Nonlinear Parabolic Equations with Regularized Derivatives
Nonlinear Parabolic Equations with Regularized Derivatives
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具有正则导数的非线性抛物型方程
DOI：
10.1006/jmaa.1999.6326
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发表时间：
1999
影响因子：
1.3
通讯作者：
M. Oberguggenberger
中科院分区：
文献类型：
--
作者：
Ya;M. Oberguggenberger
正则导数导数方法将演化型偏微分方程转化为一族普通积分微分方程组，其解在时间上向前和向后存在，并且可以更容易地获得。简而言之，空间偏导数被 w 的卷积所取代，其中 w 是收敛到狄拉克测度的缓和器族 x x e e 。这已被用于在理论上和数值上处理 Ž 不适定问题 Ashyralyev 和 w x w x w x。 w x Sobolevskii 1、Murio 14、Van 和Hao 19。 Rosinger 18 将他的无条件稳定和显式数值格式理论建立在这种 w x 方法的基础上。科伯恩等人。 5人使用这种技术来研究守恒定律的熵解。此外，它与 Colombeau 的非线性广义函数 w x 理论有关 6]8 。在这种情况下，在包含施瓦茨分布空间的广义函数代数中寻求解。正则化导数方法提供平滑的近似解，以及对所有导数的估计，
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,