7.9: Problems

Exercise $\PageIndex{1}$

Find the solution of each initial value problem using the appropriate initial value Green’s function.

1. $y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6$.
2. $y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0$.
3. $y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0$.
4. $x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0$.

Exercise $\PageIndex{2}$

Use the initial value Green’s function for $x^{\prime \prime}+x=f(t), x(0)=4, x^{\prime}(0)=$ 0 , to solve the following problems.

1. $x^{\prime \prime}+x=5 t^{2}$.
2. $x^{\prime \prime}+x=2 \tan t$.

Exercise $\PageIndex{3}$

For the problem $y^{\prime \prime}-k^{2} y=f(x), y(0)=0, y^{\prime}(0)=1$,

1. Find the initial value Green’s function.
2. Use the Green’s function to solve $y^{\prime \prime}-y=e^{-x}$.
3. Use the Green’s function to solve $y^{\prime \prime}-4 y=e^{2 x}$.

Exercise $\PageIndex{4}$

Find and use the initial value Green’s function to solve

$$x^{2} y^{\prime \prime}+3 x y^{\prime}-15 y=x^{4} e^{x}, \quad y(1)=1, y^{\prime}(1)=0 .\nonumber$$

Exercise $\PageIndex{5}$

Consider the problem $y^{\prime \prime}=\sin x, y^{\prime}(0)=0, y(\pi)=0$.

1. Solve by direct integration.
2. Determine the Green’s function.
3. Solve the boundary value problem using the Green’s function.
4. Change the boundary conditions to $y^{\prime}(0)=5, y(\pi)=-3$.
   1. Solve by direct integration.
   2. Solve using the Green’s function.

Exercise $\PageIndex{6}$

Let $C$ be a closed curve and $D$ the enclosed region. Prove the identity

$$\int_{C} \phi \nabla \phi \cdot \mathbf{n} d s=\int_{D}\left(\phi \nabla^{2} \phi+\nabla \phi \cdot \nabla \phi\right) d A .\nonumber$$

Exercise $\PageIndex{7}$

Let $S$ be a closed surface and $V$ the enclosed volume. Prove Green’s first and second identities, respectively.

1. $\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V$.
2. $\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V$.

Exercise $\PageIndex{8}$

Let $C$ be a closed curve and $D$ the enclosed region. Prove Green’s identities in two dimensions.

1. First prove

   $$\int_{D}(v \nabla \cdot \mathbf{F}+\mathbf{F} \cdot \nabla v) d A=\int_{C}(v \mathbf{F}) \cdot d \mathbf{s}\nonumber$$

2. Let $\mathbf{F}=\nabla u$ and obtain Green’s first identity,

   $$\int_{D}\left(v \nabla^{2} u+\nabla u \cdot \nabla v\right) d A=\int_{C}(v \nabla u) \cdot d \mathbf{s} .\nonumber$$

Exercise $\PageIndex{9}$

Consider the problem:

$$\frac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \frac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0 .\nonumber$$

1. Solve by direct integration.
2. Compare this result to the Green’s function in part $b$ of the last problem.
3. Verify that $G$ is symmetric in its arguments.

Exercise $\PageIndex{10}$

Consider the boundary value problem: $y^{\prime \prime}-y=x, x \in(0,1)$, with boundary conditions $y(0)=y(1)=0$.

1. Find a closed form solution without using Green’s functions.
2. Determine the closed form Green’s function using the properties of Green’s functions. Use this Green’s function to obtain a solution of the boundary value problem.
3. Determine a series representation of the Green’s function. Use this Green’s function to obtain a solution of the boundary value problem.
4. Confirm that all of the solutions obtained give the same results.

Exercise $\PageIndex{11}$

Rewrite the solution to Problem 15 and identify the initial value Green’s function.

Exercise $\PageIndex{12}$

Rewrite the solution to Problem 16 and identify the initial value Green’s functions.

Exercise $\PageIndex{13}$

Find the Green’s function for the homogeneous fixed values on the boundary of the quarter plane $x>0, y>0$, for Poisson’s equation using the infinite plane Green’s function for Poisson’s equation. Use the method of images.

Exercise $\PageIndex{14}$

Find the Green’s function for the one dimensional heat equation with boundary conditions $u(0, t)=0 u_{x}(L, t), t>0$.

Exercise $\PageIndex{15}$

Consider Laplace’s equation on the rectangular plate in Figure 6.3.1. Construct the Green’s function for this problem.

Exercise $\PageIndex{16}$

Construct the Green’s function for Laplace’s equation in the spherical domain in Figure 6.5.1.

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