A PDE, or partial differential equation, is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is commonly used to describe physical phenomena such as heat transfer, fluid dynamics, and quantum mechanics.
D/Dr represents the partial derivative with respect to the spatial variable r, while D/Dt represents the partial derivative with respect to time. In other words, D/Dr describes how a variable changes in space, while D/Dt describes how it changes over time.
The process of solving a PDE involves finding a function that satisfies the equation. This can be done analytically using techniques such as separation of variables or using numerical methods such as finite difference or finite element methods.
Boundary conditions are additional information that is provided to help determine a unique solution to a PDE. They specify the behavior of the solution at the boundaries of the domain and can be either Dirichlet (prescribing a value) or Neumann (prescribing a derivative).
Yes, PDEs are widely used in many fields of science and engineering to model and solve real-world problems. They have applications in areas such as physics, chemistry, biology, economics, and climate science.