A small time solutions for the Korteweg-de Vries equation

In this paper a heat balance integral (HBI) method is applied to the one-dimensional non-linear Korteweg-de Vries (KdV) equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times. It is shown that the HBI solutions obtained by the method may be used effectively at small times when the exact solution of the KdV equation is not known.

[1]  David Langford,et al.  The heat balance integral method , 1973 .

[2]  Graham E. Beli.  Solidification of a liquid about a cylindrical pipe , 1979 .

[3]  J. Ll. Morris,et al.  A Hopscotch method for the Korteweg-de-Vries equation , 1976 .

[4]  John W. Miles,et al.  The Korteweg-de Vries equation: a historical essay , 1981, Journal of Fluid Mechanics.

[5]  K. Goda,et al.  On Stability of Some Finite Difference Schemes for the Korteweg-de Vries Equation , 1975 .

[6]  J. M. Sanz-Serna,et al.  Petrov-Galerkin methods for nonlinear dispersive waves , 1981 .

[7]  An efficient finite‐difference scheme for multidimensional Stefan problems , 1986 .

[8]  Robert M. Miura,et al.  Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation , 1968 .

[9]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[10]  Graham E. Bell,et al.  A refinement of the heat balance integral method applied to a melting problem , 1978 .

[11]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[12]  S. W. Schoombie.  Spline Petrov—Galerkin Methods for the Numerical Solution of the Korteweg—de Vries Equation , 1982 .

[13]  M. E. Alexander,et al.  Galerkin methods applied to some model equations for non-linear dispersive waves , 1979 .

[14]  A. C. Vliegenthart,et al.  On finite-difference methods for the Korteweg-de Vries equation , 1971 .

[15]  P. Duck,et al.  Application of the heat-balance integral method to the freezing of a cuboid , 1977 .

[16]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[17]  N. Zabusky.  A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction , 1967 .