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, \[\int_{D}\left(v \nabla^{2} u+\nabla u \cdot \nabla v\right) d A=\int_{C}(v \nabla u) \cdot d \mathbf{s} .\nonumber \]
3. Use Green’s first identity to prove Green’s second identity, \[\int_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d A=\int_{C}(u \nabla v-v \nabla u) \cdot d \mathbf{s} .\nonumber \]

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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