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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11 views

What is the physical interpretation of $u_x(0,t)$, $u_x(1,t)$ and a negative diffusivity constant in 1D Heat Equation on $(0,1)$?

I'm looking for help with the following problem. We have the heat equation $u_t=Du_{xx}$ modelling the temperature across a conducting material, defined for $0<x<1, t>0$ with boundary ...
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21 views

Boundary points of the set D={(x,y): |y |>= x}

What are the boundary points of the set in xy-plane $$ D=\{(x,y): x^2+y^2<1, |y| \geqslant x\} $$ Should it be $bdry(D)=\{(x,y):x^2+y^2=1, |y|=x\}$ or $bdry(D)=\{(x,y):x^2+y^2=1, |y|\geqslant x\}$ ...
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Show that, under the hypothesis of Corollary 11.2, if $y_2$ is the solution to $y''= p(x)y'+ q(x)y$ and $y_2(a) = y_2(b) = 0$, then $y_2 ≡ 0$.

Theorem 11.1: Suppose the function $f$ in the boundary-value problem $y''=f(x,y,y')$, for $a ≤ x ≤ b$, with $y(a) = α$ and $y(b) = β$, is continuous on the set $$D=\{(x,y,y') | ~for~ a ≤ x ≤ b, ~...
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20 views

Solve Wave Equation Initial-Boundary-Value-Problem

I am trying to solve the following problem and this is my working so far. I'm struggling to get to the general solution for $X(x)$ as I'm not sure of the $\lambda$ value. Please could someone point ...
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18 views

Feynman-Kac for $\mu(t,x) = -\frac{1}{1-t}$ and $\sigma(t,x) = 1 $

Consider the following PDE on $[0,T]\times \mathbb{R}$: $$ \begin{cases} \dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2}...
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Solve a Laplace Boundary Value Problem

This is the easiest example from the book, but in the book everything is showed. So how would you solve this problem if you got it on a exam. I dont understand how to solve for $y$ after getting $x$. ...
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72 views

Uniqueness of PDE via energy functional

Assume the pde: $$ u_{tt}(t,x) = c^2u_{xx}(t,x) + \sigma u_{txx}(t,x) -\mu u_{t}(t,x), \quad x \in [0,L], t>0 $$ $$ u_x(t,0) = u(t,L) = 0 $$ $$ u(0,x) = \phi(x), u_t(0,x) = \theta(x), x\in[0,L] $$ ...
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40 views

Shooting Method

I tried to solve this differential equation: $$y''+\frac{1}{x}y'-\frac{9}{x^2}y=0\text{, where }x \in (1,2), y'(1)=\alpha, y(2)=\beta$$ I should use shooting method but I have a problem with ...
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Heat equation with Gaussian boundary condition

Let $$ S(x,t)= \frac{1}{2\sqrt{\pi t}}e^\frac{-x^2}{4t},\quad -\infty < x < \infty,\quad t>0$$ a. Find the solution to the equation $$u_t = u_{xx},\quad -\infty < x < \infty,\...
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Question about PDE, diffusion equation.

Let $u$ be sufficiently smooth and satisfy $$ \begin{cases} u_t − u_{xx}= f(x,t) & \{(x,t) : 0 < x < 1, 0 < t < 1\},\\ u(0,t) \ge 0, u(1,t) \ge 0 & 0 \le t \le 1\\ u(x, 0) \...
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Initial-Boundary Value Problems

$$u_{tt}-4u_{xx}=sin(3x\pi)-7sin(5x\pi),0\le x\le1,t\ge0$$ $u(0,t)=0$, $u(1,t)=0$ $u(x,0)=0$, $u_t(x,0)=0$ I'm having a hard time solving this question. My first attempt was writing $u(x,t)$ as $\...
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An Interesting Boundary Value Problem leading to a third order Eigenvalue system [How to proceed forward ?]

I have the following elliptic PDE (describing temperature in a plate, w in thermal contact with two fluids h and c ): $$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^...
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31 views

In Laplace's equation, why is it that there is only one solution for a particular boundary value?

I know that there is a unique solution to Laplace's equation that has particular boundary value. But I do not understand why this is the case. Thanks for your help in advance! PS: this is a follow-up ...
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44 views

Why do we need to know the value of a function at the boundary to determine that there is one, unique solution to Laplace's equation?

I'd like to draw your attention to the First Uniqueness Theorem in context of Laplace's equation. It states that if we know the value of a function $V$ at surface of a volume, then the solution to ...
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Analytical Solution of Picewise-Non linear 4th order ODE

I'm strugling for a while on that. Is there a way to approach analytical the following BVP? $$ \frac{d^4y}{dx^4}+[4\lambda_1^4H(y)+4\lambda_2^4H(-y)]y=0 $$ Where $H(x)$ stands for the Heaviside Step ...
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First-order linear partial differential equation with boundary conditions

I am trying to solve this boundary-value problem: $$u_x(x,y) + u_y(x,y) + u(x,y) = 0$$ $$u(0,y) = 1$$ $$u(x,0) = 1$$ Image version I tried to use the method of characteristics, but it seems that ...
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One-dimensional heat equation with nonlinear Robin-type boundary condition

I am trying to find an analytical solution to the following heat equation with nonlinear Robin-type boundary condition: $$ \frac{\partial}{\partial t} u(t, x) = D \frac{\partial^2}{\partial x^2} u \...
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I got stuck while calculating the Green's function in a non-trivial boundary problem

I need to find the Green's function for an ODE boundary problem. I know how to do it when the boundary conditions are trivial (for example, y(0)=0) but I have an exercise where the conditions are not ...
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39 views

Desmos help - making a finite length line with origin on a sine wave move along it and vary angle orthogonal to sine wave line

I'm trying to model something that seems fairly simple, but it's trickier than I thought on https://www.desmos.com. On desmos I created a function: $f(x)=y$ $y=(0 \leq x \leq 2)\sin( \pi x+b )$   ...
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Finding the general solution to $-u''(x)=f(x)$ for general $f(x)$

Problem: Use variation of parameters to verify that $$u(x)=\frac{L-x}{L}\left(\gamma_0+\int_{0}^{x} yf(y) \ dy\right)+\frac{x}{L}\left(\gamma_L+\int_{x}^{L}(L-y)f(y) \ dy\right) \ \ \text{for} \ 0\...
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62 views

Wave equation with Neumann boundary condition

I have the following problem: $$ \begin{array}{ll} &u_{tt}(x,t)=4u_{xx}(x,t),&x>0, t>0\\ &u_x(0,t)=-\cos(t),&t>0\\ &u(x,0)=e^{-x},&x>0\\ &u_t(x,0)=2e^{-x},&...
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Boundary condition for a well-posed linear, inhomogeneous, second-order PDE

Unfortunately, my exposure to partial differential equations in mathematical physics has been very limited and hence this question. Consider a general linear, inhomogeneous, second-order, partial ...
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25 views

Problem of separation of variables for Dirichlet boundary data of Laplace's equation in polar coordinates

Need help here with figuring out boundary conditions for this problem. Also, for (i), I do know a general way or method but here I am confused since from both equations how do I find out my desired ...
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34 views

Heat equation with zero Neumann boundary condition

I have a question ask to find Fourier cosine series, so my approach as follow: Given $\phi(x) = x^2$ and $x \in [0,1]$. For $n \neq 0$, directly applying the formulas, we have $$A_n = 2 \int_0^1 x^2 ...
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Vortex solution of Laplace equation (XY model)

The hamiltonian of XY model, which is closely connected with BKT - transition is following: \begin{equation} H=\frac{J}{2} \int \text{d}^2 r \, \nabla \varphi \cdot \nabla \varphi, \quad \...
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Solution parabollic PDE system

Let $\Omega := [0,\infty)\times [0,\infty)$ be a domain and $u, w \in C^\infty (\Omega)$. Consider the system \begin{align} \frac{\partial u}{\partial t} &= D_u \frac{\partial^2 u}{\partial x ^...
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46 views

IVP with boundary condition (incorrect statement)

Suppose, I would like to solve initial value problem (IVP) $$ (1):\quad \begin{cases} x'(t)=f_1(t,x,y),\\ y'(t)=f_2(t,x,y), \end{cases} $$ with initial conditions $$ (2):\quad x(0)=x_0,y(0)=y_0 $$ ...
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26 views

Boundary Value Problem using shooting method and Picard's method for successive approximations

Hi Guys i am working on the following question $$y''+2y'=e^{-x}$$ $$y(0)=1,y(1)=4$$ The Question states Obtain a numerical solution to the given boundary value problem when x = 0.25, x =0.5 and x =0....
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Finding Green Function of Legendre Diff. Equation

I need help to find the Green function of \begin{align} (1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} = 0, \hspace{10pt} y(0)=y'(1)=0 \end{align} Based on the book, it is a Legendre DE with order zero, ...
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Find eigenvalues and normalized eigenfunctions of the boundary value problem

$y''+\lambda y = 0$ $y(0) - y'(0)=0 $ $y(\pi)-y'(\pi)=0$ I was able to find the eigenvalues are $\lambda = -1, n^{2} (n \in \mathbb{Z}, n \geq 1)$ However I am not sure as to how to get the ...
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Dirichlet problem for Laplace equation with square domain

I'm studying for PDE qualifying exams and came across a problem that is giving me issues. The problem is: Given the square $\Omega=\{|x|<1,|y|<1\}$, let $u\in C^2(\Omega)\cap C^1(\overline{\...
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Are the boundary conditions of this PDE inconsistent?

Recently, I was attempting to solve the following boundary-value problem for the wave equation with a student that I tutor: $$\frac{\partial^2 u}{\partial x^2}=\frac{1}{4}\frac{\partial^2 u}{\partial ...
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23 views

Non-Linear BVP - ODE - using finite differences

I have the following ODE $EI\frac{{{d^2}y}}{{d{x^2}}} = [\frac{{ - qLx}}{2} + \frac{{q{x^2}}}{2}] \cdot {[1 + {(\frac{{dy}}{{dx}})^2}]^{\frac{3}{2}}}$ $\begin{array}{l} y(0) = 0\\ y(L) = 0 \end{...
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Under what condition does this boundary value second order ODE have unique solution?

This is a problem from the textbook The Way of Analysis by Robert Strichartz. So all solutions to $x''(t)=-x(t)$ will have the general form $A\cos t+B\sin t$. Then the problem asks for which values of ...
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Solving the Poisson Equation with Mixed Boundary Conditions

I am trying to solve the 2D Poisson equation $$ \triangle u(x, y) = -1$$ with mixed boundary conditions on the edges of a rectangle $[0, a]\times [0, b] $: $$u(x, b) = 0 \qquad u_y(x, 0) = 0$$ $$u_x(...
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Are there maximum principles related to the third boundary condition?

I am working on this problem \begin{equation} \begin{cases} u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\ u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0, \end{cases} \end{...
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26 views

Poisson's Equation on an Ellipse with Constant Source

Suppose I have the problem: $$\nabla^2\phi=1$$ but at the ellipse $\bigg(\dfrac{x}{a}\bigg)^2+\bigg(\dfrac{y}{b}\bigg)^2=1$, $\nabla^2\phi=0$. I have an inclination that the solution may be concentric ...
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Analysing an infinitesimal element in modelling the temperature distribution along a 1D rod

Consider the following problem whereby a rod is attached on both ends $x = 0$ and $x = L$ whereby each end has a temperature. Let the temperature be a function of distance, i.e. $u(x)$. We want to ...
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Is PDE ill-posed when the boundary condition contains singularities?

For PDE with certain boundary conditions, if the boundary condition contains singularities (due to corners or sudden changes in BC), can we say the problem is ill-posed? If not, how should we describe ...
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26 views

Second order boundary value problem system

Let $A$ be $n \times n$ real matrix and consider the following boundary value problem system $$y''(x)=Ay(x) \\ y(0)=y(1)=0,$$ where 0 is the null vector in $\mathbb{R}^n$. The matrix $A$ is not ...
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20 views

Heat equation with inhomogenious boundary conditions

I'm trying to find the analytic solution for this problem: $\frac{∂u}{∂t} =\frac{∂^2u}{∂x^2}, 0\leq x\leq 1,t\geq 0,$ $u|_{t=0}=x+sin(3\pi x), 0\leq x\leq 1,$ $\frac{∂u}{∂x}|_{x=0}=1, t\geq 0,$ $...
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17 views

In a second order ODE, can boundary conditions be applied inside the domain?

Let's say that I have a function $f(x)$ defined in $[0,1]$. $f(x)$ is the solution to a second order linear ODE $$F(f''(x), f'(x), f(x), x) = 0$$ Due to the order of the equation, the ODE needs $2$ ...
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29 views

Solving PDE with boundary conditions that depend on the same variable

Solve the Laplace equation: $$\frac{\partial^2U}{\partial X^2}+\frac{\partial^2U}{\partial Y^2}=0,$$ for: $$0<x+y<1,$$ $$0<x-y<1,$$ and boundary condtions $$U(x,-x)=0,$$ $$U(x,1-x)=0,$$ $$...
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28 views

Homogeous Boundary Value Problem (Wave Equation)

I am trying to solve the following boundary value problem, where $0<x<\pi$: $$\frac{\partial^2 u}{\partial t^2}=9\frac{\partial^2u}{\partial t^2}$$ $$\frac{\partial u}{\partial x}(0,t)=\frac{\...
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18 views

Real analysis. Continuity of a product [duplicate]

Let $X$ $\subset$ $\mathbb{R}$ and $f,g:X\rightarrow\mathbb{R}$ two continuous bounded functions. Is the product $fg$ a uniformly continuous function? Prove or give a counterexample. I would say it ...
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7 views

Truncation error of Crank-Nicolson Scheme with derivative boundary condition

I am handling a heat equation defined on $(t,x) \in [0,T]\times[0,1]$ with Robin boundary condition: \begin{equation} \begin{cases} -u_t + \frac{1}{2}u_{xx} = 0 \\ u(t,0) = u_1(t) \\ ...
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24 views

Help to proof that squared integrable functions form a hilbert space.

Marcus here. I am currently trying to understand the concept Hilbert space, specifically for squared integrable functions. The definition of a hilbert space is "A inner product space $( \langle \cdot ...
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17 views

Book for Boundary Value ODEs

Can someone suggest me a book on Boundary Value Problems in ODEs, which start from the general theory, and then go on to specialize for self-adjoint values? All the books I have found discuss the self-...
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28 views

How many boundary conditions are needed for the Navier-Stokes equation of simple fluid flow in a pipe?

(While this question pertains to fluid dynamics in specific, I am posting it here as I think the basis of my confusion lies in the basic mathematics.) For solving the Navier-Stokes equation for ...
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36 views

Exact solution of the transport equation, Neumann boundary

Please, can anyone give me some examples of the exact solution of the following Pde with the corresponding functions f and g. I need these to know if my code of an approximation scheme is correct or ...

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