In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

中文翻译：

---

时空分数阶扩散方程初值的拟边值方法

在本文中，我们考虑解决一个不均匀的时空分数扩散方程的逆初值问题。该问题不适当地提出，提出了拟边界值方法来处理该反问题，并获得反初值问题的正则化解的级数表达式。我们通过使用先验正则化参数和后验正则化参数选择规则来证明正则化解与精确解之间的误差估计。一维和二维情况下的一些数值结果表明，该方法是有效且稳定的。