To create a good numerical scheme to solve PDE, we need to understand the nature of the PDE. We can assign PDE’s into one of the 3 major categories: elliptic, parabolic and hyperbolic. Need to identify class of PDE either physically and/or mathematically.

The PDE is: 1. elliptic if the matrix is (positive or negative) definite, i.e. its eigenvalues are non-zero and all have the same sign. 2. hyperbolic if the matrix has non-zero eigenvalues and all but one have the same sign. 3. parabolic if the matrix is semi-definite (i.e. exactly one eigenvalue is zero, while the others have the same sign) and

Classification is based on the eigenvalues of : parabolic if any eigenvalues are zero; otherwise: elliptic if all eigenvalues are the same sign; hyperbolic if all eigenvalues except one are of the same sign; ultrahyperbolic, otherwise. The reason for …

As we shall see, there are fundamentally three types of PDEs – hyperbolic, parabolic, and elliptic PDEs. From the physical point of view, these PDEs respectively represents the wave propagation, the time-dependent diffusion processes, and the steady state or equilibrium pro-cesses.

We call a PDE for u(x;t) linear if it can be written in the form L[u] = f(x;t) where f is some function and Lis a linear operator involving the partial derivatives of u.

In more than two dimensions we use a similar definition, based on the fact that all eigenvalues of the coefficient matrix have the same sign (for an elliptic equation), have different signs (hyperbolic) or one of them is zero (parabolic)

The simplest form of an eigenvalue prob-lem is L[u] = λu for (x,y) ∈ D; B[u] = 0 for (x,y) ∈ ∂D. In a more complex setup, the eigenvalue may enter into the PDE, and even into the boundary condition, in a more complicated way. Examples of eigenvalue problems include: Natural frequencies of vibrating strings and

The Classification of PDEs •We discussed about the classification of PDEs for a quasi-linear second order non-homogeneous PDE as elliptic, parabolic and hyperbolic. •Such Classification helps in knowing the allowable initial and boundary conditions to a given problem. •It also helps in the effective choice of numerical methods. 2 2 2 22 f ...

The PDE is classified according to the signs of the eigenvalues λi(xk) λ i (x k) of the matrix of functions Aij(xk). A i j (x k). Elliptic: λi(xk) λ i (x k) are nowhere vanishing. All have the same sign. Ex: Poisson, Laplace, Helmholtz.

The PDE is classified according to the signs of the eigenvalues λ i (x k) of the matrix of functions . A i j (x k). Elliptic: λ i (x k) are nowhere vanishing. All have the same sign. Ex: Poisson, Laplace, Helmholtz. Parabolic: One eigenvalue vanishes everywhere (usually time dependence), the others are nowhere vanishing and have the same sign.