Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations. By a Boundary value problem, we mean a system of differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.

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Find curve that minimizes lenght, with integral constraint

I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ...
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Effect of boundary conditions on general solution

I am having problems integrating given boundary conditions on a wave-equation. The problem is as stated below. I am no expert in solving PDE's, so please forgive if I oversee something obvious or &...
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Show that is continuous and coercive

I have the following boundary value problem, where $k, q \in \mathbb{R}$ are given. $$-u'' + ku' +qu = f,\qquad u(0)=u(1)= 0$$ What I want to prove is that the bilinear form, associated with this BVP,...
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Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$

Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$ Without the last condition, we have $u = y.$ I'm trying to prove that if in addition $u>0$ on the upper half-...
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Parabolic Pde with unkown boundary conditions

I have the following parabolic partial differential equation: \begin{equation} \frac{\partial^2 \phi}{\partial x^2} - \alpha \sin{x} \frac{\partial \phi}{\partial t} + \beta(\cos{x} - \gamma) \phi = ...
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1 answer
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Lax-Milgran theorem with the Poisson equation

Let $ \Omega $ be a bounded domain in $\mathbb R^3$ with smooth boundary. Consider the Poisson equation $$ -\Delta u=f $$ where $ f\in C_0^{\infty}(\Omega) $ and $f$ is null outside $\Omega$. I'm not ...
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A boundary value problem of a Harmonic potential

A 2D electrostatic (i.e. harmonic potential) boundary value problem is shown in the figure. The solid lines are conductors (all are parallel), the two conductors with potential $V$ are infinitely long,...
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1D Transient Heat Equation with an Inhomogeneous Boundary Condition

I am trying to solve the one-dimensional transient heat equation with a specified flux in one end ($x=0$) and perfect insulation on the other ($x=L$): $$\frac{\partial T(x,t)}{\partial x} = \alpha \...
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Neumann B.C with an extra term in Matlab

I have written a heat equation with the Neumann boundary condition. I used heat equation with Neumann B.C in matlab. Now how can I add the term $\beta A \frac{\partial u}{\partial x}$ to the equation ...
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reference for poisson equation on unbounded domain

Let $\Omega\subset\mathbb{R}^3$ be some unbounded exterior domain, with $C^{\infty}$ smooth boundary. Consider the Poisson equation with Neumann boundary condition \begin{align} &\Delta u = f &...
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Neumann & Dirichlet heat kernel resp. semi-group on compact manifolds

Given a smooth compact Riemannian manifold $(M,g)$ with boundary $\partial M$. Consider the initial & boundary value problem of the heat equation $$ \tfrac{\partial}{\partial t} u(x,t) = \Delta u(...
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Inhomogeneous Boundary Value Problem. How would I solve?

The heat equation: $$\begin{align} \frac{\partial u}{\partial t} &= {9} \frac{\partial^2 u}{\partial x^2}\,, \qquad 0<x<{3}, \quad t \gt 0\, \\ \end{align}$$ Has boundary and initial ...
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Existence of Weak Solution of the Neumann Problem with non zero boundary data

I need to show that there is a unique solution modulo constants to the following Neumann problem: $\Delta u=0$ in $\Omega$ and $\frac{\partial u}{\partial \nu}=h\in H^{-1/2}(\partial \Omega)$ where $\...
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Using D'Alembert's formula

I am trying to solve the boundary value problem $u_{tt} = 4u_{xx},\; x > 0, t > 0$ $u(x,0) = \frac{x^2}{8},\; u_t(x,0) = x,\; x\geq 0$ $u(0,t) = t^2,\; t \geq 0$ I attempted to use D'Alembert's ...
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A condition on a fifth order differential equation

This may be very obvious but I am stuck trying to solve a boundary value problem. I am trying to solve the following differential equation : $$F^{3}F^{(5)}+F=1 \space (Eq.1)$$ I have been advised to ...
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Application for the trace inequality.

I know two theorems about the trace inequality. Suppose that $\Omega$ is a bounded domain with smooth boundary. One is that: $$ \gamma_0(H^1(\Omega)) = H^{\frac{1}{2}}(\partial \Omega) $$ where $\ \ ...
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Solving a Fourth-Order Linear Homogeneous Differential Equation

I like to solve the ordinary fourth order homogeneous differential equation given by $\displaystyle \frac{d^{4}\theta}{d z^{4}} + \lambda \cdot \theta = 0$ with a constant coefficient $\lambda$. ...
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1 answer
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Need help - Robin condition for a 1d wave equation on the first quardant

Given $\alpha \neq 0$ and $$u_{tt}-c^2u_{xx} =0 \quad x,t>0,$$ $$u(x,0) = f(x), \quad x\geq 0,$$ $$u_t(x,0)=g(x), \quad x\geq 0,$$ $$u_x(0,t)+\alpha u(0,t)=0, \quad t\geq 0.$$ I know it’s a Robin ...
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How to solve this 2nd ODE with simple inequality constraints numerically?

I have a 2nd ODE (derived from an elastic rod deflated naturally under its self-weight): $$ y'' + K (x - 1) \cos(y) = 0 $$ K is a constant coefficient. The variables range between $x \in [0, 1], y \...
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1 answer
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Solving BVP with variable coefficient on MATLAB

I have the second order boundary value problem $$ y''=\frac{k}{2\sqrt{t}}\left[Py'-2yy'-y(P-y))\frac{1}{2t} \right], $$ with boundary conditions I will specify later in the code. Setting $y=y_1$ and $...
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Small parameter expansion of solution to ODE

I'm working with an ODE of the form \begin{equation} \begin{split} C'(z) &= (z^2 + \epsilon^2)C'' \\ C(a) &= b \\ C(1) &= 1 \\ \end{split} \end{equation} where $0 < a,b < 1$, and $\...
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Solving a BVP with the wave equation by using the Laplace Transform

What follows is an example on how to solve a BVP using the Laplace Transform. I will copy and paste the example and its solution from the book that I am using and then I will ask my question. When ...
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Are BVP and IVP interchangable?

My question is: can the same differential problem (PDE, Action minimization...) be treated as a Boundary Value Problem or as an Initial Value Problem, depending on the nature of the constraints I ...
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2 votes
1 answer
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How to tell if PDE boundary conditions are homogeneous or not?

Say we have Laplace's equation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$ These are the boundary conditions. Now, to solve this by separation of variables, we need ...
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2 answers
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Solving a 2nd Order ODE using Finite Difference Method when Mixed Boundary Conditions are given

The problem I'm looking at is $$y'' + 3.05 y' -2.85 = 0 $$ with the boundary conditions $y(0) = 1$ and $y'(1) = 0.0305$. After obtaining the algebraic set of equations using FDM, I'm not sure how the ...
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Regularity for elliptic boundary value problem on the Sobolev space $H^{2,1}(M)$ - Literature request

I am looking for some literature, where the following regularity result is treated: Let $\overline{M}$ be a compact smooth manifold with boundary of dimension $3$, $M$ the interior of $\overline{M}$ ...
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Laplace's equation with the inner unclosed edge

Hello I'm Owen and I'm new here. Recently I was working on a boundary element problem about the Laplace equation. The basic Laplace equation is as follows: $$ \left\{\begin{array}{ll} \Delta \varphi=0 ...
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Solving second order differential equation with finite difference method (boundary conditions)

Hello I am trying to solve this second order differential equation: $$r\frac{d^{2}y}{dr^{2}}+\frac{dy}{dr}=0$$ where $0\le r\le 1$, with the following boundary conditions: $$r=0, y=200$$ $$r=1, y=36$$ ...
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1 answer
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Finite difference method for periodic ODEs

I've been learning about how to solve ODEs using the finite difference method by reducing them to systems of linear equations, like this: $$y^{\prime}(x)=y(x)^2+4y(x)-12$$ This can be reduced to a ...
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Exact solution of a BVP of second order

Im solving a BVP which is $y^{\prime\prime}(t)=-y^{\prime2}(t)+y(t)(y^{2}(t)-\frac{3}{2}y(t)+\frac{1}{2})$ with boundary conditions $y(0)=1$ and $y(1)=2$. I need to find the exact solution for this ...
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3 votes
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Solving 1D heat equation with IC and BCs

Suppose I have the heat equation, with IC and BCs: $${\partial T \over{\partial t}}=k{\partial^2 T \over{\partial x^2}}$$ $${\partial T \over{\partial x}}(0,t)=0, \hspace{5mm}T(L,t)=B$$ $$T(0,0)=A, \...
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Definition of Energy Functional

I am currently learning about boundary value problems and how to calculate their weak formulations. I came across something called "Energy Functional" some call it "The Energy of a ...
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2 votes
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Heat equation with Dirichlet and Neumann boundary conditions

The Problem Let $\Omega\subseteq\mathbb{R}^d$ be open and bounded with $C^2$-boundary. Let $T>0$ and $u\in C^{1,2}\left((0,T)\times\Omega\right)$ be a solution of the heat equation $$\partial_t u-\...
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1 vote
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How to solve a set of PDEs derived from elasticity?

The problem is discribed firstly, and a possoble strategy which might work (yet don't know how exactly) is suggested. How to gain an analytical solution? Suggestion on numerical method is equally ...
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2 votes
1 answer
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Solving Simple PDE by Green's Function, Very Confused By Some Mistake

Suppose I want to solve $u_{xy} = xy$ via Green's Function. This will correspond to the associated PDE $G_{xy} = \delta(x - x_G,\ y - y_G)$, and I want my boundary conditions for this Green problem ...
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  • 1,666
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Showing that a solution actually solves the Dirichlet-Poisson PDE problem

I'm looking at the "Dirichlet-Poisson problem" section of this Wikipedia page. It writes that, under proper assumptions, $$u(x) = \mathbb{E}[g(X_{T_{D}})] + \mathbb{E}\left[\int_{0}^{T_{D}}f(...
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1 vote
2 answers
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Boundedness of solutions to an ODE and their derivatives

The Scenario: Consider the ODE given by $$ \left\{ \begin{alignat*}{99} &-g'' + g = F(x,f(x)) \qquad &&x \in [0,L] \\ &g(0) = g(L) = 0 \end{alignat*} \right. \tag{1} $$ where: $F$ is ...
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2 votes
3 answers
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What rationale am I missing with this simple boundary value problem?

Consider $$u_{xy} = xy,\ u(0,\ y) = 0,\ u_x(x,\ 0) = 0$$ Antidifferentiating the PDE with respect to $y$ yields $u_x = \frac{xy^2}{2} + a(x)$, and antidifferentiating this equation with respect to $x$ ...
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Any resources/bibliography for periodic solutions and periodic boundary conditions of a PDE?

can anyone recommend good books and/or articles for the mathematical analysis of periodic solutions of a PDE, for example something like $$u(t)-u''(t)=f(t) $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ ? ...
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Boundary Conditions for a system of PDE

Given a system of following PDEs: $$ u_{x} + v_{y} + 3u-v=0 \\ u_{y} - w_{x}+uw=0 \\ v_{x}-w_{y}=0 $$ I found that the given system of equations is of mixed elliptic-hyperbolic type with the ...
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Second order of boundary value problem system using repeated Cauchy integral?

Let $(S)$ be the folowing boundary problem such that : $$(S)\begin{cases} y''=f(x,y(x)), \quad a<x<b \\ y(a)=\alpha,\quad y(b)=\beta \end{cases}$$ We call $ y\in C^2([a,b],\mathbb{R})$ every ...
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Green's Function Computation

I want to calculate Green's Function to solve $\triangle u = f(x,\ y)$, using Laplace Transforms. My plan was to tailor boundary conditions to the problem which simplify the computation. Because the ...
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  • 1,666
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Finding Boundary Conditions That Make This Green's Function Nice

This question used to be included here as a second point, but I figured the two were better asked as two separate questions. When people use the phrase "the ____ operator is self-adjoint" (i....
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Sufficient condition for existence of Neumann problem for Laplace equation.

First, which of the following, concerning the solution of the Neumann problem for the Laplace equation, $$\Delta u=0~ \text{on}~~\Omega, \frac{\partial u}{\partial n}=f(x,y),\text{on}~\partial\Omega$$ ...
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1 vote
1 answer
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Steady-state heat conduction in a cylinder with discontinuity in thermal conductivity

Consider a solid cylinder of length L and radius a in which the thermal conductivity has a jump discontinuity at a point along its axis. The two bases of the cylinder are maintained at zero ...
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Show that $y$ satisfies Integral equation.

Let $y$ satisfies the boundary value problem $$y''(x) + \lambda y(x) = 0, 0 < x < 1,$$ $$ay(0) = by(1), by'(0) = ay'(1),$$ where $a, b$ are constants. Show that $y$ satisfies the integral ...
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Laplace Transform and Fourier Transform With Neumann Boundary Conditions

Suppose I'm doing a Laplace Transform or Fourier Transform with respect to $t$ to solve a PDE, and I have Neumann (partial derivative) boundary conditions, such as $$u_t(a, t) = f(t)$$ Can I just ...
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  • 1,666
0 votes
1 answer
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Finite elements method: why test fuction vanishes on boundaries

I am trying to understand why (and exactly when) the test functions must vanish at the boundaries when Dirichles conditions are applied to a PDE. The context is the learning of the finite element ...
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1 vote
1 answer
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Biharmonic problem with boundary conditions on Laplacian

Let the problem, where $\Omega$ is an open set of $\mathbb{R}^3$ and $h_1$ and $h_2$ are regular given functions \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \left\{ \begin{array}[c]...
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  • 441
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Numerical treatment of radiation boundary condition for 1D implicit heat conduction

I have a question regarding the $\textbf{radiation boundary treatment}$ for the 1D heat conduction equation using the $\textbf{implicit}$ finite difference method. If I use the $\textbf{explicit}$ ...
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