What is the difference between Lumped and Distributed systems?
The elements building a lumped system are thought of being concentrated at singular points in space. The classical example is an electrical circuit with passive elements like resistor, inductance and capacitor. The physical quantities current and voltage are functions of time (only). E. g. the current at a capacitor with capacity $C$ is given by
$$
i(t) = C\frac{\mathrm d v(t)}{\mathrm d t}
$$
Where $C$ is a constant (and so are $R$ and $L$). This leads to ordinary differential equations.

In contrast, the elements in distributed systems are thought of being distributed in space, so that physical quantities depend on both time and space. The classical example is the electrical line where inductance, capacity and resistance are not constant but functions of length $x$. This leads to partial derivatives of $i(t,x)$ and $v(t,x)$ in $t$ and $x$.