A PDE (partial differential equation) greater than order 2 is a type of mathematical equation that involves multiple partial derivatives of a function. This means that the equation takes into account the rate of change of a function with respect to multiple variables. In comparison, lower order PDEs only involve first or second order derivatives. The higher order PDEs are more complex and often have more real world applications.
PDEs greater than order 2 have a wide range of real world applications, including in physics, engineering, and finance. Some specific examples include modeling heat transfer in materials, analyzing fluid dynamics in aerodynamics, and predicting stock market trends.
Solving PDEs greater than order 2 can be a challenging task, as there is no general method that can be applied to all types of equations. However, there are various techniques and methods that can be used depending on the specific equation and its boundary conditions. These include separation of variables, Fourier series, and numerical methods such as finite difference or finite element methods.
While PDEs greater than order 2 can be very useful in modeling and predicting real world phenomena, they also have some limitations. One of the main limitations is the difficulty in obtaining exact solutions for many complex equations. This often requires the use of numerical methods, which can introduce some error. Additionally, PDEs may not accurately capture all aspects of a real world system, leading to some discrepancies between the predicted and actual behavior.
PDEs greater than order 2 are closely related to other areas of mathematics such as calculus, linear algebra, and functional analysis. They also have connections to other types of equations, such as ordinary differential equations and integral equations. Understanding these relationships can provide insight into the behavior of PDEs and aid in solving them.