With regard to partial differential equations, can anyone tell me what the following mean:

What does Quasi-linear mean? What does Non-linear mean? What does Linear mean?

I know I can go to a book, but I want a simpler answer than most books provide.

Thanks all.

Hi,

a subtle point but:

1. Linear is when the dependant varibles and derivatives of dependant variables are all to the first power and each term in the equation may be multipied only by a constant.

2.Quasi-linear is when the derivatives are linear ie: d(v)/d(x) but the may be multiplied by other variables such as u*d(u)/d(x)

The Navier Sokes equations are quasi linear according to this definition but are often simply called non-linear. I am not sure if it makes too much of a difference to most of us, pedantry aside?

Regards,

Bak_Flow

Ok, nonlinear is easy: any equation that is not linear is nonlinear ;-)

Linear means, as BAK_FLOW said that the coefficients of your problem do not depend on the variables themselves.

For example, the PDEs

u_t + a*u_x = 0 and

u_t + a(t)*u_x

are linear, but the PDEs

u_t + u*u_x = 0 and

u_t + a(u)*u_x = 0

are both nonlinear (if a(u) does indeed depend on u). Actually, they are also both quasilinear. Quasilinear means: the PDE looks linear, but is actually not. So if you have a coefficient matrix appearing in your PDE, than the PDE is quasilinear. Whether it is linear or nonlinear depends on the coefficients. If they depend on u, the equation is nonlinear, otherwise it is linear.

Yours,

Philipp

Excellent...thanks guys! Can anyone shed some light on the Entropy condition and methods used to ensure that the Entropy condition is not violated in solutions to certain hyperbolic pde's? Also, what books/s would you guys recommend to someone wishing to get a deeper understanding of pde's and their analytical solutions...something for a Engineer (rather than very theoretical), but not sacrificing completeness.

Thanks.