7.9: Problems
- Page ID
- 90965
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Find the solution of each initial value problem using the appropriate initial value Green’s function.
- \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\).
- \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0\).
- \(y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0\).
- \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0\).
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.
- \(x^{\prime \prime}+x=5 t^{2}\).
- \(x^{\prime \prime}+x=2 \tan t\).
For the problem \(y^{\prime \prime}-k^{2} y=f(x), y(0)=0, y^{\prime}(0)=1\),
- Find the initial value Green’s function.
- Use the Green’s function to solve \(y^{\prime \prime}-y=e^{-x}\).
- Use the Green’s function to solve \(y^{\prime \prime}-4 y=e^{2 x}\).
Find and use the initial value Green’s function to solve
\[x^{2} y^{\prime \prime}+3 x y^{\prime}-15 y=x^{4} e^{x}, \quad y(1)=1, y^{\prime}(1)=0 .\nonumber \]
Consider the problem \(y^{\prime \prime}=\sin x, y^{\prime}(0)=0, y(\pi)=0\).
- Solve by direct integration.
- Determine the Green’s function.
- Solve the boundary value problem using the Green’s function.
- Change the boundary conditions to \(y^{\prime}(0)=5, y(\pi)=-3\).
- Solve by direct integration.
- Solve using the Green’s function.
Let \(C\) be a closed curve and \(D\) the enclosed region. Prove the identity
\[\int_{C} \phi \nabla \phi \cdot \mathbf{n} d s=\int_{D}\left(\phi \nabla^{2} \phi+\nabla \phi \cdot \nabla \phi\right) d A .\nonumber \]
Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green’s first and second identities, respectively.
- \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\).
- \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).
Let \(C\) be a closed curve and \(D\) the enclosed region. Prove Green’s identities in two dimensions.
- First prove
\[\int_{D}(v \nabla \cdot \mathbf{F}+\mathbf{F} \cdot \nabla v) d A=\int_{C}(v \mathbf{F}) \cdot d \mathbf{s}\nonumber \]
- Let \(\mathbf{F}=\nabla u\) and obtain Green’s first identity,
\[\int_{D}\left(v \nabla^{2} u+\nabla u \cdot \nabla v\right) d A=\int_{C}(v \nabla u) \cdot d \mathbf{s} .\nonumber \]
Consider the problem:
\[\frac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \frac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0 .\nonumber \]
- Solve by direct integration.
- Compare this result to the Green’s function in part \(b\) of the last problem.
- Verify that \(G\) is symmetric in its arguments.
Consider the boundary value problem: \(y^{\prime \prime}-y=x, x \in(0,1)\), with boundary conditions \(y(0)=y(1)=0\).
- Find a closed form solution without using Green’s functions.
- 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.
- Determine a series representation of the Green’s function. Use this Green’s function to obtain a solution of the boundary value problem.
- Confirm that all of the solutions obtained give the same results.
Rewrite the solution to Problem 15 and identify the initial value Green’s function.
Rewrite the solution to Problem 16 and identify the initial value Green’s functions.
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.
Find the Green’s function for the one dimensional heat equation with boundary conditions \(u(0, t)=0 u_{x}(L, t), t>0\).
Consider Laplace’s equation on the rectangular plate in Figure 6.3.1. Construct the Green’s function for this problem.
Construct the Green’s function for Laplace’s equation in the spherical domain in Figure 6.5.1.