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Questions tagged [pde]

Questions on partial (as opposed to ordinary) differential equations.

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$u_x(x,y) + 3u_y(x,y) - (3x-y)u(x,y) = -2 $

Question : I am willing to solve the PDE :$$u_x(x,y) + 3u_y(x,y) - (3x-y)u(x,y) = -2 $$ Thoughts : The characteristics are : $$\frac{\mathrm{d}x}{1} = \frac{\mathrm{d}y}{3} \implies u_1 = 3x-y$$ ...
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5 views

Derivation of weak formulation for surface Laplacian

For the surface Laplacian $\Delta_S= \nabla_S \cdot \nabla_s $ and a twice differentiable function $u$ on the surface $S$ we have $$\Delta_S u=\Delta u-\kappa\frac{\partial u}{\partial n}-n^TH(u)n,$$ ...
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Integrate the canonical form of a second order PDE

I'm given the following PDE $$u_{xx} - 2\sin x u_{xy} - (3 + \cos^2x)u_{yy} + u_x + (2 - \sin x - \cos x)u_y = 0$$ with conditions $$u(x, \cos x) = 0, \quad u_y(x, \cos x) = e^{-x/2}\cos x.$$ It's a ...
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14 views

Gradient estimate and divergence theorem in an open ball (PDE)

I need to prove the estimate $|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$ where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$ I found this ...
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A question regarding the proof of the second existence theorem for weak solutions of elliptic equations

A question arised when I was self-studying the PDE book of Evans. Let $Lu = - \sum _{i,j=1}^n (a^{ij} u_{x_i} ) _{x_j} + \sum_{i=1}^n b^{i} u_{x_i} + cu$ be a uniformly elliptic operator on a bounded ...
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1answer
30 views

Finding solution of semi-linear PDE using Method of Characteristics

I am given the PDE: $$u_x+u_y+u=e^{x+2y} \quad u(x,0)=0$$ I tried to do this using the method of the characteristics in the following way. First I find the particular solution of this inhomogenous ...
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19 views

Does there exist any solution for this inequality?

Let $\Omega,\Omega^*$ be disks in $\mathbb{R}^2$, such that $\Omega^*\subsetneq\Omega$ and their boundaries meet at one point (so they are tangent at that point; consider $N((1,0),1)$ and $N((2,0),2)$ ...
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10 views

Positive domain of the Cauchy problem

If I consider the standard Cauchy Problem in one-dimension of the form: \begin{equation} \begin{cases} a(x,t) \frac{\partial^2u}{\partial^2x} + b(x,t)\frac{\partial u}{\partial x} + c(x,t) u - \frac{...
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31 views

Is the nonlinear Schrödinger equation solved?

Consider the following initial value problem: $i\psi_t = -\psi_{xx} - 2|\psi|^2\psi$ with $x\in[0,2\pi)$ and $t\geq 0$, and $\psi(x,0) = \frac{3}{2}\left(1 - \frac{1}{10}\cos(x-\pi)\right)$. The ...
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6 views

Convergence of a truncated sequence

Let $k>0$ and define the truncated function $T_k(u)=min\{u,k\}$ for a positive $u\in H_{0}^1(\Omega)$ where $\Omega$ is a bounded domain. Let $M$ be a symmetric matrix satisfying $$ |M(x)\leq \...
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1answer
17 views

Phase shift between two oscillating solutions of equal period

Given is an oscillating function $y(t)$ of period $T$ with $y(t) \, \ge \, 0 \, \forall \, t \in \Bbb R$. Consider the differential equation $\frac{\partial g}{\partial t}(t) = a f(y(t)) - \gamma g(...
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Numerical Integration with Green's Function

I'm trying to solve the 1D heat equation on a semi-infinite domain with no heat sources and inhomogeneous boundary conditions: $$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2T}{\partial x^2}$$ ...
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17 views

Heat equation for a torus in cylindrical coordinates

I'm trying to solve an old exam problem and it is just one thing in the separation of variables which confuses me. I'm supposed to find the temperature inside a torus which I'm supposed to view as a ...
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Proving that a specific Volterra integral operator is not positive

I want to prove that the operator $$ A: L^2[0,1] \to L^2[0,1], \quad A(u)(s) = \int_0^1 |t-s| u(t) dt $$ is not positive, i.e. $\langle Au, u \rangle \geq 0$ does not hold for every $u \in L^2[0,1]$. ...
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20 views

Weak solution to a parabolic equation

I am facing problem to understand the notion of weak solution defined in page 3 of the article attached below: https://math.aalto.fi/~jkkinnun/papers/kuusi.pdf My first question is: 1) On page 3 can ...
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18 views

Reaction-diffusion equation involving product of convection terms

I was transforming a system of PDEs to simplify my analyses. In particular, I changed the two variable $(a,b)$ system into a $(u,v)$-system with $v$ being $a+b$ and $u$ being the fraction $\frac{b}{a+...
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1answer
43 views

Numerical Methods with MATLAB - A linear hyperbolic system

I am studying Numerical Methods for Conservations Laws with MATLAB by first time and I've tried to follow an example and calculate the solutions for the following Riemman Problem: $$\begin{bmatrix}u\\...
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28 views

Is the 1D Heat Equation with Dirichlet and Neumann Boundary Conditions on the Same Side Well Posed?

If I take the 1D heat equation \begin{align} \frac{\partial u}{\partial t}=\kappa \frac{\partial^2u}{\partial x^2} \end{align} on some finite interval, say $0<x<1$, and then specify Dirichlet ...
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33 views

Formal solution diffusion equation

I'm not a mathematician, so please bear with me if I write things down in a non-rigorous manner. I read in either a mathematical finance or physics book (can't remember) that the formal solution of a ...
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1answer
36 views

Local systems as solution of PDE

I want to understand the correspondence between locally constant sheaves and vector bundles with flat connection on a manifold $X$. Given a local system $\mathcal L$, it is clear how to define a ...
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39 views

Unstable numerical solution of ODEs and PDEs

Choosing the right step size for a stiff ODE or a non-linear ODE and PDEs is an important factor. While studying a paper on choosing appropriate step size in numerically solving ODE, I questioned: ...
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Quadratic finite element in 2d: Calculation of the right hand side

I'm trying to implement a solver for the 2d heat equation using FEM. I already having a working example for linear elements, there the approximation for the right hand side $f$ is calculated via a ...
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72 views

How to solve the heat equation?

We consider the following initial-boundary value problem: \begin{cases} \begin{align} u_t-u_{xx} &= -u^{q} & x \in \Omega, t > 0 \newline \dfrac{\partial u}{\partial\nu} &= u^{p} & ...
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1answer
21 views

Gamma ($\Gamma$) Convergence of Functionals

Consider a set $X$, and consider a sequence of functionals on $X$, that is maps $F_n: X \to \mathbb{R}$. We say that $F_n$ "$\Gamma$" converges to $F$, if the limit satisfies: $F(x) \leq \inf\{\...
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41 views

Is zero a solution for PDE (D'Alembert Formula)

I have the PDE: $u_{tt}=u_{xx},\quad 0<x<1,\,0<t<\infty \\ u(0,t)=0 \\ u(1,t)=\sin\,t \\ u(x,0)=u_t(x,0)=0,\quad 0<x<1$ When i used the D'Alembert formula i got $0$ because $f(x)$ ...
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47 views

Questions about fourier transformation

I understand to purpose of the fourier transformation is to transfer a PDE into an ODE. By solving the ODE and then take the inverse transform, I can get the answer. I have the following PDE ($\phi $ ...
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57 views

Should I worry about closed form solutions in research?

I am an MSc student studying operations research (specifically bioinvasions) and it seems that a lot of the mathematics in this area focuses quite heavily on PDE's, Optimal Control Theory, and ...
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25 views

How to prove the existence of u?

Given $T \in D^{'}(R^{1})$,prove the existence of $u$, satisfying $\frac{du}{dx}=T$. I totally have no idea of how to prove this. Could anybody give me a hint? Remark:$D^{'}(R^{1})$ is the dual ...
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Getting rid of non-linearity of PDE to use Hopf's Lemma

I'm reading through Partial Differential Equations by Lawrence C. Evans and am having an issue understanding a part of a proof regarding Hopf's Lemma. The version of Hopf's lemma proved in 6.4.2 ...
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1answer
27 views

Is the homogeneous BCs have no effect for solving PDE with D'Alembert?

Suppose i have wave equation with 2 BCs (Boundary Conditions), one of them is Non homogeneous and 2 homogeneous ICs (Initial Conditions). If I have a non-homogeneous boundary condition do I have to ...
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23 views

Wave Equation with boundary condition?

Given PDE with BCs and ICs $u_{tt}=u_{xx},\quad 0<x<1,\,0<t<\infty \\ u(0,t)=0 \\ u(1,t)=\sin\,t \\ u(x,0)=u_t(x,0)=0,\quad 0<x<1$ I've read the textbook about wave equation and ...
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17 views

Solve diffusion equation with mixed Neumann and Dirichlet boundary condition [on hold]

I have a question about solving the heat equation with Mixed Dirichlet - Neumann boundary condition. That is: Given $\frac { \partial T } { \partial t } = D _ { 0 } \frac { \partial ^ { 2 } T } { \...
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Apde: Clarification to non-homogenous equations: SOV

would be appreciative if someone could just double check my understanding a little here. so when solving the general 1 dimension Heat equation IBVP: we do so as follows We'll solve a General ...
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Solving PDE produced for $C^1$ isometric embedding and its complexity analysis

I am a beginner in differential geometry and I am investigating $C^1$ isometric embedding of Riemannian manifolds to Euclidean space for computer vision. I know that the PDE produced for a metric $g$ ...
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11 views

General criterion on uniqueness of elliptic pde

Is there any general condition on function $c:\Omega\rightarrow\mathbb{R}$ that serves $\Delta u+cu=0$ has only trivial smooth solution on $u|_{\partial\Omega}=0$, where $\Omega$ is a closed disk, not ...
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1answer
38 views

2D heat equation with BC as zeroes ( Solved completely: no need to answer this)

I need help with this question, I dont know what is wrong with this problem or is there any typo in the original question, By using $$u(x,y,t)=X(x)Y(y)T(t)$$ I am getting $$X(x)=0$$ which yields $$u(x,...
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1answer
20 views

Show that a function is a solution to a differential equation

My problem Given $$x(t)=x_Te^{-a(T-t)}-\int_t^T e^{-a(v-t)} b\,dv$$ , $$x_T=x(T)=constant$$ Show that x(t) is a solution to $$\dot{x}(t)=ax(t)+b $$ a and b are constants My attempt to solution $$\...
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0answers
26 views

Generalized maximum principle

I search an reference where wi found the following generalized maximum principle let $G$ be a bounded domain, $u$ be a positive solution of the Dirichlet problem for the Poisson equation $\Delta u =f$...
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0answers
21 views

Eigenfunction Expansion and Fourier Series

What is the difference of Eigenfunction Expansion solution and Fourier Series solution in solving Partial Differential Equation? Are they the same? Please to tell me about what it is the eigen value ...
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1answer
51 views

Existence of smooth vector field.

Let $\Omega$ be a smooth domain in $\mathbb R^n$ and let $x\in\partial\Omega$. Let $n(x):\partial\Omega\to\mathbb S^{n-1}$ be the normal vector field. Then there exist a vector field $\phi\in C_c^\...
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1answer
124 views

Self-similar solution of the momentum equation

I have used the steam functions $u = \psi_{y}$ and $v = -\psi_{x}$ to transform the momentum equations to the following form $$\rho\left(\psi_{y}\psi_{xy} - \psi_{x}\psi_{yy}\right)=-p_{x}+\mu\left(\...
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32 views

Green function of a 3D wave equation

I was trying to find green function of a 3D wave equation using Fourier transform method. $$\left(\nabla- \frac{\partial^2}{\partial t^2} \right)G(\bar{x},t) = \delta(\bar{x})\delta(t)$$ Using ...
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13 views

What does ordered pair notation mean in the context of Generalized Fourier Series? [closed]

I remember my professor using this in lecture and understanding it at the time but I forgot and it came up on the last test and I didn't remember what it was, and I can't find it anywhere in my notes, ...
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32 views

Uniqueness of solutions to nonhomogenuous heat equation

I'm working on an old problem from a previous exam. Suppose $\alpha \geq 0$ and $T>0$ be given. Prove uniqueness of classical solutions to the initial boundary value problem \begin{cases} ...
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40 views

Is a partial differential equation satisfied after reduction to a subspace?

I have a $n$th-order non-linear partial differential in $m$-real variables $x_1,x_2, \ldots, x_m$. Assume a function $f$ satisfies this differential equation. I denote this by $$D f(x_1, x_2, \ldots, ...
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1answer
33 views

Question about one step in the proof of the weak maximum principle for the heat equation

I'm confused about one step in the proof of the weak maximum principle for the heat equation in McOwen. Theorem (Weak Maximum Principle): Let $u\in C^{2;1}(U)\cap C(\overline{U})$ satisfy $\Delta u \...
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35 views

Evans pde proof of global approximation theorem

I have a question regarding step 3 of the proof (shown in picture below). I am wondering if there is a need for us to consider the subset $V\Subset U$ instead of directly using $U$ when showing the ...
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16 views

Inhomogeneous Lie derivative equation on a Lie group

Let $G$ be a connected Lie group and let $\xi_i$, $i=1,...,n$ be a basis of its Lie algebra (say, of left-invariant vector fields). We let $B_i$, $i=1,...,n$ be given symmetric sections of $TG \otimes ...
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Green's formula for tangential operators - How is this equality being deduced?

Source: http://www.diva-portal.org/smash/get/diva2:652933/fulltext01.pdf On page 11 it says: For tangential operators Green's formula becomes $$(\nabla_{\Sigma}\cdot w ,v)_{\Sigma}=(n_{\Gamma}\cdot ...
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11 views

(PDE) Parabolic Equation and their BCs and ICs

I have this PDE with the BCs and ICs $ u_t=4u_{xx} \\ u(0,t)=u(2,t)=0\quad \text{For All}\quad t>0 \\u(x,0)=f(x) \\ f(x)=10x,\quad x<0,5 \\ f(x)=8,5-7x,\quad 0,5\le x<1,5 \\ f(x)=-8+4x,\...