difference between linear, semilinear and quasilinear PDE's

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?

• You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
– esmo
Jul 29, 2018 at 9:11
• Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications" Jul 29, 2018 at 9:32

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

• could you please give a similar explanation for the case of second order pde ? Sep 30, 2018 at 12:38
• @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify? Jan 16, 2019 at 13:19

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: $$a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$$.

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

• (Just because the question has a lot of views) I am fairly sure that Navier Stokes is considered as being semilinear (see e.g. the definition in Evan's book, second page of the first chapter). Nov 8, 2021 at 8:07

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.