非线性发展方程在特殊C ~*-算子代数中的逼近
On the Approximation of Nonlinear Evolution Equations in Particular C*-Algebras of Operators
摘要：
本文讨论了形式为$A(u(t))+f(u(t))=u'(t)$的非线性发展方程数值解的稳定性和收敛性，这个问题的解的数值分析将使用某些特定算子代数的方法来进行，这些算子代数有时由单位C* 的单位子代数表示。由一些基本算子生成的算子代数，比如$\mathbf{1},a,\mathcal{D}(\cdot)\in\mathcal{L}(H^m(G))$，在某种适当的意义上与发展方程中的算子$A(\cdot)\in\mathcal{L}(H^m(G))$相关，还研究了算子代数不验证关于所选范数的C*-恒等式的特殊情况，当应用基本C*-代数技术对某些类型的问题执行数值解的某些估计时，在所有这些工作中，像$H^m(G)$这样表达式将用$G\subset\subset\mathbb{R}^n$表示指定的可离散希尔伯特空间。
英文摘要：
In this article we deal with the stability and convergence of numerical solutions of nonlinear evolution equations of the form $A(u(t))+f(u(t))=u'(t)$, the numerical analysis of solutions to this problems will be performed using some methods from particular algebras of operators which are sometimes represented by unital subalgebras of the unital C*-algebras of operators that are generated by some basic operators say $\mathbf{1},a,\mathcal{D}(\cdot)\in\mathcal{L}(H^m(G))$ that in some suitable sense are related to the operator $A(\cdot)\in\mathcal{L}(H^m(G))$ in the evolution equations, particular cases where the operator algebras do not verify the C*-identity with respect to the norm chosen are also studied, when applicable basic C*-algebra techniques are implemented to perform some estimates of numerical solutions to some types of problems, in all this work expressions like $H^m(G)$ will represent a prescribed discretizable Hilbert space with $G\subset\subset\mathbb{R}^n$.
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