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Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations.

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146 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
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151 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f\left(u\right) = 0, \\ u\big\vert_\Gamma = u_0 $$ by Newton’s method when its convergence is global ...
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355 views

Green Solution to Laplace Equation with Robin Boundary Conditions

Let's say that I know a solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=0\quad \mathbf{x}\in\mathbb{R}^2$$ And I need a solution for the laplace equation in the ...
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226 views

Examples on conceptual problems for eigenvalues in differential equations

I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs. Most of the problems in 5 different books that I ...
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232 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial t}\\\dfrac{\...
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49 views

Using Separation of Variables to Solve a Laplace Eigenproblem

Let $r,\theta$ be the usual polar coordinates in $\mathbb{R^2}$, let $\Omega$ be the unit disc $r<1$ and recall that the Laplacian is given by $$\nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}\...
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63 views

Laplace equation in unit square

I would like to solve the $\triangle u(x,y) = 0$ in the unit square, with periodic BC when $x=0,1$ and Neumann condition when $y=0,1$ $$\partial_y u(x,0) = \begin{cases} A \quad &\text{for } 0\...
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180 views

R.H.S. of Poisson equation localized $\Rightarrow$ Solution localized

Let $\Omega\subset\mathbb{R}^d$ a connected open set (which is not necessarily bounded). Assume that $f\in C_0^\infty(\Omega)$ with $\operatorname{supp}(f)\subset K,$ with $K$ compactand let $u$ be ...
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112 views

Solution to Singular Free Boundary PDE

As part of my research, I have come across the following problem and I am trying to tackle it. Let $(X_t)_{0 \leq t \leq T}$ be a mean controlled Brownian Motion with the following dynamics \begin{...
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151 views

Are there existence results for the heat equation on unbounded Lipschitz domain?

I am looking for a reference/ ideas on the following problem. Let $\Omega\subset\Bbb R^2$ be a Lipschitz domain (if it helps, the domain can be piecewise smooth with only one "kink", for example the ...
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380 views

Confusion about superposition principle of the PDE and Boundary Condition of an ODE.

I want to solve a PDE like this: $\frac{\partial y}{\partial t}=a\frac{\partial ^2y}{\partial x^2}-b\frac{\partial y}{\partial x}-c y,(a,b,c\in \mathbb{R})\tag{1}$ with the boundary conditions: $ ...
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516 views

Developing solution for electrodynamics problem

Although it is a question related to physics, since the point it really matters is its mathematical aspect, I post this question on MSE. There's an additional exercise from Introduction to ...
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117 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D heat equation (Laplace equation) $$\nabla^{(3)}T_s=0$$ where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
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99 views

Partial differential equation with a nowhere differentiable boundary

Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. ...
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49 views

Is there a way to combine solutions of Laplace equation from two different Dirichlet boundary conditions?

Let $\phi_1, \phi_2: \mathbb R^2\to\mathbb R$ be solutions of Laplace equation with Dirichlet boundary: $$ \nabla^2\phi_1 = 0,\quad\quad \phi_1(\mathbf x) = a_1 \quad\mbox{if}\quad\mathbf x\in\Omega_1 ...
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955 views

Weak form of steady Navier-Stokes equations with special boundary condition

Suppose we want to solve the steady low-Mach-number Navier-Stokes equations coupled with a passive scalar $\xi$, which read: \begin{align} \nabla \cdot \rho \mathbf{v} & = 0,\\ \rho \mathbf{v} \...
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106 views

Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
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694 views

Conditions for solvability of Poisson's equation with Neumann boundary condition

Suppose I have: $$\begin{cases}-\Delta u= f, &\text{ on } \Omega\\ \nabla u \cdot n = g &\text{ on } \partial \Omega\\ \int_\Omega u = \operatorname{const}. \end{cases}$$ I'm supposed to ...
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27 views

Curious Question on Inhomogeneous Boundary Conditions of a PDE

I am given the PDE, $\ u_t=u_{xx},$ with boundary conditions $u(0,t)=A, \ u(1,t)=B$ and $u(x,0)=f(x)$. I have found the solution of this PDE is $$u(x,t)=A+(B-A)x+\sum_{n=1}^{\infty}B_ne^{n^2\pi^2 t}\...
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How to solve wave equations with boundary condition $u_x(0,t)=h(t)$?

\begin{align*} u_{tt}-c^2u_{xx}=0, x>0\\ u(x,0)=u_t(x,0)=0\\ u_x(0,t)=\frac{t}{1+t^2},t>0 \end{align*} According to the textbook, I should look for solutions in the form $u(x,t)=F(x-ct)$ and ...
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73 views

Uniqueness of the potential flow past a cylinder

I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \...
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90 views

Converting between Solution forms using Green's Functions in Linear Differential Equation

EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ...
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61 views

PDE Similarity Solutions Boundary Value Problem and Solution: Explanation of Steps Requested

I have the following similarity solutions problem and solution: Problem $u_t = ku_{xx}$ for all $x > 0$, with $u_x (0, t) = 1$, $u(x, t) \to 0$ as $x \to \infty$, and $u(x, 0) = 0$ for $x &...
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33 views

Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem 2

This problem is related to this question. If you can answer this, then you might be able to also answer the other question, so please have a look at it. I am trying to solve the following problem: $$...
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83 views

Heat equation with time-dependent transport term

$\bullet$ I have the following heat equation with a time-periodic transport term: $$\kappa u_{xx} - a \sin(\omega t)u_x = cu_t$$ I'm considering a 1D domain over $-l<x<l$. I'd like to be able ...
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75 views

Estimating the $L^p$ norm of a second derivative of a solution of the Laplace equation

Consider Dirichlet boundary value problem on unit disk : $u_{xx}+u_{yy}=0$. Then, is there a constant $c>0$ satisfying the following? -For every solution $u$ that $\int_0^{2\pi}\left|\frac{d^2u(e^...
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192 views

Heat equation using the Laplace Transform

The Question: The temperature $u(x,t)$ in a semi-infinite conductor occupying $x \in [0,\infty)$ satisfies the equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \qquad x,t&...
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165 views

Galerkin method + FEM - clarification for Poisson equation with mixed boundary conditions

I will be refering to this link, but I am interested in slightly easier equation: $$ -\Delta c = f, \quad (x, y) \in \Omega $$ with the following (mixed) boundary condition: $$ \begin{aligned} 1. &...
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195 views

Solving ODE with Neumann boundary conditions

How can I solve the ODE with boundary conditions? I have solved it without boundary conditions. I have no idea with such boundary conditions. ODE: $${\partial ^2 Y(X) \over \partial ^2 X} = \beta Y(...
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189 views

Green's function of elliptic equations.

I am interested in the following two problems: \begin{equation} \left\{ \begin{array}{ll} - \text{div} A(x)\nabla G(x,y) = \delta(x-y) & \text{in} \ \Omega \\ G=0 & \text{on} \ \partial \Omega ...
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58 views

Numerical solution of a BVP with a boundary value at infinity

Lets say we want to solve $y''=x y$ with boundary conditions $y(0)=1$ and $y(\infty)=0$ I know that this is the airy equation but how would one solve such an equation numerically?
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158 views

fundamental solution of Fokker Plank equation for the Langevin equation of motion

I am studying the original paper by Ornstein and Uhlenbeck on the theory of Brownian Motion. After constructing the Fokker-Planck equation for the Langevin equation of motion, the authors arrive at ...
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477 views

Energy method for one dimensional wave equation with Robin boundary condition

Show that the initial-boundary value problem \begin{align} & {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\ & u\left( x,0 \...
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137 views

How to solve this differential equation with Fourier Transform?

Consider the differential equation $$\dfrac{\partial w}{\partial t} = -\alpha \dfrac{\partial w}{\partial x} + D \dfrac{\partial ^2 w}{\partial x^2}$$ together with the boundary conditions that $$\...
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207 views

solving PDE with state-dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
3
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574 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string satisfies ...
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170 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
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465 views

Analytical solutions of Thomas Fermi equation

The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
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191 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon \bigg(\frac{dy}{dt}...
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30 views

Parabolic PDE with zero Neumann condition

I'm working with the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0$$ Given the Neumann boundary condition above, if I ...
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72 views

Laplace [Heat] type equation with source terms

$$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} =\beta_h e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \, \mathrm{d}x + \beta_c e^{-\...
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107 views

Eigenvalues keep giving trivial solutions everytime.

I am trying to find the eigenvalues of this Eigen BVP. $\mu$ is the eigenvalue parameter $$ \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \...
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37 views

Can this Helmholtz PDE with Robin boundary conditions be solved analytically?

Consider the following Helmholtz problem in the infinite triangle $y>0,\;x>y$ with parameters $Q<0$, $P\ge0$, $P<|Q|$. $$\left\{\begin{align} &\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\...
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0answers
47 views

Weird eigensystem of ODE with regular singularity

Consider the eigenvalue problem of the following 2nd-order ODE $$(x/2+a)^2y(x)-xy'(x)-x^2y''(x)=\lambda^2y(x),$$ in which $y\in(-\infty,+\infty)$ and parameter $a>0$. It has a regular singularity $...
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53 views

Finding the Green Function

I'm having some trouble finding the Green function of the following differential equation: $$ \frac{d[x y'(x)]}{dx} = f(x)\\ 0 \leq x \leq 1\\ y(1) = 0 $$ $y(0)$ is finite.
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Integrating over the domain of influence for the solution to the wave function on the half-line with source

Problem: Find a solution to $$v_{tt}-c^2v_{xx} = f(x,t)$$ $$v(x,0)=\phi(x)$$ $$v_t(x,0)=\psi(x)$$ $$v(0,t) = h(t)$$ on $0<x<\infty$ I know that the general solution to the wave function in the ...
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77 views

Laplace/Poisson-like equation with “interior” holes

I am trying to solve following PDE: $$ \frac{\partial^{2} \theta}{\partial x^{2}} + \frac{\partial^{2} \theta}{\partial y^{2}} - m^{2} \cdot \theta = 0 \\ 0 \leq x \leq a, 0 \leq y \leq b $$ with ...
2
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0answers
58 views

Unidirectional Non-Linear Wave Motion PDE Problem: $u_t + uu_x = 0$, with $u(x, 0) = f(x)$

I'm completely stuck on the following problem: Consider an example of unidirectional non-linear wave motion: $u_t + uu_x = 0$, with $u(x, 0) = f(x)$. i) Show that the characteristic ...
2
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0answers
172 views

Gradient Blowup for a Parabolic (Heat) Equation

Let $u(x,t)$ be a solution to the following parabolic PDE: With $\alpha \in (0,1)$, \begin{align} \partial_t u(x,t) &= \alpha (1-t)^{\alpha - 1} \partial_x u(x,t) + \frac{1}{2} \partial_{xx} u(x,...
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votes
0answers
28 views

Construction of a bounded map which is the identity or the null operator on the space of square integrable functions using the Helmholtz decomposition

Let $\Omega$ be a bounded open subset of $\mathbb{R}^3$ and $n$ the outward-pointing unit normal to the boundary of $\Omega$. We know that the space $(L^2(\Omega))^3$ of square integrable functions ...