The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter $p\in [0,1]$ is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

具有参数特征的一般双向分数阶导数的扩散反应方程分析 $p\in [0，\mathrm{1个}]$本文进行了研究。首先，我们提出一种Petrov-Galerkin方法，导出适当的弱公式，并证明其弱解的适定性。此外，基于双面雅可比多项式，推导了彼得罗夫-加勒金方法的先验误差分析。此外，严格建立了后验误差分析。更准确地说，我们开发了一种新颖的后处理技术，以通过添加少量计算来增强Petrov-Galerkin方法，并分析渐近精确的后验误差估计量。最后，我们用数值例子证明了理论结果。