Question

Which of these is a quasi-linear partial differential equation?

(a) \(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\)

(b) \(\frac{\partial^2 u}{\partial x^2}+a(x,y)\frac{\partial^2 u}{\partial y^2}=0\)

(c) \(\frac{\partial u}{\partial x}\frac{\partial ^2 u}{\partial x^2}+\frac{\partial u}{\partial y}\frac{\partial^2 u}{\partial y^2}=0\)

(d) \((\frac{\partial ^2 u}{\partial x^2})^2+\frac{\partial^2 u}{\partial y^2}=0\)

I have been asked this question by my school teacher while I was bunking the class.

Enquiry is from Partial Differential Equation in chapter Mathematical Behaviour of Partial Differential Equations of Computational Fluid Dynamics

Answer 1

Correct answer is (c) \(\frac{\partial u}{\partial x}\frac{\partial ^2 u}{\partial x^2}+\frac{\partial u}{\partial y}\frac{\partial^2 u}{\partial y^2}=0\)

Explanation: \(\frac{\partial u}{\partial x}\frac{\partial ^2 u}{\partial x^2}+\frac{\partial u}{\partial y}\frac{\partial^2 u}{\partial y^2}=0\) is a quasi-linear partial differential equation. \(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\) and \(\frac{\partial^2 u}{\partial x^2}+a(x,y)\frac{\partial^2 u}{\partial y^2}=0\) are linear partial differential equations. \((\frac{\partial ^2 u}{\partial x^2})^2+\frac{\partial^2 u}{\partial y^2}=0\) is non-linear.