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A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod. (English) Zbl 1037.65085
Summary: A nonstandard finite difference scheme is constructed to solve an initial-boundary value problem involving a quartic nonlinearity that arises in heat transfer involving conduction with thermal radiation. It is noted that the positivity condition is equivalent to the usual linear stability criteria and it is shown that the representation of the nonlinear term in the finite difference scheme, in addition to the magnitudes of the equation parameters, has a direct bearing on the scheme’s stability. Finally, solution profiles are plotted and avenues of further inquiry are discussed.
MSC:
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K55Nonlinear parabolic equations