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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12 views

Non-Linear ODE with some consideration

π‘Ž(𝑑)π‘Žβ€³(𝑑)+π›Όπ‘Žβ€²(𝑑)βˆ’π›½π‘Ž(𝑑)π‘Žβ€²(𝑑)βˆ’π›Ύπ‘Ž(𝑑)2+πœ‰=0 π‘Ž(0)=0π‘Žβ€²(0)=0π‘Žβ€³(0)=0 All constant is positive and π‘Ž is just depending on time. This equation I have although is a little bit weird but I feel ...
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1answer
27 views

Do solutions to the forced Stokes equations for zero-Reynolds number flows always exist?

I'm in the process of solving a forced Stokes flow problem in two dimensions, the governing equations of which are given by $$ \nabla p = \eta \Delta \mathbf{v} + \mathbf{f}, \quad \nabla\cdot \mathbf{...
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1answer
22 views

Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?

My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
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20 views

Discretization of the Neuman boundary condition.

I am trying to derive the approximation $$ u_x(0,t) \approx \frac{-3u(0,t) + 4 u(h,t) - u(2h,t)}{2h} $$ and $$ u_x(1,t) \approx - \left(\frac{-u(1-2h,t) + 4 u(1-h,t) - 3u(1,t)}{2h}\right) $$ I ...
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32 views

What is the period of the solution of a wave equation boundary value problem

In my studies of numerical PDEs, I was given this problem We consider a vibrating string that satisfies the wave equation $u_{tt}=u_{xx}$ on the unit interval with boundary conditions $u(0,t)=0$, $u(...
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1answer
35 views

Are these boundary conditions consistent $y(x,0)=\cos(\pi x)$ and $y(0,t)=0$

I was asked to solve $$\dfrac{\partial^2 y}{\partial t^2}=9\dfrac{\partial^2 y}{\partial x^2}$$ with the boundary conditions $y(x,0)=\cos(\pi x)$,$\dfrac{\partial y}{\partial t}(x,0)$ and $y(0,t)=y(4,...
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16 views

Existence of weak solutions to initial value problems of second-order hyperbolic differential equations on unbounded domains?

Evans gives a theorem for the existence of weak solutions for second-order hyperbolic differential equations. Specifically, he says if $U$ is an open, bounded subset of $\mathbb{R}^n$, we call $U_T = (...
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1answer
50 views

Why is this operator called a lift in a PDE context?

Let $L$ be some differential operator and consider the PDE $Lu = 0$ with some boundary conditions. Let $W$ denote the space that solutions live in and $B$ the space that boundary conditions live in. ...
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37 views

Holomorphic function with given boundary values

I have a complex valued function on the boundary of a set, and want to know if it has a holomorphic extension to the entire set (holomorphic in interior, continuous up to the boundary). In other words:...
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14 views

finding variational equivalent of a BVP

Find the variational equivalent of the BVP $$-au''(x)+xu'(x)+u(x)=f(x)$$ subject to $u(0)=u'(L)=0$ . We start with $$\delta\int_0^Lfudx=\int_0^Lf\delta udx=\int_0^L(-au''+xu'+u)\delta udx$$ By parts ...
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2answers
33 views

Solving a wave equation with boundary conditions on a circle?

How might one solve a wave equation which had boundary conditions on a circle? i.e. given $$(\partial^2_x+\partial^2_y)\phi(x,y)=0$$ And known values, $f$ on a circle: $$\phi(\cos(\theta),\sin(\theta))...
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23 views

How to solve a boundary value problem?

for this question I'm having trouble with finding a particular equation, and solving this equation. Here is what I have so far. Can anyone please help me out? Solve $y'' + \pi^2y = 6x, 0<x<1$ $y(...
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1answer
35 views

Name of vector which includes value and its derivatives

What would I call a vector that describes a quantity (Temperature in this case) and its first and second derivatives at a specific location? It seems that a vector like this would have a formal name. ...
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11 views

Boundary condition for derivative of trigonometric behaving function

I want to find a trigometric function for which: $u(0)=u(1)=0$ and $\frac{\partial u}{\partial y}(y=0)=\frac{\partial u}{\partial y}(y=1)=0$. Lets look at e.g. $u=C1 cos(a y+\phi_0)+C2 sin(a y +\phi_1)...
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13 views

Classic reference text for studying the Dirichlet to Neumann map of an elliptic operator.

I was wondering if anyone could provide good reference texts for studying the Dirichlet to Neumann map of an elliptic operator? Ideally it would be one of the later chapters of an introductory/general ...
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23 views

Does this second-order system of nonhomogeneous ODE have bounded solutions?

The equations of motion for a bead on a smooth, freely rotating rod with unit length are $$ \left\{\begin{aligned} 0&= \omega_0^2\sin\theta - x\dot{\theta}^2+\ddot{x}\\ 0&=3\omega_0^2\cos\...
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18 views

Wave equation on half space with nonzero Dirichlet boundary condition

We consider the wave equation for $u(t,x)$ $$ \partial_t^2 u - \Delta u = 0 \, \text{ in } (0 , \infty) \times \mathbb R^n_+, $$ where $\mathbb R^n_+ := \{ (x_1, \cdots , x_n ) \in \mathbb R^n: x_n &...
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27 views

Solving inhomogeneous ODE - using Duhamels principle

I want to solve following ODE: $a (-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2})+c x_2 (-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2})-d(\frac{\partial^4 u(x_2)}{x_2^4}-2b\frac{\partial^2 u(x_2)}{x_2^2}+b^2u(...
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Creating a score that fits within the value boundaries of an existing model

First let me say that I'm sorry if I don't explain this in the clearest way. I'm a researcher, and not trained in mathematics. I'm looking at an existing algebraic model and trying to create an ...
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43 views

Vibrating Circular Membrane with Vibrating Boundary

Looking online, I have found many solutions to variations of 'vibrating circular disc problem', that is solving the following PDE \begin{align*} \begin{cases} u_{tt} - c^2\nabla^2 u = 0\\ u(R, \varphi,...
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Is this BVP solvable using Neumann boundary conditions?

I have the BVP $$u_{xx}+sin(x) = 0 \qquad \forall x\in[0,2\pi]$$ with dirichlet boundary conditions $$u(0)=u(2\pi)=0$$ Obviously $sin(x)$ is a solution. But if we instead have Neumann boundary ...
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3answers
32 views

Analytic solution of the heat equation with a source term

I have the heat equation with Dirichlet boundary conditions $$u_t(t,x)=u_{xx}(t,x)+\sin(x)$$ $$u(t,0)=u(t,2\pi)=0$$ $$u(0,x)=u_0(x)$$ Now, without the source term I could write the solution as $$u(t,x)...
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21 views

How to solve this Boundary Value Problem? (Using Fourier)

$u_{tt} = u_{xx} - u$ Boundary condition: $u(x,0)=f(x), u_t(x,0)=0,$ $f(x)$ is a Schwartz function. I tried to make $u(x,t)=X(x)T(t)$, then I get $\frac{X}{X''}=\frac{T}{T'' + T}$. Let $\frac{X}{X''}...
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9 views

Control chart boundaries for the proportion of words incorrectly typed by a typist per hour

This a problem presented in Hoel's Probability Book Chapter 3. Suppose you wish to construct a control chart for the proportion of words incorrectly typed by a typist per hour. If she typed 1200 words ...
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19 views

Harmonic function with Dirichlet and Neumann boundary

Suppose $D^+_1 \subset \mathbb{R}^2$ is the upper half part of the unit disk and $u$ solves $$\left\{ \begin{aligned} \Delta u &=0& \quad &\text{ in $D^+_1$};\\ \partial_2 u&= f(x,u)&...
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1answer
48 views

Verify that integral satisfies boundary conditions

Consider $$u(x_0,y_0,z_0)=\frac{z_0}{2\pi}\int_{\mathbb R^2}\frac{f(x,y)}{\left[(x-x_0)^2+(y-y_0)^2+z_0^2\right]^{3/2}}\,dx\,dy.$$ Show that (i) $u\to f(x_0,y_0)$ in the limit as $z_0\to0$ and (ii) $u\...
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23 views

Diffusion (Heat) equation at the equilibrium

Let's consider the diffusion (heat) equation at the equilibrium: $D \rho''\left(x\right)= \phi \rho\left(x\right)+\eta\left(x\right)$. On a domain $x\in\left[-L,L\right]$, for instance $L=\infty$. My ...
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2answers
102 views

Show that this ODE with boundary conditions has a unique solution

We have the ODE $$-\ddot{x}+q(t)x=g(t),\quad x(a)=x_a,\, x(b)=x_b$$ with $g\in C([a,b],\mathbb{R}$ and $q\in C([a,b],\mathbb{R}_0^+)$. Show that there is a unique solution. First of all, I ...
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18 views

Poisson equation in a cylinder

I need to solve the problem $\nabla^{2} u(r,\theta,z)=Q(r,\theta,z)$ inside a circular cylinder $(0 < r < a, 0 < \theta < 2\pi, 0 < z < H)$ subject to $u = 0$ on the sides. I'm ...
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21 views

Wave equation near two boundaries in n dimensions

Consider a function $u(t, x, y)$ defined on $t \in (t_i, t_f) \ , x > 0 \ , y \in \mathbb{R}^{n - 2}$ satisfying the wave equation (here $\Delta_y$ is the Laplace operator w.r.t. the variable $y$) \...
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32 views

Coupled reaction diffusion equation ODE.

How would one go about solving this with the B.C.s considering the derivatives are not defined at the boundary?
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1answer
18 views

Find the Eigenvalues and Eigenfunctions for the Boundary problem

I recently found this answer to a similar problem I'm currently working on. The problem is the following... Find the eigenvalues and eigenfunctions for $y^{\prime \prime}+\lambda y=0$ with the ...
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21 views

Asymptotic expansion linear elliptic PDE in 2D

I am looking for the asymptotic expansion of a solution to the boundary value problem $$ \Delta u + \varepsilon^\alpha f(u)=0 $$ with $\alpha \geq 1$ and $f$ polynomial (e.g. $f(u)=u$, $f(u)=u^2$, ...)...
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101 views

uniqueness of system of PDE using Lax Milgram

Consider the following question. (5) Consider the system (1) $$ \begin{array} -\Delta u(x)-v(x)&=&f(x) && x \in \Omega \\ u(x)-\Delta v(x) & = & g(x) && x \in \Omega \\...
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28 views

Sturm-Liouville Form with boundary conditions

I am having some problems understanding the Sturm-Liouville Theory for ordinary values when we have a non-symmetric system and non-homogenous conditions. I understand that we use a weighting function ...
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2answers
79 views

Prove that $f\left ( x \right )- x^{2021}$ always has at least one root $x_{0}\in\left ( 0, 1 \right )$

Given positive continuous function $f\left ( x \right )$ on the interval $\left [ 0, 1 \right ]$ so that $\int_{0}^{1}f\left ( x \right ){\rm d}x< \frac{1}{2022}.$ Prove that $f\left ( x \right )- ...
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46 views

1D Wave PDE with Nonzero Initial and Boundary Conditions

I'm not sure how to start this PDE since the initial and boundary conditions are nonzero. May someone point me in the right direction? This is the problem: $$u_{tt} = u_{xx}$$ $$u(x,0) = \frac{1}{2+ \...
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31 views

Solution of the 2D diffusion equation in a rectangular domain

I have a very specific problem concerning the solution of the two-dimensional diffusion equation in a rectangular domain. The problem I have is that I don't understand the way to get to the final ...
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23 views

Finding norm of operator on $W^1_2 [0;1]$

Let $H = W^1_2[0;1]$ (Sobolev space) and operator $A: H \to H$, $(Ax)(t) = t\cdot x(t)$. I'm trying to find $||A||$ w.r.t norm $||x||^2 = \int\limits_0^1 \left( x^2(t) + \dot{x}^2(t)\right) dt$. What ...
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16 views

Using Lax Milgram lemma and energy estimates on the real line

I just want to check something. I want to use the energy estimates on the real line for an elliptic operator $L$ acting on $L^2(\mathbb{R})$. (The energy estimates are related to the Lax-Mi https://...
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2answers
232 views

Impose PDE itself as Boundary Condition?

Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$. Solution methods often require me to impose boundary conditions. Often, these arise naturally from ...
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7 views

How to generalize a boundary value problem?

A boundary value problem can be of k-order with k conditions where it could be linear and non-linear equations. Can you give me a neat way to represent the equation or a good free/open source? write a ...
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1answer
32 views

Minimizing $|\frac{f'}{f}|$ and $|\frac{f''}{f}|$ under constraints.

Let's give constraints first. $f$ is a $C^2$ function on $[0,K]$, and we require that $f(0)=f(K)=1$, $f'(0)=C$, $C>0$ is a given constant. The goal is to minimize the quantities $$\left|\frac{f'}{f}...
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8 views

What does it mean agreement in norm mean in differential equations?

I am taking a course in numerical analysis in physical systems. Trying to understand the next definition: Absorbing boundary condition: Suppose $ u$ solves the well-posed differential equation $$ Lu=f,...
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1answer
70 views

How to maximize $\| {\bf U x}\|^2$ when ${\bf U}$ is a upper triangular matrix

When ${\bf U}$ is an $N$-by-$N$ complex-valued upper triangular matrix whose diagonal elements are positive real values, how to obtain an $N$-by-$1$ vector ${\bf x}=[x_1\cdots x_N]^T$ with $|x_n|=1$, ...
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46 views

Can I use separation of variables to solve the heat equation on an infinitely long rod

The heat equation in 1D is: $$ \frac{\partial T(x,t)}{\partial t} = \alpha \,\frac{\partial^2T(x,t)}{\partial x^2} \, , \quad \alpha > 0 $$ My IC's and BC's \begin{cases} T'_x(0,t) = T'_x(L,t) = ...
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1answer
55 views

Is this Laplace BVP well posed? If not, why?

Consider the boundary-value problem for the Laplacian $\nabla^2\phi(x,y)=0$ within a semi-infinite strip $0<x<a$, $0<y<\infty$, with the following boundary conditions $$\partial_x\phi(0,y)=...
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1answer
69 views

Conceptual Question About the Nature of PDE Boundary Conditions

In this $1D$ boundary value explanation, it is stated that "Taking the example of a straight line, whose slope at the boundary points is decidedly not $0$ [the animation shows a mostly-unlabeled ...
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1answer
32 views

Applying simple boundary conditions to a multi-variable function.

If I have the following general differential equation solution (to Laplace equation): $$ \phi(r,\theta)= A+B\ln(r)+\sum_{n=1}^{\infty}\left(C_nr^n+\frac{D_n}{r^n}\right)\left( E_n \cos(n\theta) + F_n \...
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1answer
43 views

Integral of Legendre polynomials with tangent

I have encountered the following relationship$^{[1][2]}$, stated without proof both times $$\int_0^\gamma dt \tan(t/2)\cdot [P_n(\cos(t))+P_{n-1}(\cos(t))]=\frac{1}{n}[P_{n-1}(\cos(\gamma))-P_{n}(\cos(...

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