论文标题
提高具有均匀边界条件的分数Sobolev空间中广义亚伯方程的规律性
Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions
论文作者
论文摘要
在有界间隔上的广义（或耦合）ABEL方程已在H $ \ ddot {\ text {o}} $ lderian空间中进行了很好的研究，这些空间在端点处允许可集成的奇异点在其他功能空间中相对不足。近年来，此类运算符出现在分数微分方程的BVP中，例如通常在分数Sobolev空间框架中研究的分数扩散方程，用于弱溶液和数值近似；他们的分析在将弱解决方案转换为真实解决方案的过程中起着关键作用。 本文开发了概括的亚伯运算符的映射属性$α{_AD_X^{ - s}}}+β{_xd_b^{ - s}} $在分数sobolev space中，其中$ 0 <α，β$，β$，$α+β= 1 $，$ 0 $，$，$，$，$和$和$ $ and $ {_xd_b^{ - s}} $是分数riemann-liouville积分。它主要与$（α{_ad_x^{ - s}}}+β{_xd_b^{ - s}}）U = f $的规则性属性相关。也就是说，我们研究了$ u（x）$的规律性行为，而让$ f（x）$变得更顺畅，并施加均质边界限制$ u（a）= u（b）= 0 $。
The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\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 $u(a)=u(b)=0$.
