High order PDEs (partial differential equations) are mathematical equations that involve multiple variables and their partial derivatives. They are used to describe physical phenomena in fields such as physics, engineering, and mathematics.
A PDE is considered trivial if it has a solution that can be easily determined by inspection, without the need for complex mathematical techniques. This usually occurs when the PDE is linear and has simple boundary conditions.
The main difference between high order PDEs and low order PDEs is the number of independent variables and their corresponding derivatives involved. High order PDEs have more independent variables and derivatives, making them more complex to solve.
Two trivial high order PDEs include the wave equation and the heat equation. Both of these equations involve two independent variables and their corresponding derivatives, and have simple boundary conditions that can be easily solved.
High order PDEs are important in scientific research because they can accurately model and describe complex physical phenomena. They are also used in a wide range of fields, including fluid mechanics, electromagnetism, and quantum mechanics, making them essential for advancing scientific knowledge and technology.