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

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Partial Differential limits question.

I have a PDE that I solve to be: $$u(x,t) = f(x+at) + g(x-at)$$ I need to apply initial conditions, $u(x,0)=r(x)$ and $u_t(x,0)=s(x)$ From this I get: $$f(x)+g(x)=r(x)$$ and $$a(f'(x)-g'(x) = ...
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45 views

A second-order non-linear differential equation

Which are the possible solutions for this second-order non-linear différential equation : $$\frac{\partial Z}{\partial q} \frac{\partial^2 Z}{\partial p \space \partial t} - \frac{\partial ...
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74 views

When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?

Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth. Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
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135 views

how to prove the uniqueness solution of the following PDE

how to prove the uniqueness solution of the three dimensional wave: $c^2\frac{\partial^2u}{\partial t^2}=\Delta u $ with satisfy the boundary conditions : $u(x,y,z,t) = F(x,y,z,t) $ on $S$ and ...
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80 views

Integro-differential equation in one dimensional linear thermo-elasticity

I have this system of coupled pde's: \begin{equation} \frac{\partial^2\theta}{\partial x^2}=\frac{\partial \theta}{\partial t}+\sqrt{a}\frac{\partial^2 u}{\partial x\, \partial t} \end{equation} ...
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132 views

Differentiable but not Absolutely continuous

Please give an example (if it exists) for a function which is differentiable everywhere but not absolutely continuous.
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61 views

Boundedness of a Solution Operator

Let $v \in L^2(\partial \Omega)$ and define $S(v) := y$ where $y$ satisfies $-\Delta y + y = 0 $ in $\Omega$ and $\frac{\partial y}{\partial \nu} = v$ on $\partial \Omega$. I need to show that $S$ is ...
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67 views

Analytical solution to PDE (2)

Does the PDE $$u_{tt} = u_{xxx}$$ with $x \in [0,1]$ and $t \in [0,T]$ with initial conditions $$u(x,0)=\sin(\pi x),\, \partial_t u(x,0)=0$$ and boundary condition $$u(0,t)=u(1,t)=0$$ have a ...
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54 views

theorem in capacity theory

I am trying to understand the proof of a theorem of capacity theory the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, ...
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79 views

$L^2(U)$ compact embedded in $H^{-1}(U)$?

Let $U$ be an open subset of $R^d$. We already knew that $L^2(U)$ is a subset of $H^{-1}(U)$. Question: is this a compact embedding?
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33 views

Computing a derivative of map from $V \to V^*$ (PDEs and regularity)

I am reading Rogers and Renardy book on parabolic regularity. There they consider a PDE $$\dot u = A(t)u + f(t)$$ where $A(t):V \to V^*$ is an operator. In the regularity result, they need $A \in ...
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35 views

find the solution to the PDE

Consider the problem, where α>0 $$ U_t=2Uxx, 0≤x≤2, t≤0 $$ $$ Ux(0,t)=-α , Ux(2,t)=0 $$ $$ U(x,0)=f(x)= -αx(1-(1/4)x) $$ By trying a function of the form u(x,t)=Ax² + Bx + Ct to ...
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340 views

Solving Partial Differential Equation

Solving Partial Differential Equation $u_{xyy} (x,y,z) = 2 \sin x $ Is this correct? $$ \frac{\partial^3 u}{\partial y \, \partial y \, \partial x} (x,y,z) = 2 \sin x $$ Integrate respect to y ...
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2answers
113 views

Solving a general partial differential equation by using the method of characteristics

I want to solve $u_t + b \cdot \nabla u + cu =0$ with initial condition $u(x,0) = g(x)$. I started by using the method of characteristics. Define: $$ z(s) := u(x + bs, t+s)$$ Then: $$ \frac{dz}{ds} = ...
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758 views

How to solve the non-homogeneous PDEs: $u_x + u_y=2u,\ u(x,0)=h(x)$?

I have this first-order non-homogeneous partial differential equation with initial condition: $u_x + u_y=2u,\ u(x,0)=h(x)$ The following was what I tried: By the method of characteristic curves, we ...
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461 views

Laplace’s equation in three dimensions

How to solve Laplace’s equation in a cube?? $\left\{\begin{matrix} u_{xx}+u_{yy}+u_{zz}=0, 0 <x,y,z< \pi \\ u(\pi,y,z)=g(y,z)\\ u(0,y,z)=u(x,0,z)=u(x,\pi,z)=u(x,y,0)=u(x,y,\pi)=0 ...
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83 views

if $\Delta u \geq c$ for some $c>0$ then $u$ has a max on the boundary

Let $D=\{(x,y): \vert(x,y)\vert \leq 1\}$ and let $u:D\rightarrow \mathbb R$ be continuous function with three continous derivatives in the interior of $D$. Show that if there is a number $c>0$ ...
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540 views

Question from Evans' PDE book

How do you do the second part of question 8, chapter 5, of Evans' PDE book (first edition)? I have proven the inequality for smooth, compactly supported functions using integration by parts, and I ...
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95 views

Laplace transform in solving 2d wave equation

I have the following wave equation $\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=\dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}$ with boundary conditions at $x=0,\ ...
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265 views

Hörmander's theorem on hypoelliptic PDEs - existence of solution?

I've seen many authors state that Hörmander theory implies the existence of a $C^\infty$ solution. For example, on Wikipedia it says: The great achievement of Hörmander's 1967 paper was to show ...
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70 views

Simple inequality with complex number

who know how to prove the estimate $Re(\frac{1}{\zeta^2})\gt\frac{4}{81t^2}$ (pr something like this)? You have to prove that if $\zeta=t+(\frac{t}{2})\exp(i\theta)$ then ...
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68 views

Does formulating a PDE's analytical solution (if possible) imply its existence?

Or are there counterexamples? Especially some that are not quite artificial by e.g. taking the limit of a PDE series?
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478 views

Method of Eigenfunction Expansion

The solution of a PDE can be represented by a Fourier cosine series $$ u(x,t)=\sum_{n=1}^\infty A_n(t)\cos\frac{n\pi x}L. $$ Applying a given initial condition $$ u(x,0)=100, $$ lets us solve for ...
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171 views

direct differentiation in PDE

How can we check by direct differentiation that the formula $u(x, t) = \varphi(z)$, where $z$ is given implicitly by $x − z = ta(\varphi(z))$, does indeed provide a solution of the PDE $u_t + a(u)u_x ...
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104 views

teaching a little nonlinear PDE to an undergraduate

I would like to teach a little nonlinear PDE to an undergraduate who is taking a course in second-order linear boundary value problems. I have never taught nonlinear PDE before, although it is my ...
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128 views

Solving a PDE possibly with method of characteristics or other methods

For a PDE $$(x-y^{2}) u_{x} + u_{y} = 0$$ I've tried to use method of characteristics. But I've failed to do so. It was because of the term $x-y^{2}$; I don't know how to integrate this on the ...
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108 views

Sobolev Approximation Theorem Example

The approximation theorem states that if $U$ is bounded, $u \in W^{1,p}(U)$ for some $1 \leq p < \infty$ then there are functions $u_m \in C^\infty(U) \cap W^{k,p}(U)$, such that $u_m \rightarrow ...
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102 views

alternative form of the general solution of $u_{xx}+u_{yy}=0$

It is well-known that one of the form of the general solution of $u_{xx}+u_{yy}=0$ is $u(x,y)=C_1(x+iy)+C_2(x-iy)$ , where $C_1$ and $C_2$ are arbitrary functions. However it is not quite convenient ...
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About the statement of PDE Problem

Take the Dirichlet Problem as example: The difinition of Dirichlet Problem from wiki Given a function $f$ that has values everywhere on the boundary of a region in $\mathbb{R}^n$, is there a ...
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63 views

PDEs - Satisfying equations

Let $u = f(x-ut)$ where $f$ is differentiable. Show that $u$ (amost always) satisfies $u_t + uu_x = 0$. What circumstances is it not necessarily satisfied? This is a question in a tutorial sheet ...
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172 views

How to solve the partial differential equation $a\frac {\partial L}{\partial a}+b\frac {\partial L}{\partial b}= L $

I am trying to express the eliptic integral in series expression that depends on $a,b,\alpha$ and without integral $$L(\alpha)=\int_0^\alpha\sqrt{a^2\sin^2 t+b^2 \cos^2 t}\,dt $$ $$\frac ...
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248 views

Numerical Solution of Systems of PDE

Could someone give me some reference to the Numerical solution of a System of PDEs of following type.. (which also encompasses strongly elliptic system of PDEs) in 2D or 3D. $$\left\{ ...
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245 views

How to solve linear partial differential equation?

Given that $P, Q, R$ is a function of $x, y, z$, we have $$ P {\partial z \over \partial x} + Q{\partial z \over \partial y} = R \hspace{2 cm} (1)$$ The book says that $ u = \phi(v) $ is the solution ...
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75 views

Boundary term while performing partial differentiation.

This is the part of the proof for the finite speed of propagation of wave equation that i am learning. I am not understanding how the boundary term come in the first step ie while finding $\dot e(t)$ ...
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2k views

solving pde by matlab

Can you help me please? I need to solve wave equation using Matlab. How to solve this PDE by Matlab? $u_{tt}=c^{2}u_{xx} $ for $0<x<l$ B.C: $u_x(0,t)=0$; $u(l,t)=0$ I.C: $u(x,0)=f(x); ...
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1answer
72 views

Why we have “$\Delta \Gamma = \delta$”, in $\mathbb{R}^{n}$, where $\Gamma$ is fundamental solution and $\delta$ is the dirac measure?

Why we have "$\Delta \Gamma = \delta$", in $\mathbb{R}^{n}$, where $\Gamma$ is fundamental solution and $\delta$ is the dirac measure?
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370 views

Double-Well Delta Potentials - Schrödinger Equation

Page 177 on Davies' book- Spectral theory of diff operatrs contains the following computation problem: Calculate the negative eigenvalues and the corresponding eigenfunctions of the following ...
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103 views

stability definition for nonhomogeneous pde

I see the definition for stability for any solution operator $E$ is that $$||Eu^{n}||^2=||u^{n+1}||^2\leq C||u^{n}||^2$$ for some constant $C$ and some pde $u_t=Lu$. However, I can show that ...
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94 views

PDE with series

Consider this PDE The solution to the PDE is So what I am having trouble is solving it using this method. I am going to say that my $u(x,t) = \sum_{n=1}^{\infty} u_n(t) \sin(nx)$ and $x \sin(t) ...
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213 views

Mean value property satisfied of continuous functions.

If $u$ is only continuous and satisfies Mean value property , is it true that $u$ is harmonic in $\Omega \subset \mathbb{R}^n$ . $\Omega$ is bounded and open. What basically here should I know to ...
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264 views

Fundamental solution of the wave operator

What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks. The operator I'm talking about is ...
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332 views

Mean value property of Heat equation.

I am learning Mean value property(MVP) of heat equation . MVP of laplace equation was relatively easy to understand i think its because of the spherical symmetry . But i am not able to appreciate the ...
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77 views

Problem on scaling, shifting etc.

Given $\int_{B_1 (0)}|u|^2dx \le C \int_{B_1 (0)} |\triangledown u|^2 dx$, where $C \in (0, \infty ) $, $B_1(0) \subset R^d $ and $u \in H_0^1(B_1(0))$. I have few questions , a) how can i via ...
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259 views

technique to find an arbitrary function when solve PDE

I need to solve a transport equation in the form $$ \frac{\partial f}{\partial t} + a(t) \frac{\partial f}{\partial x} = b f + c$$ with $a(t) = A t$. From a reference book I took a solution $f = -c/b ...
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2k views

Advection diffusion equation

The advection diffusion equation is the partial differential equation $$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v \frac{\partial C}{\partial x}$$ with the boundary ...
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286 views

Finding a geodesic on a plane using polar coordinates

This is from my homework on PDE. I need to find a geodesic on a plane using polar coordinates. Now, I know $dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$, but I get stuck while converting ...
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302 views

Wave equation in a medium PDE

Suppose there is a string in a medium that applies a resistant force per unit length proportional to the velocity of the string. How do you write the equation of string vibrations?
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408 views

Change of variables of PDE

I have a particle of mass $m$ that moves in 2-d in the potential $V(x,y)=\frac{1}{2}m\omega^2(6x^2-2xy+6y^2)$. I have to use the coordinates $u=\frac{x+y}{\sqrt 2}$ and $w=\frac{x-y}{\sqrt2}$ to show ...
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1answer
163 views

wave equation with 2 neuman conditions

I am looking for a solution to a wave equation $\frac{\partial^2 u}{\partial \tau^2} = \frac{\partial^2 u}{\partial \xi^2}$ in which $t_c\tau = t$, $L\xi = x$, and $t_c = L/v_c$ is the ...
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131 views

First order partial differential equation question

The population density $u$ of a species at age $y$ and time $t$ is given by the equation $u_{t} + u_{y} = \frac{-u}{(L-y)}$ $t\geq 0$ and $0 \leq y < L$ Initial conditions are $y = 0, u(t,0) = ...