Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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0
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15 views

Question about heat equations?

I have to solve the heat equation with $u(0,t) = 0$, the end at $x=2$ insulated $\forall t \gt 0$ and initial condition $u(x,0)= 20\sin\frac{\pi x}{4}$. I interpreted the 2nd b.c to mean that ...
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0answers
4 views

Find a condition on $\lambda(k)$ such that the normal mode functions satisfy the given equation.

Let $k=(k_1,k_2)$, $x=(x_1,x_2)$ and $\alpha>0$. Find a condition on $\lambda(k)$ such that the normal mode functions $u(x)=a(k)e^{i(k_1x_1+k_2x_2)+\lambda(k)t}$ satisfy the following equation ...
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0answers
9 views

The riemann problem for p-system [on hold]

Consider the equations u_t -v_x = 0 v_t + (\farc {v^2} {u} +u^2)_x =0 We now study the Riemann problem, for general ...
1
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0answers
14 views

Partial Differential Equations, how to find a change of variables

I am trying to understand how to find variable changes for partial differential equations. I know characteristics method gives you a valid variable change when you have a condition given on a ...
3
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0answers
44 views

Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = ...
4
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0answers
18 views

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
-2
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0answers
23 views

Cantilever fourth order PDE with homogeneous solution and separation of variables [on hold]

The equation for a driven cantilever, with damping is: $EI\frac{\partial^4u}{\partial x^4}+\sigma\frac{\partial u}{\partial t}+\rho A\frac{\partial^2 u}{\partial t^2}=F(x,t)$ a) Solve for the ...
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1answer
19 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
2
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2answers
40 views

Find the Fourier Transform of $2x/(1+x^2)$

I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
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1answer
26 views

Evans PDE problem 9,Chapter 6

Let $U\subset \mathbb{R}^n$ be an bounded domain with smooth boundary. Assume $u$ is a smooth solution of $$ Lu=-\sum_{i,j=1}^na^{i,j}u_{x_ix_j}=f \ \text{in} \ U, \ \ u=0 \ \text{on} \ \partial ...
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0answers
15 views

solution of $u_t=\Delta \left(u^m\right)$ [on hold]

How can I find solution of $u_t=\Delta \left(u^m\right)$ where $m>1$?
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0answers
29 views

Overview of nonlinear analysis, ODE and PDE, dynamical systems, and mathematical physics and their relationships

(Apologies in advance for my naive question.) The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a ...
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1answer
17 views

are those boundary $ C^1$

I am studying PDE and my question is that if we have a open unit disk with $ [0,1)$ on x axis removed, is the boundary of this set $C^1$ ? And on the other hand, is the boundary of a open rectangle ...
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0answers
26 views

a question on an eigen value [on hold]

What is the first eigen value of the following problem?. \begin{eqnarray} \Delta u &=& -\lambda u,\ \ \ \ \ \text{on }\Omega,\\ u &=& 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ on } ...
3
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0answers
20 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
1
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1answer
23 views

Intuition behind estimates on derivatives of a harmonic function

In Evans' PDE book he gives the following theorem. Assume $u$ is harmonic in $U$. Then, $$ |D^{\alpha}u(x_0) | \le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ When asking my professor for some ...
2
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1answer
22 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,t) | \leq \frac ...
1
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1answer
13 views

Verifying Duhamel Principle for Heat Equation

From separation of variables, we get a solution to the homogeneous problem for the heat equation $$u_t - u_{xx} = 0$$ $$u(0,t) = u(L,t) = 0$$ $$u(x,0) = f(x)$$ of the form $$u(x,t) = \int_0^L f(y) ...
2
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0answers
27 views

How this solution can be obtained for the Biharmonic equation?

I'm trying to solve the Biharmonic equation $\nabla^4\psi = 0$ on the plane, subject to the boundary conditions $$\dfrac{\partial \psi(x_s,y_s)}{\partial y}=0, \\ \dfrac{\partial ...
1
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0answers
19 views

Heat on a wedge

I'm trying to solve the PDE given by $$\begin{array}\ u_t = \nabla^2 u, & 0 \lt r \lt 1, & 0 \lt \theta \lt \alpha \lt 2\pi\end{array} \\ u(r,\theta, 0) = f(\theta) \\ u(1,\theta,t) = u(r,0,t) ...
2
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1answer
21 views

How can I prove this Bessel function relation

Prove $$J_{s \pm 1}(z) = \frac{s}{z}J_s(z) \mp J_s'(z)$$ from $$J_s(z) = \sum^\infty_{j=0} \frac{(-1)^j}{\Gamma(j+1)\Gamma(j+s+1)}\Big(\frac{z}{2}\Big)^{2j+s}$$ I proved $J_{s - 1}(z) = ...
1
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0answers
21 views

How to get first order term in wave equation

Like in heat equation $U_{t} + U_{xx} + bU_{x} + U = 0$. Just let $v(t,x) = e^{t}u(t,x+bt)$, then we turn the problem into a standard one. Now the question is how to turn the wave equation $U_{tt} - ...
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0answers
13 views

A PDE question using variation of parameters

Variation of parameters: Consider IBVP \begin{align} u_t − u_{xx} = f(x, t) \qquad & \text{on } \Omega = (0, \pi) \times \Bbb R^+\\ u(x, 0) = \varphi(x) \qquad & \text{on } (0, \pi)\\ ...
1
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1answer
22 views

Laplace Equations with Neumann boudary-value problem

The problem is that, Assume U is connected, use the maximum principle to show that the only smooth solutions of $-\Delta u=0$ in U and $\frac{\partial u}{\partial \nu}=0$ on $\partial U$ are $ u ...
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0answers
22 views

Does the differential operator in the heat equation have a name?

Does the operator $$\frac {\partial}{\partial t} - k\nabla^2$$ have a name?
1
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1answer
21 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
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0answers
23 views

Find the supremum of the function

Hi I'm trying to figure out for which values of $w$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is going to be all ...
-2
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0answers
25 views

Hyperbolic conservation laws An Illustrated Tutorial [on hold]

Consider the equations Ut−Vx=0 Vt+[V²U+U²]x=0 We now study the Riemann problem,for general initial date. (U,V)={(U₀,V₀),(U₁,V₁)ifX<0ifX>0 for u>0
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0answers
15 views

The riemann problem for p-system [on hold]

Study The riemann problème: Ut−Vx=0 Vt+[V²U+U²]x=0 (U,V)={(U₀,V₀),(U₁,V₁)ifX<0ifX>0
1
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2answers
39 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
2
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1answer
39 views

How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE $$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$ using Charpit's method. The ...
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0answers
29 views

The Riemann problem [on hold]

Study the Riemann problem: $$U_t − V_x =0$$ $$V_t + [\frac{V²}{U} + U²]_x =0$$ $$(U,V) =\begin{cases}(U₀ ,V₀),& \text{if} X < 0\\(U₁ ,V₁)&\text{if} X > 0\end{cases}$$ How do you find ...
0
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30 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
1
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1answer
26 views

nonhomogenious partial differential equation

How to solve this transport equation? $\dfrac{u_t}{u}-\dfrac{t}{x}\dfrac{u_x}{u}=-\dfrac{fx^2+2g}{ax^2+2b}$ in $(0,\infty)\times \mathbb{R}-\{0\}$ $u(t_0,x)=u_{t_0}(x) $ on $\{t=t_0\}\times ...
0
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1answer
15 views

The expansion of harmonic function at infinity

If $u$ is a harmonic function on $\mathbb R^n$ outside some compact set such that $u$ goes to $1$ at infinity. Then does $u$ have the following expansion $$ u=1+\frac{a}{|x|^{n-2}}+O(|x|^{1-n})\quad ? ...
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0answers
27 views

The Riemann problem for the P-system [on hold]

exambles of the riemann problems
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0answers
9 views

Reference Request for Penalty Method for Optimal Control?

Is there a good book or review article to read about the methods like penalty method, method of duality and method of relaxation in problems of calculus of variations and their relations to optimal ...
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0answers
11 views

Observe approximation order of numerical solution of a Partial Differential Equation

When solving a Partial Differential Equation numerically, I estimated the approximation orders theoretically as follows, $$ u(x,t)= u_{h,k} + C_1 h^{p} +C_2 k^{q}, $$ where $ u_{h,k} $ is a numerical ...
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0answers
22 views
+50

Why are nodes and nodal sets called this way?

Nodes of standing waves are points where they are zero. Generally, nodal sets of Laplacian eigenfunctions are the sets of points where they are zero. Why is this the name for them (that is, why is ...
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1answer
15 views

What is the dy/ds characteristic equation for this PDE?

$$u_t+\text{yu}_x+\frac{1}{2}\text{(u(u-1)})_y\text{=0}$$ The initial condition is given as $$\text{u(x,y,0)=}u_0\text{(x,y)}$$ I know what $$\frac{\text{dt}}{\text{ds}}\text{ ...
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1answer
43 views

How do I solve this PDE (diffussion equation) using the sepration of variables method?

$$\frac{\partial u}{\partial t} =\nu\large(\frac{\partial^2u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \large), 0 < r < a, t >0.$$ Subject to the conditions $$\frac{\partial ...
1
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0answers
15 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
1
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1answer
19 views

Difficult boundary conditions for the PDE $U_{xx}=U_t$

I am given $U_{xx}=U_t$, $U_x(1,t)=U(1,t)$, and $U_x(0,t)=0$. I use separation of variables and set $U(x,t)=X(x)T(t)$, then $$X''T=XT'$$ $$\frac{X''}X=\frac{T'}T=-\lambda$$ for some $\lambda$. And ...
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0answers
30 views

How to solve this pde ? mass transfer problem [on hold]

How to solve this pde:$$v_{z}\frac{\partial C_{A}}{\partial z}=\mathcal{D}_{AB}\frac{\partial^{2}C_{A}}{\partial x^{2}} $$ with these boundary conditions: at $z=0$, $C_{A}=0$ at $x=0$, ...
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0answers
17 views

Laplace's Equation with Neumann B.C. on disc

I am trying to solve Laplace's equation with the following boundary conditions: $On\hspace{0.1in}the\hspace{0.1in}disc\hspace{0.1in}of\hspace{0.1in}radius\hspace{0.1in}1,\hspace{0.1in} \triangledown ...
2
votes
1answer
52 views

Quasi linear partial differential equation

I am trying to solve the pde given by: $$\rho_t + (\rho v)_x = 0$$ Subject to conditions $v = f(x-vt)$ and $\rho(x, 0) = g(x)$. I have used the product rule on the second term to observe that ...
0
votes
2answers
33 views

Can we integrate from $0$ to $x$ / $t$?

I want to find all the solutions of the following problem: $$w_x(x,t)=3x^2t+t, x,t \in \mathbb{R}\\w_t(x,t)=x^3+x, x, t \in \mathbb{R}$$ I have tried the following: $$w_x(x,t)=3x^2t+t \Rightarrow ...
0
votes
1answer
22 views

What are the solutions to $\partial_{x_i x_j} F(x) = 0$?

Let $F\in C^2(\mathbb{R}^d, \mathbb{R})$. I'm looking for solutions to $\frac{\partial^2 F}{\partial x_1 \partial x_2}(x) = 0$ or $\frac{\partial^2 F}{\partial x_1 \partial x_1}(x) = 0$. My guess is, ...
1
vote
0answers
18 views

Boundary condition not compatible with initial condition…why?

Equation $$\frac{\delta^2 p}{\delta t^2} = \frac{\delta^2 p}{\delta x^2} + \frac{\delta^2 p}{\delta y^2}$$ Boundary conditions: $p = 0$ for $x = 0, \pi$ $p = 0$ for $y = 0, \pi$ Initial ...
3
votes
2answers
51 views

Find solution of problem - Method of characteristics

I want to find the solution of the problem: $$(t+u(x,t))u_x(x,t)+tu_t(x,t)=x-t, x \in \mathbb{R}, t>1 \\ u(x,1)=1+x, x\in \mathbb{R}$$ I have tried the following: $$(x(0), t(0))=(x_0, 1)$$ ...