Approximation of Semilinear Fractional Cauchy Problem
Approximation of Semilinear Fractional Cauchy Problem
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DOI：
10.1515/cmam-2015-0001
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发表时间：
2015-04
影响因子：
1.3
通讯作者：
Ru Liu;Miao Li;S. Piskarev
中科院分区：
文献类型：
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作者：
Ru Liu;Miao Li;S. Piskarev
本文给出了求解柯西问题（𝐃t α u）（t）=Au（t）+J1-α ft，u（t），t∈[0，T]，0 <α<1，u（0）=u 0，$（\mathbf {D}_{t}^{\alpha }u）（t）= A u（t）+ J^{1-\alpha } f\big（t，u（t）\big），\quad t \in [0，T]，0 < \alpha <1，\qquad u（0）= u^0，$ with operator A，在Banach空间E中，给出了生成解析紧分解族{S α（t，A）} t≥0 ${\lbraceS_{\alpha }（t，A）\rbrace 0}}$的新的可解性.证明了预解式的紧收敛蕴涵了半离散逼近到精确解的收敛性。我们给出了一个一般的近似方案，其中包括有限差分和投影方法的分析。
Abstract The semidiscretization methods for solving the Cauchy problem (𝐃 t α u)(t)=Au(t)+J 1-α ft , u ( t ),t∈[0,T],0<α<1,u(0)=u 0 ,$(\mathbf {D}_{t}^{\alpha }u)(t) = A u(t) + J^{1-\alpha } f\big (t,u(t)\big ), \quad t \in [0,T], 0 < \alpha <1,\qquad u(0) = u^0,$ with operator A, which generates an analytic and compact resolution family {S α (t,A)} t≥0 ${\lbrace S_{\alpha }(t,A)\rbrace _{t\ge 0}}$ , in a Banach space E are presented. It is proved that the compact convergence of resolvents implies the convergence of semidiscrete approximations to an exact solution. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.