order of a system of PDEs
The order of a system of partial differential equations (PDEs) is defined as the highest order of derivative present in the equations.
For example, consider the following system of PDEs:
∂u/∂t = ∂^2u/∂x^2 + ∂^2u/∂y^2
∂v/∂t = ∂v/∂x + ∂v/∂y
The highest order of derivative in these equations is the second derivative (∂^2u/∂x^2 and ∂^2u/∂y^2), so the order of this system is 2.
It is important to note that the order of a system of PDEs refers to the highest order of derivative with respect to any independent variable in the equations.
The order of a system of partial differential equations (PDEs) refers to the highest order of the derivatives present in the equations. To determine the order of a PDE system, follow these steps:
Step 1: Identify each equation in the system.
Step 2: Determine the highest derivative in each equation. This includes both spatial derivatives (e.g., ∂²u/∂x²) and temporal derivatives (e.g., ∂u/∂t).
Step 3: Find the maximum order among all the derivatives in the system.
Step 4: The maximum order found in step 3 is the order of the PDE system.
For example, consider the following system of PDEs:
∂²u/∂x² + ∂u/∂t = 0
∂v/∂x + ∂²v/∂t² = 2
In this case, the first equation contains a second-order spatial derivative (∂²u/∂x²) and a first-order temporal derivative (∂u/∂t), while the second equation has a first-order spatial derivative (∂v/∂x) and a second-order temporal derivative (∂²v/∂t²). The highest derivative order in this system is 2 (from the second derivative with respect to time in the second equation), so the order of the PDE system is 2.