Nonlinear Parabolic Equations with Regularized Derivatives

The method of regularized derivatives derivatives transforms a partial differential equation of evolution type into a family of ordinary integro-differential equations, whose solutions exist forward and backward in time and can be obtained in a much easier way. Briefly stated, spatial partial derivatives ­ are replaced by convolution with ­ w , where w is a family x x e e of mollifiers converging to the Dirac measure. This has been used to treat Ž ill-posed problems theoretically and numerically Ashyralyev and w x w x w x. w x Sobolevskii 1 , Murio 14 , Van and Hao 19 . Rosinger 18 has based his ` theory of unconditionally stable and explicit numerical schemes on this w x method. Cockburn et al. 5 have used this technique to study entropy solutions of conservation laws. Further, it is of relevance to the nonlinear w x theory of generalized functions of Colombeau 6]8 . In this setting, solutions are sought in algebras of generalized function containing the space of Schwartz distributions. The method of regularized derivatives provides smooth approximate solutions, together with estimates on all derivatives,

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