A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain \(\Omega \times [0,T]\), where \(\Omega \) is an open bounded domain in \(\mathbb {R}^d\) for some \(d\ge 1\), and the order \(\alpha (x)\) of the Riemann-Liouville fractional derivative may vary in space with \(1/2< \alpha (x) < 1\) for all x. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution u of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution \(u_h(t)\), is constructed. Error estimates for \(\Vert (u - u_h)(t)\Vert _{L^2(\Omega )}\) and \(\int _0^t \left| \partial _t^{1-\alpha } (u-u_h)(s)\right| _1^2 \,ds\) are proved for each \(t\in [0,T]\) under the assumptions that the following quantities are finite: \(\Vert u(\cdot , 0)\Vert _{H^2(\Omega )}, |u(\cdot , t)|_{H^1(\Omega )}\) for each t, and \(\int _0^t [\Vert u(\cdot , t)\Vert _{H^2(\Omega )}^2 + |\partial _t^{1-\alpha }u|_{H^2(\Omega )}^2]\), where u(x, t) is the unknown solution. Furthermore, these error estimates are \(\alpha \)-robust: they do not fail when \(\alpha \rightarrow 1\), the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications).

在时空域\（\ Omega \ times [0，T] \）上考虑Fokker–Planck类型的时间分数初始边界值问题，其中\（\ Omega \）是 \ （\ mathbb {R} ^ d \）对于某些\（d \ ge 1 \）的存在，并且黎曼-利维尔分数阶导数的阶数（\ alpha \（x）\）在空间中可能随\（1/2 <x \ alpha（x）<1 \）对于所有 x。在某些连续时间随机游动模型的公式化中自然会出现这些问题。在合理的假设下证明了问题的任何解决方案u的唯一性。一种半离散数值方法，使用空间中的有限元产生解\（u_h（t）\）被构造。\（\ Vert（u-u_h）（t）\ Vert _ {L ^ 2（\ Omega}} \）和\（\ int _0 ^ t \ left | \ partial _t ^ {1- \ alpha}的错误估计（u-u_h）（s）\ right | _1 ^ 2 \，ds \）在以下量是有限的假设下针对每个\（t \ in [0，T] \）进行证明：\（\ Vert u （\ cdot，0）\ Vert _ {H ^ 2（\ Omega}}，| u（\ cdot，t）| _ {H ^ 1（\ Omega}} \）每个 t和\（\ int _0 ^ t [\ Vert u（\ cdot，t）\ Vert _ {H ^ 2（\ Omega）} ^ 2 + | \ partial _t ^ {1- \ alpha} u | _ {H ^ 2（\ Omega}} ^ 2] \），其中u（x，  t）是未知解。此外，这些误差估计为\（\ alpha \）-稳健：当经典的Fokker-Planck问题\（\ alpha \ rightarrow 1 \）时，它们不会失败。对于不存在问题的漂移项的特殊情况，可以获得更清晰的结果（在某些应用中很有意义）。