-1
votes
1answer
38 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature $T(x,t)$ satisfies $$\frac{\partial T}{\partial t}= k ...
0
votes
0answers
28 views

Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$ u_{yy} + (2-x)u_y - 2xu = 1 $$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$ y'' + (2-x) y' - 2x y = 1. $$ For the ...
0
votes
0answers
19 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
votes
0answers
9 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
2
votes
1answer
47 views

Neumann's problem necessary and sufficient condition (Evans PDE)

Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution of Neumann's problem \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f & \text{in } U \\ \frac{\partial ...
0
votes
0answers
10 views

Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$ (1) Calculate the coefficients of the Full ...
0
votes
1answer
37 views

Cauchy-Reimann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $\frac{\delta u}{\delta x} = \frac{\delta ...
1
vote
1answer
26 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
1
vote
1answer
18 views

Find adjoint operator $L^*$ for the 3rd order operator $Lu(x,t)=u_t(x,t)-u_x(x,t)+\gamma^2u(x,t)$, with $wLu-uL^*w$ (1) is divergence expression.

I'm working out of the Zauderer PDEs book and am having some trouble on the adjoint operators section (3.6). Specifically this problem: "Obtain the adjoint operator $L^*$ for the third order operator ...
1
vote
2answers
57 views

Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$.

This is an exercise of my course of PDE: Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$. My Attempt A function $g$ is called harmonic if $\Delta ...
0
votes
1answer
22 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
0
votes
0answers
17 views

Partial Differential Equation vibrating string

Find the displacement of a string stretched between two fixed points at a distance 2c apart when the string is initially equilibrium position and points of the string are given initial velocities v ...
-4
votes
0answers
30 views

Prove u=0 for a harmonic function u.

Let $\Omega$ be an open connected set in $\mathbb{R}^n$ with the boundary $\partial\Omega$ of class $C^2$. Let $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ be a harmonic function in $\Omega$, such ...
0
votes
0answers
41 views

Show that a system has at most one solution using the energy method?

We are given the one dimensional heat equation with source $f$: $${\partial u \over \partial t} - k {\partial ^2 u \over \partial ^2 x} = f(t,x)$$ over $x\epsilon(0,l)$ and for $t\gt0$ with Neumann ...
1
vote
1answer
54 views

Having trouble integrating for use of energy method to prove uniqueness

We are given $u_{tt} - c^2u_{xx} + ru_t$. To prove only one solution exists, I am taking w = $u_1 - u_2$, assuming they are both solutions to the given wave equation. So: $u_{tt} - c^2u_{xx} + ...
1
vote
1answer
34 views

minimizes the functional to solve a pde

I am trying to do this exercise: Let $\Omega$ a open bounded domain in $R^n$. Consider the Dirichlet problem $$ \left\{ \begin{array}{ccccccc} -\Delta u = \lambda \sin (u) + f , \ \text{in ...
2
votes
1answer
79 views

Solve as a series the equation $u_t = u_{xx}, u_x {(0,t)}=0 , u{(1,t)} =1, u{(x,0)} = x^2$.

This question is from section 5.6 of Partial Differential Equations: An Introduction 2nd Edition by Walter Strauss 2008. I have approached this question by using the separation of variables $u(x,t) = ...
2
votes
1answer
33 views

Partial Differential Equation $u_t=3u_{xx}$

Solve the partial differential equation $$u_t=3u_{xx}, u(0,t)=0, u(x,0)=\cos{x}\sin{5x}$$ Attempt: Using separation of variables, let $u(x,t)=f(x)g(t)$, so $$f(x)g'(t)=3f''(x)g(t)$$ ...
0
votes
0answers
19 views

Find examples where $\omega (\overrightarrow x)$…

Help figure examples where The set of all $\omega$ - points of $\phi_t (\overrightarrow x$) is called the $\omega$ - limit set of $\phi_t (x)$. 1) $\omega (\overrightarrow x) = \emptyset$ for all ...
2
votes
0answers
37 views

differential equation looks like Bessel but isn't

I have this question What I did is: $U=X(x)*T(t)$ after putting it back into the function I got $-x^2*T''/T= x^2X''-2xX'+2X $ after deviding by $x^2$ remembering to check $x=0$ I get $-T''/T= ...
0
votes
0answers
17 views

Find the value of the origin inside the unit ball.

Here, $\Delta$=0 inside the unit ball in $\mathbb{R}^{3}$ and $u(1,\varphi,\theta)=sin^{2}(\varphi)$. Where is the value of u at the origin? I am not sure if $\Delta=0$ is suppose to be $\Delta u = ...
0
votes
0answers
26 views

Simple Question- Is this a Sturm Liouville regular problem?

I have the following differential system: \begin{align} &(1-t^2)x'' -2tx' +\lambda x= 0 \\&x(0)=0 \\&x(l)+x'(l)=0 \end{align} I have to decide if it is an homogeneous regular Sturm ...
1
vote
2answers
42 views

Wave equation, initial conditions

The displacement of an infinite string obeys the wave equation: $$ \frac{\partial ^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ Find the solution in the form: $$u(x,t) = f(x-ct) + ...
0
votes
1answer
23 views

show that every integer and every angle … pendulum moving around the circle at least n times but not n+1 times

For the nonlinear damped pendulum, show that for every integer $n$ and every angle $θ_0$ there is an initial condition $(θ_0 , v_0)$ with a solution that corresponds to the pendulum moving around the ...
2
votes
1answer
32 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & ...
1
vote
2answers
37 views

Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
0
votes
0answers
37 views

Adjoint boundary-value problem of the biharmonic function

I have to find the adjoint boundary-value problem for \begin{align} \Delta^2 u(x) &= f(x), && x \in \Omega, \\ \Delta u(x) &=0 && x\in \partial \Omega, \\ \nabla u(x) ...
0
votes
1answer
45 views

Partial differential equation .

I'm new to the forum and I hate to be the guy who asks for people to solve my homework but I've been trying to solve the follow exercise and I've been stuck for hours :(. Partial equation is probably ...
2
votes
1answer
53 views

Validating a PDE problem solution

I have the following problem, which I have tried to solve myself and I would like someone to verify that my answer is valid. The problem is the following: By separation of variables, derive the ...
1
vote
1answer
42 views

Understanding partial differential equation requirement

I'm reading about separation of variables in my Fourier series book and there is one requirement in a problem I don't understand. Here it is: I don't understand, why can't the $\sqrt{-A}$ take a ...
1
vote
1answer
27 views

Solve the initial value problem $u_x^2u_t-1=0$, $u(x,0)=x$.

Solve the initial value problem $u_x^2u_t-1=0$, $u(x,0)=x$. This becomes $u_x^2u_t=1$, $u(x,0)=x$. I was thinking that this was a nonlinear wave equation at first, but the $x$ component is ...
0
votes
0answers
43 views

Solve PDE by Cauchy's Method of Characteristics

Solve the following partial differential equation by the method of characteristics $\displaystyle zp+yq=x$ $\displaystyle x_o(t)=t, y_o(t)=1, z_o(t)=2t$ Solution (A):Using initial values of $x_o, ...
1
vote
0answers
170 views

Prove that $\vec x = \vec0$ is an asymptotically stable fixed point for this linear system.

Consider the linear vector field $\vec x' = A\vec x$, $\vec x \in R^2$ , where A is an $2$ x $2$ constant matrix. Suppose all eigenvalues of A have negative real parts. Prove that $\overrightarrow x = ...
0
votes
1answer
34 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
1
vote
0answers
83 views

Adjoint boundary problem

Suppose $A\subset\mathbb{R}^2$ is open and bounded and has a boundary $\partial A$ which is $C^1$-smooth. Let $N$ be the outward normal unit vector field of $A$ and $T$ the counterclockwise pointing ...
1
vote
0answers
26 views

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$.

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$. I know I have to solve for implicit $u$ first and then $r$, but I don't know if I'm using the right method here. What I've ...
2
votes
0answers
179 views

Find all of the equilibrium points and describe the behavior of the $x' = sin(x), y' = cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
0
votes
0answers
17 views

Solve $yu_x+xu_y=0$ where $u(0,y)=\exp(-y^2)$. Where in the $xy$-plane is the solution uniquely determined?

Solve $yu_x+xu_y=0$ where $u(0,y)=\exp(-y^2)$. Where in the $xy$-plane is the solution uniquely determined? Now I think I got the first part right by considering the PDE as the dot product ...
2
votes
2answers
77 views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad ...
1
vote
0answers
50 views

separation of variable in a nonhomogeneous heat equation

I want to solve the following of the heat equation using separation of variable: But have one problem a the end of the method, Thx for your help. $ \left\{\begin{matrix} u_{t}-u_{xx}=tx \; \; ...
2
votes
1answer
31 views

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$ First, I notice that this equation is not differentiable at x = 0. Therefore, the ...
0
votes
0answers
17 views

On taking the fourier transform of a PDE

Suppose I have the following PDE:$$u_t=-u_x +F(x,t)$$ When I take it's fourier transform, does the forcing function become a variable of t and the fourier variable?
1
vote
0answers
30 views

Show that for $u_t(x,t)+u_x(x,t)=F(x,t)$, $u(x, x/c)=f(x)$ is a characteristic initial value problem if $f^\prime(x)=\frac{1}{c}F[x,\frac{x}{c}]$.

Show that for $u_t(x,t)+u_x(x,t)=F(x,t)$, $u(x,\frac{x}{c})=f(x)$ is a characteristic initial value problem if $f^\prime(x)=\frac{1}{c}F(x,\frac{x}{c})$. I did a problem similarly to this one, but I ...
0
votes
1answer
22 views

Show that the equation $a(t)v_x(x,t)+b(x)v_x(x,t)=0$ has the general solution $v(x,t)=F[B(x)-A(t)]$ where $B^\prime(x)=b(x)$ and $A^\prime(t)=a(t)$.

Show that the equation $a(t)v_x(x,t)+b(x)v_t(x,t)=0$ has the general solution $v(x,t)=F[B(x)-A(t)]$ where $B^\prime(x)=b(x)$ and $A^\prime(t)=a(t)$. This problem is similar to a homework problem I'm ...
0
votes
1answer
51 views

Construct an example of a first-order differential equation on $\mathbb{R}$ for which there are no solutions to any initial value problem.

Construct an example of a first-order differential equation on $\mathbb{R}$ for which there are no solutions to any initial value problem. Could anyone please get me started on this. I am struck as ...
0
votes
0answers
29 views

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $a > 0$ and $x(0) = 0.$

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $a > 0$ and $x(0) = 0.$ Let $f(x) = X'$ I, then, take the derivative of $f(x)$ which gives me $f'(x) = ...
1
vote
1answer
28 views

For which constants does the following converge to a delta function?

let $g_n(x,y)=\frac{c_n}{1+n^2(x^2+y^2)}$ for which constants, $c_n$ , does the function converge to a delta function as n becomes arbitrarily large? My intitial thought, corroborated by plots in ...
0
votes
0answers
24 views

Help deriving the differential, variational, and minimization problems (forms) conditions necessary on terms to satisfy

I have some hopefully quick questions on different formulations of $$ (\beta(x)u'(x))' = f(x) $$ I am looking for the differential form (D), the minimization form (M), and the variational form (V) ...
0
votes
1answer
13 views

Verify solution to PDE

Define $u:\mathbb{R}^2\to \mathbb{R}$ by $u(x,y)= e^{x\sin y}f(x-y)$, where $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function. Show that $$\frac{\partial u}{\partial x}+\frac{\partial ...
1
vote
1answer
55 views

Find an explicit conjugacy between their flows. $X' = \begin{bmatrix}-1& 1 \\ 0 & 2\end{bmatrix}X$ and $Y' $…

Find an explicit conjugacy between their flows. $X' = \begin{bmatrix}-1& 1 \\ 0 & 2\end{bmatrix}X$ and $Y' = \begin{bmatrix}1& 0 \\ 1 & -2\end{bmatrix}Y$ I have no clue how to ...