4) Classification of PDEs Flashcards
What is the order of a PDE
The highest order of derivative that occurs in the equation
What is a scalar PDE and a system of PDES
Scalar PDE: If we have a single PDE for a single unknown dependent variable u
System of PDES: . If we have m > 1 coupled equations involving m unknown dependent variables u, v, ….w
What are homogenous and inhomogenous PDES
Homogeneous: A PDE without free terms; it includes only the dependent variable and its derivatives, set equal to zero
Imhomogeneous: A PDE that includes terms independent of the dependent variable
What is a Linear PDE
A PDEs where the coefficients multiplying the derivatives of the unknown functions only depend on the independent variables
What is a Quasi-Linear PDE
A PDE that is linear in its highest derivative
What is a Semi-Linear PDE
The coefficients multiplying the highest derivatives are
functions of the independent variables only
What are the 3 types of PDEs
Consider a PDE of the form -
A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy + D(x, y, u, ux, uy) = Q(x, y)
* If B^2 − AC = 0 the PDE is parabolic.
* If B^2 − AC < 0 the PDE is elliptic.
* If B^2 − AC > 0 the PDE is hyperbolic
What is a mixed type PDE
If B^2 − AC changes sign over the (x,y) space
How are second-order semi-linear PDEs transformed into canonical forms, and what are these forms
How do you change coordinates from ξ and η back to x and y
How are hyperbolic PDEs solved using characteristics and coordinate transformations
How are elliptical PDEs solved using characteristics and coordinate transformations
How are parabolic PDEs solved using characteristics and coordinate transformations
What are Dirichlet and Neumann Boundary Conditions in the context of PDEs
Dirichlet Boundary Conditions: Specify the function’s values at the boundary
Neumann Boundary Conditions: Specify the derivative of the function at the boundary
What are Robin Boundary Conditions in the context of PDEs
What defines Laplace’s equation as an elliptic PDE, and how do boundary conditions affect its solutions
