Second Order PDEs
We will cover following topics
Linear PDEs Of The Second Order With Constant Coefficients
A second order lnear PDE with constant coefficients is given by:
\[a u_{x x}+b u_{x y}+c u_{y y}+d u_{x}+e u_{y}+f u=g(x, y)\]where at least one of \(a\), \(b\) and \(c\) is non-zero.
- If \(b^{2}-4 a c>0\), then the equation is called hyperbolic.
The wave equation \(a^{2} u_{x x}=u_{t t}\) is an example.
- If \(b^{2}-4 a c=0\), then the equation is called parabolic.
The heat equation \(\alpha^{2} u_{x x}=u_{t}\) is an example.
- \(b^{2}-4 a c < 0\), then the equation is called elliptic.
The Laplace equation \(u_{x x}+u_{y y}=0\) is an example.
Canonical Forms
A second-order linear PDE can be reduced to its canonical form by an appropriate change of variables \(\xi=\xi(x, y)\) and \(\eta=\eta(x, y)\).
If the Jacobian \(J=\begin{vmatrix}{\xi_{x}} & {\xi_{y}} \\ {\eta_{x}} & {\eta_{y}}\end{vmatrix}\) is non-zero, we can solve for \(x\) and \(y\) as functions of \(\xi\) and \(\eta\).
Using these transformations, the second order PDE \(\qquad a(x, y) u_{x x}+2 b(x, y) u_{x y}+c(x, y) u_{y y}+d(x, y) u_{x}+e(x, y) u_{y}+f(x, y) u=g(x, y)\), can be reduced to the below canonical form:
\[A w_{\xi \xi}+2 B w_{\xi \eta}+C w_{\eta \eta}+D w_{\xi}+E w_{\eta}+F w=G(\xi, \eta)\]where
\[A =a \xi_{x}^{2}+2 b \xi_{x} \xi_{y}+c \xi_{y}^{2}\] \[B =a \xi_{x} \eta_{x}+b\left(\xi_{x} \eta_{y}+\xi_{y} \eta_{x}\right)+c \xi_{y} \eta_{y}\] \[C =a \eta_{x}^{2}+2 b \eta_{x} \eta_{y}+c \eta_{y}^{2}\] \[D =a \xi_{x x}+2 b \xi_{x y}+c \xi_{y y}+d \xi_{x}+e \xi_{y}\] \[E =a \eta_{x x}+2 b \eta_{x y}+c \eta_{y y}+d \eta_{x}+e \eta_{y}\] \[F =f(x(\xi, \eta), y(\xi, \eta))\] \[G =g(x(\xi, \eta), y(\xi, \eta))\]PYQs
Linear PDEs Of The Second Order With Constant Coefficients
1) Solve the partial differential equation:
\[(2D^2-5DD'+2D'^2)z=5sin(2x+y)+24(y-x)+e^{3x+4y}\]where \(D\equiv \dfrac{\partial}{\partial x}\), \(D'\equiv \dfrac{\partial}{\partial y}\).
[2018, 15M]
2) Solve \(\left(D^{2}-2 D D^{\prime}-D^{2}\right) z\)=\(e^{x+2 y}+x^{3}+\sin 2 x\) where \(D \equiv \dfrac{\partial}{\partial x}\), \(D^{\prime} \equiv \dfrac{\partial}{\partial y}\), \(D^{2} \equiv \dfrac{\partial^{2}}{\partial x^{2}}\), \(D^{\prime 2} \equiv \dfrac{\partial^{2}}{\partial y^{2}}\).
[2017, 10M]
3) Let \(\Gamma\) be a closed curve in \(xy-plane\) and let \(S\) denote the region bounded by the curve \(\Gamma\). Let \(\dfrac{\partial^{2} w}{\partial x^{2}}+\dfrac{\partial^{2} w}{\partial y^{2}}\)=\(f(x, y) \forall(x, y) \in S\). If \(f\) is prescribed at each point \((x, y)\) of \(S\) and \(w\) is prescribed on the boundary \(\Gamma\) of \(S\) then prove that any solution \(w=w(x, y)\), satisfying these conditions, is unique.
[2017, 10M]
4) Solve the partial differential equation \(\dfrac{\partial^{3} z}{\partial x^{3}}-2 \dfrac{\partial^{3} z}{\partial x^{2} \partial y} \dfrac{\partial^{3} z}{\partial x \partial y^{2}}+2 \dfrac{\partial^{3} z}{\partial y^{3}}=e^{x+y}\).
[2016, 15M]
5) Solve: \(\left(D^{2}+D D^{\prime}-2 D^{\prime}\right) u=e^{x+y}\), where \(D=\dfrac{\partial}{\partial x}\) and \(D^{\prime}=\dfrac{\partial}{\partial y}\).
[2015, 15M]
6) Solve the partial differential equation \(\left(2 D^{2}-5 D D^{\prime}+2 D^{\prime 2}\right) z=24(y-x)\).
[2014, 10M]
7) Solve \(\left(D^{2}+D D^{\prime}-6 D^{\prime 2}\right) z=x^{2} \sin (x+y)\) where \(D\) and \(D^{\prime}\) denote \(\dfrac{\partial}{\partial x}\) and \(\dfrac{\partial}{\partial y}\).
[2013, 15M]
8) Solve the PDE \(\left(D^{2}-D^{\prime 2}+D+3 D^{\prime}-2\right) z\)=\(e^{(x-y)}-x^{2} y\).
[2011, 12M]
9) Solve the PDE \(\left(D^{2}-D^{\prime}\right)\left(D-2 D^{\prime}\right) Z=e^{2 x+y}+x y\).
[2010, 12M]
10) Solve: \(\left(D^{2}-D D^{\prime}-2 D^{\prime 2}\right) z\)=\(\left(2 x^{2}+x y-y^{2}\right) \sin x y-\cos x y\) where \(D\) and \(D^{\prime}\) represent \(\dfrac{\partial}{\partial x}\) and \(\dfrac{\partial}{\partial y}\) and \(\dfrac{\partial}{\partial y}\).
[2009, 15M]
11) Find the general solution of the partial differential equation: \(\left(D^{2}+D D^{\prime}-6 D^{2}\right) z=y \cos x\), where \(D \equiv \dfrac{\partial}{\partial x}\), \(D^{\prime} \equiv \dfrac{\partial}{\partial y}\).
[2008, 12M]
12) Solve: \(\dfrac{\partial^{3} z}{\partial x^{3}}-4 \dfrac{\partial^{3} z}{\partial x^{2} \partial y}+4 \dfrac{\partial^{3} z}{\partial x \partial y^{2}}\)=\(2 \sin (3 x+2 y)\).
[2006, 12M]
13) Solve the partial differential equation: \(\dfrac{\partial^{2} z}{\partial x^{2}}-\dfrac{\partial^{2} z}{\partial x \partial y}-2 \dfrac{\partial^{2} z}{\partial y^{2}}=(y-1) e^{x}\).
[2004, 15M]
14) Find the general solution of \(\dfrac{\partial^{2} z}{\partial x^{2}}+3 \dfrac{\partial^{2} z}{\partial x \partial y}+2 \dfrac{\partial^{2} z}{\partial y^{2}}=x+y+\cos (2 x+3 y)\).
[2003, 12M]
Canonical Forms
1) Reduce the following second order partial differential equation to canonical from and find the general solution:
\(\dfrac{\partial^2u}{\partial x^2}-2x\dfrac{\partial^2u}{\partial x\partial y}+x^2\dfrac{\partial^2u}{\partial y^2}\)=\(\dfrac{\partial u}{\partial y}+12x\)
[2019, 20M]
2) Reduce the equation \(y^{2} \dfrac{\partial^{2} z}{\partial x^{2}}-2 x y \dfrac{\partial^{2} z}{\partial x \partial y}+x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}\)=\(\dfrac{y^{2}}{\partial x} \dfrac{\partial z}{\partial x}+\dfrac{x^{2}}{\partial y} \dfrac{\partial z}{\partial y}\) to canonical form and hence solve it.
[2017, 15M]
3) Find the solution of the initial-boundary value problem:
\(u_{t}-u_{x x}+u=0\), \(\quad 0<x<l\), \(t>0\)
\(u(0, t)=u(l, t)=0\), \(\quad t \geq 0\)
\(u(x, 0)=x(l-x)\), \(\quad 0<x<l\)
[2015, 15M]
4) Reduce the second-order partial differential equation \(x^{2} \dfrac{\partial^{2} u}{\partial x^{2}}-2 x y \dfrac{\partial^{2} u}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} u}{\partial y^{2}}+x \dfrac{\partial u}{\partial x}+y \dfrac{\partial u}{\partial y}=0\) into canonical form. Hence, find its general solution.
[2015, 15M]
5) Reduce the equation \(\dfrac{\partial^{2} z}{\partial x^{2}}=x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}\) to canonical form.
[2014, 15M]
6) Reduce the equation \(y \dfrac{\partial^{2} z}{\partial x^{2}}+(x+y) \dfrac{\partial^{2} z}{\partial x \partial y}+x \dfrac{\partial^{2} z}{\partial y^{2}}=0\) to its canonical from when \(x \neq y\).
[2013, 10M]
7) Reduce the following \(2^{nd}\) order partial differential equation into canonical form and find find its general solution.
\(x u_{x x}+2 x^{2} u_{x y}-u_{x}=0\).
[2010, 20M]
8) Reduce \(\dfrac{\partial^{2} z}{\partial x^{2}}=x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}\) canonical form.
[2008, 15M]
9) Solve the equation \(x^{2} \dfrac{\partial^{2} z}{\partial x^{2}}-y^{2} \dfrac{\partial^{2} z}{\partial y^{2}}+x \dfrac{\partial z}{\partial x}-y \dfrac{\partial z}{\partial y}=x^{2} y^{4}\) by reducing it to the equation with constant coefficients.
[2001, 20M]