difference between linear, semilinear and quasilinear PDE's

Question

I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear.

However, I do not quite understand the other two.

My professor described

• "semilinear" PDE's as PDE's whose highest order terms are linear, and

• "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms.

No examples were provided; only equivalent statements involving sums and multiindices were shown, which I do not think I could decipher by tomorrow.

Can someone provide some examples of "semilinear" and "quasilinear" PDE's?

Answer 1

I think this will help you to understand the PDE $:$

Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$

Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$

Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$

Answer 2

Using Einstein convention for summation and omitting dependence on $x$ of $u$ and its derivatives,

1. linear: $a^{ij}(x)D_{ij}u+b^i(x)D_iu+c(x)u+d(x)=0$
2. semi-linear: $a^{ij}(x)D_{ij}u+b(x,u,Du)=0$
3. quasi-linear: $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$.

Answer 3

I hope these examples will help you.

Semilinear/Almost Linear PDE:

1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$

2) $U_{tt}-U_{xx}+U^3=0$

Quasi Linear PDE:

1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$

2) $U_x+UV_y=0$

3) $U_{tt}-UU_{xx}+U^3=0$

4) $U_{tt}-UU_{xx}+U=0$

5) Navier Stokes equation is also Quasi Linear Equation

Answer 4

Linear. The degree for the unknown function is one through out. And no functions of the Unknown function. Semilinear. The derivatives are linear but the unknown function is not likear. Quasilinear. Derivatives of the order are not linear. Once the whole eqn is not linear then it becomes non linear.