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Questions tagged [boundary-value-problem]

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

11
votes
2answers
269 views

Robin BC in the 1D wave equation

The problem of interest is as follows: the quantity of interest: $u(x,t)$ the wave equation: $\partial_2^2u(x,t)-c^2\partial_1^2u(x,t)=0$ where $c>0$ one Robin boundary condition at $x=0$: $\...
11
votes
1answer
410 views

Counter example for uniqueness of second order differential equation

I have a second order differential equation, \begin{eqnarray} \dfrac{d^2 y}{d x^2} = H\left(x\right) \hspace{0.05ex}y \label{*}\tag{*} \end{eqnarray} where, $\,H\left(x\right) = \dfrac{\mathop{\rm ...
9
votes
0answers
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 ...
8
votes
1answer
184 views

Solving $u_{xx} + u_{yy} = 0$ subject to $u(x, 0) = u(0, y) = 0$ $ u(x, 1) = \sin(x)$, $u(1, y) = y^2$

I tried to proceed as expected: set $u = X(x)Y(y)$, then you get $$\frac{X''}{X} = -\frac{Y''}{Y} = -\lambda$$. Assume $\lambda>0$, and $\lambda = z^2$ then you get $$X = C_1e^{-zx}+C_2e^{zx}$$, $$...
8
votes
0answers
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 ...
8
votes
1answer
357 views

Uniqueness of Solution to a Boundary Value Problem

Question Let $f:\mathbb R_+ \to \mathbb R_+$ be a function twice continuously differentiable (with derivative $f'$ and second derivative $f''$), and $a$, and $b$ be parameters in $\mathbb R_+$. ...
7
votes
1answer
2k views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
7
votes
1answer
364 views

What would be an example of Neumann boundary conditions on a two dimensional domain

The Wikipedia page says that it would involve the derivative with respect to some normal vector being constant, but I don't quite understand this. Is the value of the normal vector adjusted across ...
6
votes
2answers
2k views

Solve the boundary value problem $y''+y= -1$, $\,y(0)=y(\pi/2)=0$ with the Green's function method

Using the Green's Function method solve the boundary value problem: $$ y''+ y= -1,$$ with boundary conditions $$y(0)=0, \quad y(\pi/2)=0.$$ Verify the result by elementary technique.
6
votes
2answers
8k views

Heat Equation in spherical coordinates

Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate $...
6
votes
3answers
82 views

Show that $u_x(x, y) = f′(r) \cos(\theta)$ and hence deduce that $f′(0) = 0$, which implies the Neumann boundary condition $u_r = 0$ when $r = 0$.

Suppose that $u(x,y)$ is a continuously differentiable, circularly symmetric function, so that when expressed in polar coordinates, $x = r \cos(\theta)$, , $y = r \sin(\theta)$, it depends solely on ...
6
votes
1answer
46 views

Trouble in Checking PDE Boundary Solution Problem

I'm doing this problem with some other students, but it seems that our solution doesn't work? We have the partial differential equation $\dfrac{\partial^2 u}{\partial x^2} -2 \dfrac{\partial^2 u}{\...
6
votes
2answers
134 views

Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem

I am trying to solve the following problem: $$\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x), \ x > 0, t > 0$$ $$\phi(x, 0) = 0, \ x > 0$$ $$\phi(0,...
6
votes
1answer
209 views

Integrating a sum of delta functions?

I know that the "hand-wavy" definition of the $\delta (x)$ function is $$ \delta(x) = \begin{cases} \infty &\quad\ x=0 \\ 0 &\quad\text{otherwise} \end{cases} $$ ...
6
votes
0answers
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 ...
6
votes
0answers
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 ...
6
votes
0answers
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{\...
5
votes
3answers
575 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the Euler ...
5
votes
1answer
1k views

You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
5
votes
1answer
178 views

Solve Boundary Value Problem for $y''+ y' + e^xy = f(x)$

Consider to solve Boundary Value Problem : $y''+ y' + e^xy = f(x)$ with $0 < x < 1$ and $y(0)=y(1)=0$ with exact solution $y(x) = \sin \pi x$ $f(x)=(e^x- \pi^2)\sin \pi x + \pi \cos \pi x$...
5
votes
2answers
1k views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\...
5
votes
1answer
285 views

Understanding solution of PDE using method: separation of variables.

Could someone please help me to understand the doubts I have about the solution of this pde problem and to check the things that I've added to the solution? Oscillations of the beam are described ...
5
votes
1answer
130 views

Steady State Temperature Distribution in Unbounded Region (Laplace's Equation in Spherical Coordinates)

I'm trying to find the steady state temperature distribution in the infinite region outside a sphere of unit radius centred on the origin, where the temperature takes the value $u = f(\theta$) on the ...
5
votes
1answer
208 views

Difficult partial differential equation

Problem $$ \Omega\frac{\partial}{\partial t}A(y,t) +6\Lambda\Omega\left(y^2-y\right) \sin(t) = \frac{\partial^2}{\partial y^2}A(y,t) $$ Boundary conditions $$ \frac{\partial}{\partial y}A(t,0) =...
5
votes
3answers
335 views

Eigenvalues of Differential Equation with Boundary Condition

Here is a problem from my homework assignment that I am struggling with: Consider the differential equation $\frac{d^2\phi}{dx^2}+\lambda\phi=0 $. Determine the eigenvalues $\lambda$ if $\phi$ ...
5
votes
0answers
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}\...
5
votes
0answers
62 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\...
5
votes
0answers
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 ...
5
votes
0answers
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{...
5
votes
0answers
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 ...
5
votes
0answers
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: $ ...
5
votes
0answers
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 ...
4
votes
3answers
300 views

Extremizing a functional subject to an equality constraint

Question at hand is: Let $y\in\cal C^2([0,\pi])$ satisfying $y(0)=y(\pi)=0$ and $\int_0^\pi y^2(x)dx=1$ extremize the functional $$J(y)=\int_0^\pi\left(y'(x)\right)^2dx$$ It's an MCQ, and from ...
4
votes
1answer
5k views

Use Green's function to find solutions for the boundary value problem

Find a solution using Green's functions $$y''+y=t; y(0)=0, y(1)=1$$ So far I have $$x(t)=c_1 \cos(t)+c_2 \sin(t)$$ so $$y_1=\cos(t), y_2=\sin(t)$$ and $$W(y_1,y_2)=-1$$ When I put that in the ...
4
votes
2answers
69 views

Stuck with boundary value PDE problem

The last time I posted this I got many down votes and I don't know why. Maybe because I forgot to include my work? I got the PDE $\dfrac{\partial^2 u}{\partial x^2} -2 \dfrac{\partial^2 u}{\partial x ...
4
votes
1answer
128 views

Finding $\beta$ where $u_x(0)=\beta$ is a boundary value of a heat equation

Suppose an ice core are perfectly insulated and heated at one end at a rate $\alpha$ and cooled from the other end at a rate $\beta$. Consider the following boundary value problem $$u_t-u_{xx}=0, \ ...
4
votes
3answers
2k views

Physical meaning of boundary conditions in the diffusion equation

I want to simulate the diffusion equation numerically. $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ With the boundary condition $$ \frac{\partial u}{\partial x} \bigg|_{...
4
votes
1answer
379 views

Is Helmholtz equation in arbitrary regular polygon solvable in closed form?

A Helmholtz equation $\Delta f=-\lambda^2 f$ with Dirichlet boundary conditions can easily be solved in a square and also not too hard to solve in equilaterial triangle. In both cases the solution is ...
4
votes
1answer
93 views

Higher order Sturm-Liouville form

The differential equation book I was reading briefly mentions about the generalization S-L form and it says if we consider the BVP $L(y)=\lambda r(x)y$ where $L(y)=\displaystyle P_n(x)\frac{d^ny}{...
4
votes
1answer
665 views

How to treat corners in a finite-difference solution to a PDE

The Problem Suppose I wish to solve the following PDE $$\vec{\nabla} \cdot \vec{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_z}{\partial z} = 0$$ on the 2D domain ($0 \le x \le L$ and $0 ...
4
votes
1answer
955 views

Due to numerical inaccuracy, the solution of a boundary value problems becomes negative

I treat a toy example to get my point across. In reality I have to deal with a much more complex model. Let us consider a one dimensional boundary value problem using the bvp5c solver in Matlab. Two ...
4
votes
1answer
124 views

Choosing a boundary for integration

I have the following differential equation $$ \frac{d}{dx}\left(\mu e^{cx}f(x)\right) = -\mu\left(\frac{a xe^{-cx}}{a x+x-1}\right) $$ that I am trying to integrate to find $f(x)$ with the boundary ...
4
votes
1answer
1k views

How to solve Robin problem with general initial data on the half line?

Solve: $$u_t=ku_{xx}, \text{ for } t>0,x>0$$ $$u(x,0)=\phi(x)\text{ for }x>0$$ $$u_x(0,t)-hu(0,t)=0 \text{ for }x=0$$, where $h$ is constant. When $\phi(x)$ is $x$, and $h$ is $2$, we ...
4
votes
1answer
266 views

Why is the function $\operatorname{Log}(G(t))$ Holder continuous?

I was reading the theory about the Riemann-Hilbert problem $\Phi^+(t)=G(t)\Phi^-(t)$ where $G(t)$ is a Holder continuous function on a closed curve $c$ with index $\operatorname{Ind}_cG(t)=0$. To ...
4
votes
2answers
134 views

How solve inhomogeneous ODE with boundary conditions?

I'm trying to solve this ODE: $$y''-2xy'+3y=x^3 $$ With the conditions: $$\lim_{x\to\pm\infty}e^{-x^2/2}y(x)=\lim_{x\to \pm\infty}e^{-x^2/2}y'(x)=0$$ The homogeneous part is Hermite's equation for ...
4
votes
2answers
86 views

How/Why does factoring the linear/differential operator suggest a specific change of variables?

Find a solution of the PDE $u_{tt} - c^2 u_{xx} = 0$, (where $c$ is a constant) in the half plane $t > 0$ with initial conditions $u(x, 0) = g_0(x)$ and $u_t(x, 0) = g_1(x)$. $$\therefore \...
4
votes
1answer
114 views

Sturm-Liouville-Problem with confusing result

I have the following ODE defined on $D_x = [-1,1]$: $$y''(x)=-k^2y(x).$$ Prom the physical problem, I know that the solution is non-zero and that $y(x)$ is a even function [$y(-x)=y(x)$] vanishing ...
4
votes
2answers
80 views

Integrating over a somewhat continuous function

I have a function $q(t)$ that starts at $q(0)=q_0$ and needs to get to $q(1)=q_1>q_0$. I have a free parameter $z$ that I can wiggle around to control $q'(t)$. Namely, I have a function $$q'(t)=Q(t,...
4
votes
1answer
338 views

Finding the frequencies of vibration of a drum; PDE

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have ...
4
votes
1answer
274 views

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the initial/...