In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field.

翻译：本章介绍了分数阶耗散偏微分方程，并着重探讨其动力学行为。我们关注的主要方程类型为反应扩散方程，同时也展望了与之密切相关的其他方程类别。为简化论述，我们分别讨论分数阶时间导数和分数阶空间导数的情形。作为核心工具，我们描述了分析和数值方法，这些方法在研究非线性动力学中通常是必要的。我们首先从分数阶时间导数的稳态和局部线性稳定性分析开始，继而将视角扩展至全局层面，考虑时间分数阶偏微分方程与梯度流。接着，我们研究空间分数阶偏微分方程的稳态、线性稳定性分析及分岔。在最后的分析部分，我们讨论了空间分数阶偏微分方程行波解的存在性与稳定性。最后，我们提供了分数阶（耗散）偏微分方程的数值离散格式，并将这些技术应用于分数阶反应扩散偏微分方程的数值延拓实例。最后，我们简要总结并展望了该领域的未解问题。