分数阶索伯列夫空间中广义Abel方程在齐次边界条件下的正则性
Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions
摘要：
有界区间上的广义Abel方程（或称耦合Abel方程）在H $\ddot{\text{o}}$空间中得到了很好的研究，这类空间在端点处具有可积奇异性，而在其它函数空间中则相对不足.近年来，此类算子出现在分数阶微分方程的边值问题中，如分数阶扩散方程，通常在分数阶索伯列夫空间的框架下进行弱解和数值逼近的研究，其分析在将弱解转化为真解的过程中起着关键作用.本文研究了广义Abel算子$α{_aD_x^{-s}}+β{_xD_b^{-s}}$在分数索伯列夫空间中的映射性质，其中$0<α,β$，$α+β=1$，$ 0<s<1$和$ {_aD_x^{-s}}$，$ {_xD_b^{-s}}$是分数Riemann-Liouville积分.主要研究了考虑齐次边界条件的$(α{_aD_x^{-s}}+β{_xD_b^{-s}})u=f$的正则性。也就是说，我们调查的规律性行为的$u(x)$，同时让$f(x)$变得更平滑，并施加齐次边界限制$u(a)=u(b)=0$。
英文摘要：
The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\ddot{\text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. This article develops the mapping properties of generalized Abel operators $α{_aD_x^{-s}}+β{_xD_b^{-s}}$ in fractional Sobolev spaces, where $0<α,β$, $α+β=1$, $ 0<s<1$ and $ {_aD_x^{-s}}$, $ {_xD_b^{-s}}$ are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of $(α{_aD_x^{-s}}+β{_xD_b^{-s}})u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u(x)$ while letting $f(x)$ become smoother and imposing homogeneous boundary restrictions $u(a)=u(b)=0$.
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