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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Techniques for solving the Navier-Cauchy equations of linear elastostatics

I'm trying to get an exact solution to a specific boundary value problem in linear elasticity. Standard techniques one would use to solve the heat or wave equations, such as separation of variables ...
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Proving a regularity estimate for $−εu′′ + bu′ = f $

I've been given: $$−εu′′ + bu′ = f ,x ∈ (0, 1), u(0) = u(1) = 0$$ and have been asked to prove the regularity estimate $$∥u′′∥_{L^2(0,1)} ≤ C_R∥f∥_{L^2(0,1)}$$ I normally try to provide some working ...
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Method of characteristics - inhomogeneous pde

The boundary conditions problem: $2\frac{\partial u}{\partial x}-14\frac{\partial u}{\partial y}=3(x+y)$ Where the boundary conditions are: $u(x,-6x)=Cos((2x)^2) \quad \forall x \in \mathbb{R}$ How ...
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1 answer
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boundary conditions - inhomogenius liniar pde

I have this boundary conditions question: $2\frac{\partial u}{\partial x}-14\frac{\partial u}{\partial y}=3(x+y)$ Where the boundary conditions are: $u(x,-6x)=Cos((2x)^2) \quad \forall x \in \mathbb{R}...
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Determining the bilinear form a(. , .) is bounded.

I am trying to derive a weak formulation of the boundary value problem $-\epsilon u′′ + bu′ = f $ for $x \in (0, 1)$, $u(0) = u(1) = 0$,: where $\epsilon \in \mathbb{R}$, is such that $\epsilon > 0$...
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Boundary conditions for weight function in weak form of an ODE or PDE

I am really quite confused by how to pick a weight function's boundary conditions when evaluating the weak form of a differential equation. For example, given this ODE in the domain $0<x<1$: $$u'...
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3 votes
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2D Laplacian with 1 Non-Homogeneous Dirichlet, 1 Homogeneous Dirichlet and 2 Homogeneous Neumann BCs, Rectangular Domain

To preface, I am very new to PDEs and not overly familiar with hyperbolic trig so forgive me if there are obvious mistakes - I have tried my best to research this problem before asking this question. ...
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Can you bound the H1 norm of the solution of a Poisson problem with its L2 norm?

Let $u \in H^1(\Omega)$ be such that $$ \int_{\Omega} \nabla u \cdot \nabla v + \int_{\Omega} u v = \int_{\partial\Omega} q v \qquad \forall v \in H^1(\Omega) $$ for a given boundary datum $q \in H^{-...
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2 answers
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Integral of a two-variable function with boundary condition [closed]

A function $f(x,y)$ with $f(x=0,y)=0$. We define $g(x,y) = \int f(x,y)dy$. Does $g(x=0,y)=0$? How does one prove this?
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Condition on Sobolev function on the boundary of a domain

I encountered the following kind of a condition when studying boundary value problems: Let $\Omega \subset \mathbb{R}^2$ be a domain. Let $\varphi \in H^1 (\Omega)$ be such that $\varphi \ge 0 $ on ...
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I cant solve a problem based on using central difference approximation to discretize a boundary value problem [closed]

problem statement: I tried solving this problem a couple of days ago,but my answer didnt match any of the available options.Can someone show me how to solve this kind of problem,and give me the answer?...
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4 votes
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Variational formulation, weak formulation

I'd like to find the weak formulation of the problem $-u''+au=f$ on $(0,1)$ $u(0)=0$ $u'(1)=b$ $a>0$ and show that there exists a unique solution using Lax-Milgram. What I did: By multiplying ...
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Green function for Homogeneous diffusion equation in the semi infinite domain with inhomogenous neumann boundary conditions

The Initial Boundary Value-Problem in the semi infinite domain $0 \leq x < \infty$ is given by $$ k u_{xx} - u_{t} = 0 \\ u(x,0) = u_0 \\ -u_x (0,t) = f(t) \\ -u_x (x \rightarrow \infty , t) = ...
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Using finite element method to study a second order diff. equation with b. c., which includes a value of a solution at a certain point.

I am looking into a certain analytical result numerically. I am using FreeFem and Mathematica, but have trouble with implementing a part of differential operator. Can you recommend a different library ...
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How to determine the boundary layer based on the PDE?

My professor solved a problem and I don't understand how they explained the location of the boundary layer of the following PDE: $$\epsilon y'' -(2x+1)y' + 2y=0$$ $$y(0)=y(1)=1$$ As $\epsilon → 0$, a ...
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What are the boundary conditions on Green's function and why?

The equation $$\frac{d^2}{dx^2}\phi(x)=f(x)\tag{1}$$ has the general solution $$\phi(x)=\phi_0(x)+\int G(x,x')f(x')dx^\prime\tag{2}$$ where $$\frac{d^2}{dx^2}G(x,x^\prime)=\delta(x-x^\prime)\tag{3}$$ ...
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Existence of solution to linear boundary value problem

Consider a differential equation of the form $$ \dot \gamma(t)=F_t(\gamma(t)) $$ For $\gamma:[0,1]\to \mathbb R^{2n}$ and a smooth $F:[0,1]\times \mathbb R^{2n}\to \mathbb R^{2n}$ such that every $F_t$...
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition

I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
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Solving IBVP with Laplace Transform

It is given the following BVP \begin{align} u_{t} &= u_{xxxx},\quad\quad x \in (0, \pi), t >0 \tag1\\ u(0,t) &= u(\pi,t) = 0 ,\quad t > 0 \tag 2\\ u_{x}(0,t) &= e^{-4t},\quad t > ...
2 votes
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Eigenvalues for hypergeometric differential equation using asymptotic analysis (or WKB approximation)

I am studying a problem in high energy physics and have reduced it to the following problem regarding the confluent hypergeometric differential equation: $$ \left(\gamma \partial_\gamma^2 + \partial_\...
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Under what condition on the convection term is the 2nd order BVP $-(au')' + b u' + cu = f$ the critical point of a convex energy functional?

Consider the second-order elliptic boundary value problem $$-(a(x) u(x)')' + b(x) u' + c(x)u = f(x)$$ over interval $I = [0,1]$, together with homogeneous Dirichlet boundary conditions $u(0)=u(1) = 0$....
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1 vote
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is my analytical solution for Heat Equation correct?

Could you please check my analytical solution for accuracy and correctness? Consider the heat equation: $$ u_t(x, t)=u_{x x}(x, t) $$ where $$ 0<x<1, \quad t>0 $$ Neumann Boundary condition: $...
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Example of Green's function solution to linear homogeneous differential equation subject to inhomogeneous boundary conditions

Context Part A Generally, I like Handbook on Differential Equations by Zwillinger [1]. It seems to meet my needs most of the time. For example, from this book I was able to learn how to solve linear ...
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1 answer
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Continuity of the branch function at the boundary of the subdomains

I solved this exercise, but I am insecure of myself so I would like some comment from you about the correctness, or not, of my reasoning. Consider the set $A$ and the function $f(x, y)$ such that $$A =...
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4 votes
1 answer
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How do I prove that the BVP $y'' = y\sin\left(x\right), y\left(0\right) = a, y\left(1\right) = b$ has a unique solution?

I want to prove that the BVP $$ y'' = y\sin\left(x\right), y\left(0\right) = a, y\left(1\right) = b $$ has a unique solution. I know how to prove uniqueness: Suppose there are two different solutions $...
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1 answer
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General compatibility condition for pure Neumann problem

It is known than for a second order elliptic boundary-value problem with pure Neumann conditions, a certain compatibility condition between the data must be satisfied. For example, in the case of the ...
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Transformation of boundary conditions at Laplace Equation in polar Coordinates

It is given the following BVP \begin{align} \Delta u &=0 , \quad 0\leq r <R,\quad 0<\phi<\alpha\leq 2\pi \tag1\\ u(r,0)&=0,\quad 0\leq r <R \tag 2\\ u(r,\alpha)&=\alpha,\quad 0\...
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24 views

FEM code for PDE subdomain interface condition

I have a simple problem to solve. I have two pde, one for $u_{1}$ and one for $u_{2}$, where the domain in space for $u_{1}$ is $[-L_{0}, 0] $ , and for $u_{2}$ the space domain is $ [0,L_{0}] $. I ...
1 vote
1 answer
36 views

How do I solve $0= \frac{d}{dx}(x^2u'(x))$ in $C^1[-2,1]$ with boundary conditions

I am trying to solve the following ODE with boundary conditions in the set $C^1[-2,1]$, with $u(-2)=-1/2$ and $u(1)=1$. that is I must tell if there exist a solution $u$ in $C^1[-2,1]$ $$0= \frac{d}{...
1 vote
1 answer
38 views

Finding the value of a constant in a Boundary Value Problem

Let $t \to y(t)$ be a real-valued smooth function on any open interval containing $[0,1]$. Suppose that $y(t)$ satisfies the differential equation: $$y''(t) + w(t)y(t) = \lambda y(t)$$ where $\lambda$ ...
1 vote
1 answer
85 views

Invert Laplace Transform with Heaviside function

I'm solving the following boundary value problem $$ y \frac{\partial u}{\partial y}+\frac{\partial u}{\partial x}=1, \quad u(x, 1)=1=u(0, y) . $$ I've derived that $\bar{u}(p, y)=p^{-2}+p^{-1}-p^{-2} ...
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How can I solve $u_t = u_x + u_{xx}$ on an isolated half-rod (Robin conditions)?

This question seems so simple, yet the answer eludes me.. I'm trying to solve the advection-diffusion equation on the half-line x > 0: $$ u_t = u_x + u_{xx} $$ The advection is in the direction of ...
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2 votes
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The solution of a set of equations is about to appear but still elusive.

A set of equation is derived from an elastic problem with axisymmetry. The solution is assumed in a serial form, whose coeffecients are almost found but still remain elusive because of the expansion ...
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2 votes
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Initial boundary value problem for the 1-D Schrödinger equation

Consider the following initial boundary value problem for the 1-D Schrödinger equation for a function $u(t,x)$ in the domain $\Omega=[0,T]\times[0,L]$: $$ \begin{cases} iu_{t}(t,x)+\Delta u(t,x)=0,\...
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1 vote
1 answer
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How are higher order boundary conditions handled in a weak formulation?

I am reading Gazzola et al's book and there for a space $H^{m}(\Omega)$ they seem to allow boundary conditions that are of degree as high as $2m-1$. How is this reconciled with the fact that ...
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1 vote
1 answer
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What is the exact solution of this stationary convection-diffusion problem (1d)

I need some assistance with the following problem. I am trying to find the exact solution of a convection-diffusion equation as follows: $$-Du''(x)+0.5u'(x)=1, 0<x<\pi$$ $$u(0)=u(\pi)=0$$ where $...
3 votes
1 answer
125 views

Polyharmonic interpolation with mixed boundary conditions, variational formulation?

I am reading up on polyharmonic interpolation, though I want to use a different domain and boundaries than what is used by Duchon and Meinguet (notably the Green's functions do not even have a closed ...
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1 vote
0 answers
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Solvability of elliptic PDE with Neumann boundary condition

I'm trying to better understand the theory of second-order elliptic PDEs on a smooth compact region $\Omega \in \mathbb{R}^N$ with boundary conditions. For the Dirichlet case, there seems to be a ...
0 votes
0 answers
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Heat Wave problem

In case 2 I have stuck in the second case what do I have to do? Have I to use hyperbolic function? $$ \begin{align} u_{t}&=k u_{xx} \quad -l \leq x \leq l,\ t \geq0, k>0 \tag1 \\ u(-l,t)&...
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PDE with resetting/self referencial boundary conditions

I am trying to solve the pde of the form $$ -\rho v(x,t) - \frac{\partial v(x,t)}{\partial t} -\nu\frac{\partial v(x,t)}{\partial x}+\frac{\sigma^{2}}{2} \frac{\partial^{2} v(x,t)}{\partial x^{2}}=0 ...
1 vote
1 answer
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How to transform the boundary conditions between formulations of PDE? 2D fluid flow

Question: How do I transform the boundary conditions of the speed formulation $(u, \ v)$ into the vortex-stream formulation $(\psi, \ w)$? Cavity flow problem: A incompressible Newtonian fluid is in ...
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3 votes
1 answer
148 views

Laplace’s equation (using separation of variables)-- Is the solution correct?

I am trying to solve Laplace’s equation (using the separation of variables). I found a result below. But for some reason does not seems right to me. Any suggestions? Thank you in advance $$ \begin{...
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0 answers
65 views

How to "numerically" calculate eigenvalues of differential operator using Finite difference?

Consider a simple problem $$ \frac{d^2y}{dx^2} y = -\lambda^2 y.$$ I would like to calculate the eigenvalues of operator $\, d_x^2\, $ using finite difference, but I am not sure how to do it. ...
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0 answers
41 views

Time-evolution of a matrix of vector fields: well-defined?

Consider a $2 \times 2$ matrix of smooth vector fields on a smooth manifold $M$: $$X = \begin{pmatrix} X_{11} & X_{12} \\ X_{22} & X_{21} \\ \end{pmatrix}$$ I'm thinking about $X$ as an ...
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1 answer
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Numerically solving a BVP for a 2nd order ODE with mixed Dirichlet and Neumann boundary conditions with the finite difference method

I would like to solve a simple BVP for a second order differential equation on the domain $I=[0,\pi/2]$ using the finite difference method. $$ u''(x)+u(x)=0\\ u(0)=0\\ u'(\pi/2)=0 $$ By construction, $...
1 vote
1 answer
60 views

Calculus of Variations with first derivative boundary conditions

I am self teaching calculus of variations and have a seemingly basic question, which I cannot find the answer to. I understand that to obtain the shortest path between two points, $A$ and $B$, one can ...
0 votes
1 answer
43 views

Fourier series for a potential with boundary conditions on cylinder

I'm working with a problem from Electromagnetics where I'm supposed to calculate the potential distribution in two different intervals. However, I'm very unfamiliar with the boundary conditions and I ...
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1 vote
1 answer
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How to find Eigenfunction of this Sturm-Liouville problem

My question what do I have to do to find the eigenfunction in the first case It is given the following problem \begin{align} xy''(x)+y'(x)+ \frac{\lambda}{x}y(x) &=0, & &x\in(1,e^{2\pi}) ...
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1 answer
29 views

Difference between Neumann condition and Robin Condition

I'm a bit confused by the difference between a Neumann condition and a Robin condition. In the textbook a Neumann condition is defined as a normal derivative that is specified on the boundary. It's in ...
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Can I use this condition to solve a radially summetric equation with the finite difference method

Consider the symmetric radial heat equation: $$ f_t = \nabla^2f \Leftrightarrow f_t = f_{rr} + {1 \over r}f_r \tag 1$$ To solve this with the finite difference method using the FTCS scheme, we need ...

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