1

I just wanted to make sure my reasoning behind the classification of the following PDE's is correct (note $u=u(x,t))$:

a) $$u_t -u_{xx}=0$$
-This is second order as the highest order term has order 2
-Linear as it consists of a linear combination of $u$ and its partial derivatives

b)$$(u_{xt})^2 + u_xu_{ttt}=xt^2$$ -3rd order as the highest order term has order 3
- Non -linear because of the term $u_xu_{ttt}$ and $(u_{xt})^2$
- Not quasi-liner as the highest order partial derivative $u_xu_{ttt}$ is multiplied by a derivative of u. Hence it is not semi linear.

c)$$uu_{x}+x^2tu_{tt}=1$$ -This is second order as the highest order term has order 2
- Non -linear because of the term $uu_{x}$ - Quasi-liner as the highest order partial derivative $u_{tt}$ is linear but is multiplied by a derivative of u.
- Semi linear as the highest order derivative is not multiplied by u or any of its derivatives

d)$$u_{xx} + uu_{xxtt}=0$$ -4th order as the highest order term has order 4
- Non -linear because of the term $uu_{xxtt}$
- Quasi-liner as the highest order partial derivative $uu_{xxtt}$ is linear but is multiplied by u.
- not Semi linear as the highest order partial derivative is multiplied by u

odesinit
  • 69
  • a), c) and d) are correct. The equation b) is quasilinear, but not semilinear. – Wang Nov 30 '17 at 17:50
  • how can b) quasi-linear if its highest order term is multiplied by a derivative of u? – odesinit Nov 30 '17 at 17:54
  • A partial differential equation is said to be quasilinear if it is linear with respect to all the highest order derivatives of the unknown function. In b) the term, $u_xu_{ttt}$, is linear in the highest order derivative $u_{ttt}$. – Wang Nov 30 '17 at 18:03
  • im pretty new to this, i dont fully grasp the concept? in my course we've been told that the coefficients of the highest order derivates cannot be derivatives of u – odesinit Nov 30 '17 at 18:07
  • 2
    See this. – Wang Nov 30 '17 at 18:12
  • so am I correct in assuming that the PDE: $u_tu_{ttt}+u_{tt}=0$ is an example of a quasi-linear PDE as w.r.t. the highest order term, $u_{ttt}$ it is linear? – odesinit Nov 30 '17 at 18:23
  • 1
    Yes, it is linear in the highest order derivates $u_{ttt}$; so it is a quasilinear equation. – Wang Nov 30 '17 at 18:29

0 Answers0