# Questions tagged [pde]

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

14,408 questions
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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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### 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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### 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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### 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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### 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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### Positive domain of the Cauchy problem

If I consider the standard Cauchy Problem in one-dimension of the form: \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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\phi\in C_c^\... 1answer 124 views ### Self-similar solution of the momentum equation I have used the steam functions$u = \psi_{y}$and$v = -\psi_{x}$to transform the momentum equations to the following form $$\rho\left(\psi_{y}\psi_{xy} - \psi_{x}\psi_{yy}\right)=-p_{x}+\mu\left(\... 0answers 32 views ### Green function of a 3D wave equation I was trying to find green function of a 3D wave equation using Fourier transform method.$$\left(\nabla- \frac{\partial^2}{\partial t^2} \right)G(\bar{x},t) = \delta(\bar{x})\delta(t)$$Using ... 0answers 13 views ### What does ordered pair notation mean in the context of Generalized Fourier Series? [closed] I remember my professor using this in lecture and understanding it at the time but I forgot and it came up on the last test and I didn't remember what it was, and I can't find it anywhere in my notes, ... 0answers 32 views ### Uniqueness of solutions to nonhomogenuous heat equation I'm working on an old problem from a previous exam. Suppose \alpha \geq 0 and T>0 be given. Prove uniqueness of classical solutions to the initial boundary value problem \begin{cases} ... 0answers 40 views ### Is a partial differential equation satisfied after reduction to a subspace? I have a nth-order non-linear partial differential in m-real variables x_1,x_2, \ldots, x_m. Assume a function f satisfies this differential equation. I denote this by$$D f(x_1, x_2, \ldots, ... 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 \...
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 ...