Questions tagged [linear-pde]

Question 1

I am trying to rigorously derive the diffusion equation, given by $$ \frac{\partial u}{\partial t} = D\,\frac{\partial^2 u}{\partial x^2}, \qquad D = \frac{h^2}{2\tau}. $$ from a simple one-...

Question 2

The form $$\Phi_s(p)= \int_0^\infty e^{-px} e^{-s/x} \, dx = 2\sqrt{\frac sp} K_1(2\sqrt{sp})$$ is a standard representation for the $K_\nu(\cdot)$ Bessel function ($\nu=1$). It appears in analytic ...

Question 3

The definition I have seen for an elliptic differential operator (say with constant coefficients) is that its principal symbol (which corresponds only to the highest order terms) only can vanish at 0. ...

Question 4

Suppose I start a Brownian motion $X_t$ at position $(0,0,\varepsilon)$ in $\mathbb R^3$. Suppose also I have a cone with its apex at the origin, given by $U=\{(x,y,z):\sqrt{x^2+y^2}< z\tan\alpha\}$...

Question 5

I have the following pair of first order PDEs for the unknown function K(x,y), where f(x,y) and g(x,y) are known functions, and both x and y are spatial variables: $$ {\frac {\partial }{\partial x}}K \...

Question 6

Consider a two-dimensional Minkowski spacetime patch described by light-cone coordinates $(U,V)$ within the domain $(0,1)^2$. The leaves of our foliation are defined by the relation: $$ V(U) = e^{s/...

Question 7

I am currently faced with solving a linear second-order PDE of the form $$ v_{xy} + cv = 0, $$ where $v(x,y)$ is a function of both $x$ and $y$ and $c>0$ is a constant. Note: I use $ v_{xy} $ to ...

Question 8

I am studying the 3D wave equation $$u_{tt}=\nabla^2u$$ in an arbitrary volume $V$ with bounary $\partial V$ and I was wondering if there is an analogue of the formula we get for the Laplacian $$\...

Question 9

Edit: I had to slightly change the question so that it's a little more focused on one single issue. During my undergraduate years, I studied my fair share of functional analysis (Lebesgue spaces, ...

Question 10

Let $\Omega$ be a connected open subset of $\mathbb{R}^n$ and let $\partial \Omega$ denote its boundary. Let $\boldsymbol{n}$ be the outward unit normal on $\partial \Omega$ and let $T > 0$. ...

Question 11

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain, possibly with smooth boundary. I am aware of the following classical embedding: $$L^2(0,T; H^1(\Omega)) \cap H^1(0,T; H^{-1}(\Omega)) \...

Question 12

I want to ask about the uniqueness of functional which is able to produce E-L equation in the form of $$-\Delta u+D \varphi \cdot D u=f.$$ The answer here said that the energy functional of $$ -\Delta ...

Question 13

While studying PDE, specifically the Lagrange equation, the doctor wrote this equation and asked us to solve it. I've tried to solve it many times and failed, so I'm wondering if it has a closed form ...

Question 14

All linear scalar fields $f : \mathbb R^n \to \mathbb R$ are solutions to the following PDE: $$\langle \nabla f_p,\ p \rangle = f(p)$$ for every $p \in \mathbb R^n.$ (A more general statement can be ...

Question 15

Disclaimer: This PDE arose from a physics problem where the boundary conditions are not very well defined and I am assuming them from the symmetry of the system. I am trying to solve the following PDE ...

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Questions tagged [linear-pde]

Derivation of Diffusion Equation in 1-D

Bessel satisfies linear pde but can't find any references for it

Why is the heat operator not elliptic?

Boundary value problem for Laplace's equation / probability a BMP leaves cone via the ice cream rather than the biscuit

Pair of first order PDEs for one dependent variable (with integrability condition)

Real analytic subclass of solutions to a linear PDE?

Linear Second-Order PDE

Formal Solution to Wave Equation with arbitrary Boundary Conditions

Do the solutions of the Heat Equation form a closed subspace? Why do Fourier series work?

Source term with boundary condition for a PDE

Embedding of $L^2(0,T; H^1(\Omega)) \cap H^1(0,T; H^{-1}(\Omega))$ into $L^\infty(0,T; H^s(\Omega))$ for bounded domains

Uniqueness of type of functional which is able to produce E-L equation in the form of $-\Delta u+D \varphi \cdot D u=f$.

How to solve $(5z+4y)z_x+(4x-2z)z_y=2y-5x$ using Lagrange's method?

A PDE that characterizes linear functions?

Difficulty applying boundary conditions to PDE using the method of characteristics

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