An identity for the Heaviside function and its application in representation of nonlinear Green’s function
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An identity showing that any positive real power of the Heaviside function is almost everywhere equal to the Heaviside function of power 1 is proved in this paper. A similar identity is shown to hold for corresponding distributions. The identity is then used to attach a specific meaning to the nonlinear Green’s function of quasi-linear ordinary differential equations. As specific examples, nonlinear Green’s function is constructed for the forced cubic Duffing (explicitly) and fractionally nonlinear damped Liénard (implicitly) equations. Efficiency of the nonlinear Green’s function solution for various source functions is demonstrated numerically in comparison with the solution obtained by the well-known method of lines. Solutions obtained are unique in literature.
KeywordsMultiplication of distributions Nonlinear distributions Ivanov’s hyperdistributions Nonlinear Green’s function Duffing equation Liénard equation
Mathematics Subject Classification46F10 46T30 34A05 34A34
The work was partly supported by State Administration of Foreign Expert Affairs of China, which I acknowledge gratefully.