Introduction
Differential equations are essential in mathematics for modeling natural and human-made phenomena. They describe the rate of change of quantities and are widely used in physics, engineering, biology, economics, and more — including predicting planetary motion and understanding population dynamics.
ODEs and PDEs
Differential equations are divided into two main types: Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs).
Conservation of Mass
Water problems are represented mathematically by differential equations. Examples of such problems or processes are propagation of waves in a river channel, pollutant transport, reservoir operations, and so on.
Conservation of Momentum
If gravity and pressure forces are detailed, conservation of momentum in the $x$ direction yields:
Conservation of Energy
For phenomena in which temperature varies, in addition to the momentum equation, conservation of energy must be considered. The energy equation accounts for both kinetic and potential energy, expressed by the term:
Classifying Flow Equations
Water problems are represented mathematically by differential equations. Examples of such problems or processes are propagation of waves in a river channel, pollutant transport, and reservoir operations.
Solving ODEs and PDEs
Solution of an ODE
Characteristics and Boundary Conditions
The majority of flow phenomena in hydraulics (hydrodynamics) are governed by a system of two differential equations (as is the case, for example, for free surface flows). A system of two first-order partial differential equations, having two independent variables ($\xi, \eta$) and two unknown functions ($u$ and $v$), is written as:
Worked Examples: Hyperbolic, Parabolic, Elliptic
Hyperbolic equation example