occurring in Aerodynamics, Quart. Appl. Maths.9, 225–236 (1951).

    Google Scholar 

  3. C. W. Chu,A Class of Reducible System of Quasi-Linear Partial Differential Equations, Quart. Appl. Maths.23, 275–278 (1965).

    Google Scholar 

  4. G. B. Whitham,Linear and Nonlinear Waves, John Wiley and Sons (1974), pp. 580–585.

  5. J. M. Burgers,Mathematical Examples, Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion, Verhandel. Kon. Nederl. Akad. Wetensch.17, 1 (1939).

    Google Scholar 

  6. K. M. Case andC. S. Chu,Burgers' Turbulence Models, Phys. Fluids 12, 1799 (1969).

    Google Scholar 

  7. S. Goldstein andJ. D. Murray,On the Mathematics of Exchange Processes, Proc. Roy. Soc. A,252, 334, 348, 360 (1959).

    Google Scholar 

  8. J. D. Murray,Singular Perturbations of a Class of Nonlinear Hyperbolic and Parabolic Equations, J. Maths. Phys.47, 111 (1968).

    Google Scholar 

  9. J. D. Murray,On the Gunn Effect and Other Physical Examples of Perturbed Conservation Equations, J. Fluid Mech.44, 315 (1970).

    Google Scholar 

  10. J. D. Murray,Perturbation Effects on the Decay of Discontinuous Solutions of Nonlinear First Order Wave Equations, SIAM J. Appl. Math.19, 273 (1970).

    Google Scholar 

  11. C. M. Defermos,Solutions of the Riemann Problem for a Class of Hyperbolic Systems of Conservation Laws by the Viscosity Method, Arch Rat Mech. and Anal.52, 1 (1973).

    Google Scholar 

  12. J. D. Murray,On Burgers' Model Equations for Turbulence, J. Fluid Mech.59, 263 (1973).

    Google Scholar 

  13. D. Colton,The Approximation of Initial Boundary Value Problems for Parabolic Equations in One Space Variable, Quart. Appl. Math.33, 377 (1976).

    Google Scholar 

  14. E. R. Benton andG. W. Platzman,A Table of Solutions of One-Dimensional Burgers' Equation, Quart. Appl. Maths.30, 195 (1972).

    Google Scholar 

  15. S. H. Lehnig,Conservation Similarity Solutions of the One-Dimensional Autonomous Parabolic Equation, Z. angew Math. Phys.27, 385 (1976).

    Google Scholar 

  16. G. W. Swan,Exact Fundamental Solutions of Linear Parabolic Equations with Spatially Varying Coefficients, Bull. Math. Biol.39, 435 (1977).

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Applied Mathematics, Indian Institute of Science, Bangalore, India

    P. L. Sachdev

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  1. P. L. Sachdev
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Sachdev, P.L. A generalised cole-hopf transformation for nonlinear parabolic and hyperbolic equations. Journal of Applied Mathematics and Physics (ZAMP) 29, 963–970 (1978). https://doi.org/10.1007/BF01590817

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