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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Solution to a 1D Fokker-Planck equation with 2 absorbing boundaries and 1 continuity boundary

Consider the following simple Fokker-Planck equation: $$\partial_t f(x,t) = a \partial_x^2 f(x,t) $$ which holds on the intervals $x\in(0,c)$ and $(c,b)$. with $0<c<b$. $0$ and $b$ are ...
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Solving homogeneous differential equation with boundary condition using Green functions

I have an homogeneous equation $Lf=0$ where $L$ is an operator and $f$ a function, with boundary conditions $f(\partial\Omega)=g(\partial\Omega)$ where $g$ is a known function. In the case when the ...
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Question on Eigenvalues/Eigenfunctions in Boundary-Value Problems

A simple question, hopefully - consider, for example, the standard BVP defined as follows $$y^{\prime\prime} + \lambda y = 0\qquad y\left(0\right) = 0 \qquad y\left(2\pi\right) = 0$$ The traditional ...
gettingmathy's user avatar
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Odd extension on two-dimensional parabolic PDE with boundary conditions

When solving a parabolic PDE with boundary conditions (e.g. the reflection method of the barrier option), we can change the parabolic PDE into a heat equation by applying some substitution. For ...
sssssoku's user avatar
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Existence of solution based on convergence of finite-difference scheme

Consider the following boundary value problem $$ \begin{cases} y''(x)=p(x)y'(x)+q(x)y(x)+r(x)\\ y(0)=\alpha,\quad y(1)=\beta. \end{cases} $$ for continuous $p(x)$, $q(x)$, $r(x)$ and positive $q(x)>...
goldfish's user avatar
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Stefan Problem With Time Boundary Conditions

I have doubt on Stefan Problem with time boundary conditions, does this problem has an analytical solution or not? For example in the following article Tao, L. N., The Stefan problem with arbitrary ...
JOSE JESUS OJEDA GARCIA's user avatar
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How to solve a boundary value problem with Neumann Boundary conditions using the Finite Element Method

I have been given the question: Consider the boundary value problem: $$-u'' + u = f(x) \forall x \in (0,1), u(0) = 0, u'(1) = 0.$$ Using continuous piecewise linear basis functions on a uniform mesh ...
caroline's user avatar
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Comformal mappings of boundaries conditions

I'm reading the book "complex variables Demystified" from David McMahon and right now I'm in the chapter 12 where he explains how to solve Boundary values problems using comformal mappings ...
Juan Sin Tierra's user avatar
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How should I show the following stationary path with a natural boundary condition at $x=0$?

Let $a>0, b>0, c>0$ be constants. Show that the stationary path of the functional $$S[y]=\int_{0}^{a}(y'^2+2by)dx, \quad y(a)=0,$$ with a natural boundary condition at $x=0$ and subject to ...
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Notation for a subspace of $H^1(\mathbb{R})$

In a work of mine, I need the following subspace of $H^1(\mathbb{R})$: $$ \{u \in H^1(\mathbb{R}):u(0)=0\}. $$ I first used the notation $H_0^1(\mathbb{R})$ for it but a referee did not like that ...
Gateau au fromage's user avatar
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Why are the eigenfunctions of the Laplacian on a region with spherically symmetric boundary condition are not spherically symmetric?

For example the eigenfunctions of a spherically symmetric membrane can be found in https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane. Then again I sometimes see people in Physics saying ...
TheFibonacciEffect's user avatar
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Solving a 2-D BVP using Green's Function. Need guidance on obtaining $G(x,y,x',y')$

This is a generic example of a Poisson Equation with homogeneous BCs which I am reading the solution to in Haberman's Applied Partial Differential Equations with Fourier Series and Boundary Value ...
Researcher R's user avatar
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A superposition problem (or not?)

I have encountered a peculiar problem in my mathematical physics test which is nothing I have seen before. Solve $\Delta u(x) = 0$ inside a half-ball in $\mathbb R^3$, which is given by $|x| < R$, ...
dnes's user avatar
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Re the Fokker-Planck equation: How to reconcile boundary conditions for reflecting barriers with arbitrary initial conditions?

I'm learning about boundary conditions for the Fokker-Planck equation and am misunderstanding something fundamental. In one dimension, the Fokker-Planck equation is $$\partial_t f(x,t) = - \partial_x ...
Stig's user avatar
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Boundary Value Problems Solving with Central Difference

I have differential equation $u''(x)+9u(x) = \cos(2x), \; x \in [0, π/2]$ the boundary conditions are $u(0)=1$, $u(\pi / 2) = -1$, with $h = 0.2$ Now I replaced $u''(x)$ with centered difference ...
lovbal's user avatar
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Solve Cauchy-Euler boundary value problem $4x^2y'' + 8xy' -3y=0$ subject to initial conditions $y(0) = 0\, y(4) = 6$

I have to solve $4x^2y'' + 8xy' -3y=0$ subject to initial conditions $y(0) = 0,\, y(4) = 6$ This is a Cauchy-Euler equation so using the characteristic equation, I get the general solution $y = Ax^{-\...
number8's user avatar
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How to numerically approximate solutions to a set of two second order two-variable ODEs using a python script (boundary value problem)?

I have a set of two ODEs like so: $$ \ddot r = \dot \theta^2r \ - \frac{GM}{r^2}$$ $$ \dot r = -\frac {r \ddot \theta} {2 \dot \theta}$$ $\theta$ and $r$ are functions of time, G and M are constants. ...
AtomProgrammer's user avatar
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Reference for boundary regularity of Neumann eigenfunctions

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with piecewise $C^{\infty}$ boundary. I have seen it implicitly used in several results that Neumann eigenfunctions of the Laplacian on $\Omega$ ...
Lawford Hatcher's user avatar
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1 answer
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Brezis' exercise 8.30.7: what happens to the sequence $(u_n)$ if $k_n \to 1$?

Let $I$ be the open interval $(0, 1)$. Let $k \in \mathbb R \setminus \{1\}$. We consider the space $$ V := \{v \in H^1 (I) : v(0) = kv(1)\}, $$ and the symmetric bilinear form $a$ defined on $V$ by $$...
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Brezis' exercise 8.30.6: $u_n \to u$ in $H^2 (I)$

Let $I$ be the open interval $(0, 1)$. Let $k \in \mathbb R \setminus \{1\}$. We consider the space $$ V := \{v \in H^1 (I) : v(0) = kv(1)\}, $$ and the symmetric bilinear form $a$ defined on $V$ by $$...
Akira's user avatar
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Does this second-order boundary-value problem have a unique solution when $k=1$?

Let $I$ be the open interval $(0, 1)$. Let $k \in \mathbb R$. We consider the space $$ V := \{v \in H^1 (I) : v(0) = kv(1)\}, $$ and the symmetric bilinear form $a$ defined on $V$ by $$ a(u, v) = \...
Akira's user avatar
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Hypothesis on a parameter to ensure the unique solvability of a modified Laplace equation

Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following ...
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Solve and integral equation with symmetric kernel [closed]

I have the following integral equation with symmetric kernel $$ x(t)=\sin(\pi t)+\pi \cos (\pi t) +\lambda \int_{0}^{1} k(t,s)x(s)\,ds $$ where $k(x,t)$ is a symmetric kernel given by $$k(t,s)= \...
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Appropriate definition of the energy functional for one-dimensional wave equation with Robin BC

$$\def\d{\mathop{}\!\mathrm{d}}$$ In the domain $Q_T = (0,l)\times (0,T)$ we consider the following problem $$\begin{align} & u_{tt} - a^2 u_{xx} + b(x,t)u_x + b_0(x,t)u_t + c(x,t)u = f(x,t), \...
Stephen's user avatar
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How to implement the Neumann boundary condition when solving the heat equation using Chebyshev's pseudo-spectral method

I am studying the Chebyshev pseudo-spectral method and having problems understanding how to implement the Neumann boundary condition when trying to solve a PDE. To understand better how to implement ...
Nikola Ristic's user avatar
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Inviscid Burgers Equation on a Ring

I am interested in solving for the Inviscid Burgers Equation on a ring. $$ u_t + uu_x=0;\quad(x,t) \in [0,L)\times \mathbb{R}_+,\\ u(x,0)=\phi(x),\\ u(0,t)=u(L,t). $$ This problem I believe is well ...
user3166083's user avatar
2 votes
1 answer
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Solving PDE containing mixed partial derivatives [closed]

I want to solve the PDE $$\frac{\partial^2u}{\partial t\,\partial x} = \frac{\partial^2u}{\partial x^2},\quad -\infty < x < \infty,$$ $$u(x,0) = e^{-\lvert x\rvert}.$$ without the final answer ...
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How to estimate eigenvalues of Sturm-Liouville ODE whose solutions are not known

Suppose $p, q$ are functions that satisfy the properties for $$\frac{d}{dx} \Big( p(x) \frac{d\phi}{dx}\Big) + q(x)\phi + \lambda w(x) \phi = 0 \tag{1}$$ to be a Sturm-Liouville problem on some ...
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Solving linear second order hyperbolic PDE $\nabla \cdot \left( M \nabla u\right)=0$

Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain with a smooth boundary $\Gamma$. Furthermore, let $M$ be a matrix-valued function, where the entries $m_{ij}$ are smooth and ...
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Help Solving and Understanding a Temperature Problem

Consider the following temperature problem: $$u_t(x,t)=ku_{xx}(x,t), \;0\leq x \leq \pi,\;\; t,k >0$$ with boundary conditions: $$u_x(0,t)=u(0,t)$$ $$u_x(\pi,t)=u(\pi,t)$$ $$u(x,0)=f(x)$$ I know ...
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Polar Laplace equation on half disk

Suppose we have a semidisk, with the origin in the center of it. I have to solve the Laplace equation $$\nabla^2u(\rho,\phi)=0 \tag{1}$$ with the following boundary conditions $$u(r,0)=u(r,\pi)=T_0\...
Javier's user avatar
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Brezis' exercise 8.21.3: if $k \neq k_0$ and $f \in L^2(I)$ then problem $(2)$ admits a unique solution $u \in H^2(I)$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve a problem in Brezis' Functional Analysis, i.e., Exercise 8.21 Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \...
Akira's user avatar
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2 votes
1 answer
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Transform Lagrangian with a square root

I have an action given by, \begin{equation} S = \int^{\tau_f}_{\tau_0} d\tau \left(\frac{1}{z^3}\sqrt{-f(u,z) \dot{u}^2 + 2 \dot{u} \dot{z} + \dot{x}^2} + \frac{2 \dot{z}}{z^3} \right), \end{equation} ...
mathemania's user avatar
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Polyharmonic system

In the problem, with $\Omega \subset \mathbb{R}^3$ a smooth bounded domain, and the given functions $h_1, h_2 \in C(\partial\Omega)$ \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \...
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The existence of a solution to $-(pU')' + qU = f$ on $(0, 1)$ with boundary condition $U(1) = 0$

Let $I$ be the open interval $(0, 1)$. Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \alpha>0$ and $q(x) \geq 0$ for all $x \in \bar I$. Let $f \in L^2 (I)$. In the ...
Akira's user avatar
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Prove the boundness and monotonicity wrt parameter of solutions to this PDE

the problem is: (11) Let $Q=(0, l) \times(0, \infty)$ and $u_h=u_h(x, t)$ be a solution of $$ \begin{aligned} u_t-u_{x x} & =0 \quad \text { in } Q, \\ u(x, 0) & =0 \quad \text { in }(0, l), \\...
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1 answer
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Brezis' exercise 8.18: show that $u$ is the solution of some ODE with appropriate boundary conditions

Let $I$ be the open interval $(0, 1)$. I'm trying to solve a problem in Brezis' Functional Analysis, i.e., Exercise 8.18 Check that the mapping $v \mapsto v(0)$ from $H^1(I)$ into $\mathbb{R}$ is a ...
Akira's user avatar
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Tetration and $f(x) = \exp(\int_1^x \ln(f(t)) dt)$

Let $g(x)$ satisfy $g(1) = 1 , g'(1) = 1 , g(1+x) = \exp(g(x))$ Now it is clear that $g'(5) = g(5)g(4)g(3)g(2)$ This invites to think of the function $f(x)$ which is defined similarly and might or ...
mick's user avatar
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Can I extend the solution of my wave equation to satisfy one more zero boundary condition?

Using d'Alembert's formula, I found the solution to the following wave equation on a half-line problem: $$u_{xx} = u_{tt} \tag 1$$ $$u_x(0, t)=0 \tag 2$$ $$u(x, 0)=0 \tag 3$$ $$u_t(x, 0)=(1-x)e^{-kx} ...
Nikola Ristic's user avatar
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Proving that the infinite sum converges to $1$ as $k$ tends to infinity

While solving a PDE using the Eigenfunction expansion method, I approximated the following function with an infinite sum: $$ f(x) = (1-x)e^{-kx} = \sum_{n=0}^\infty A_n \cos(w_nx) \tag 1 $$ where $x ...
Nikola Ristic's user avatar
1 vote
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Solution to simple system of PDEs - Did I prove that only trivial solutions exist?

I am trying to solve a system of first-order linear PDEs. In the best case, I would want to solve it explicitly but proving that there exists a (unique) solution would also be helpful. Let $\Omega \...
SebastianP's user avatar
4 votes
0 answers
175 views

Solution to linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain ...
SebastianP's user avatar
1 vote
2 answers
72 views

Help Understanding Sturm-Liouville Boundary Value Problem

Consider the following Sturm-Liouville boundary value problem: $\text{Given parameters }c > 0 \text{ and } \beta > 0, \text{ let } y=y(x) \text{ for } 0 \leq x \leq c. \text{ We have }$ $$y''+ \...
RungSoup's user avatar
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references for 2nd order elliptic boundary value problems

Let $\Omega \subset \mathbf{R}^2$ open and bounded with smooth boundary $\partial \Omega$. Suppose that we have a second order differential operator $L$ on $\Omega$ with smooth coefficients. I'm ...
snape1234's user avatar
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How to approach the following boundary value problem on the unit ball: $\Delta u(x,y)=| x|$, $u(x,y)=0$ for $x^{2}+y^{2}=1$

I think the title is pretty self-explanatory but I'm elaborating just in case: let $u\in C^{2}(D(0,1),\mathbb{R})$ (where $D(0,1)$ is the (open) unit disk centered at $0$) such that the laplacian of $...
Modesto Rosado's user avatar
3 votes
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104 views

Constructing functions with lower bounds on hessian and zero boundary value

Let $Ω := \mathbb{R}^n \times [0, 1]$ and let $F: Ω \to \mathbb{R}^{n \times n}_\mathrm{sym}$ be a function on $Ω$ valued in symmetric matrices. We have two assumptions on $F$: There exists a $R > ...
Carlos Esparza's user avatar
1 vote
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Rayleigh quotient for Robin boundary condition

I'm reading Walter Strauss' PDE book. In section 11.4 he said that the Rayleigh quotient of the following Robin eigenvalue problem $$ \begin{cases} -\nabla\cdot(p\nabla u) + qu = \lambda mu, & \...
Stephen's user avatar
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Fourier transform of Poisson's equation with periodic shifted boundary conditions

I am trying to solve following 2D Poisson's equation numerically: $$ \Delta \Phi = \rho $$ When $\rho$ and $\Phi$ are periodic in both directions, this can be solved straightforward thanks to Fourier ...
steven's user avatar
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2 votes
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How do you minimize t instead of an objective function over t?

Let $\mathbf{x}(t) : \mathbb{R}\rightarrow\mathbb{R}^{n}$ be a vector-valued function, and $\{\mathbf{x}_{0}, \mathbf{x}_{t}, \mathbf{v}_{0}, \mathbf{v}_{t}, c\}$ constants. With $\mathbf{x}(0) = \...
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Complex solutions of ordinary BVP with real/complex coefficients

The classic Protter and Weinberger: Maximum principles in Differential Equations states the following uniqueness theorem (theorem 8): Let $u_1$, $u_2$ be solutions of the BVP $u''(x) + g(x) u'(x) + h(...
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