Questions tagged [boundary-value-problem]
For questions concerning the properties and solutions to the boundary-value problem for differential equations.
997 questions
1
vote
1answer
40 views
Calculus - Integral - existence problem
We are given a function $f$, $f$ is integrable (in the riemann sense) in $[a,b]$ and also $f'$ is a continuous, and $f(a)=f(b)=0$. Prove that there exists a point $c$ such that $|f'(c)| \geq \frac{4}{...
-1
votes
2answers
33 views
Laplace Equation in Rectangle: Boundary Value Problem [on hold]
Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(...
2
votes
1answer
39 views
Solving a system of ODE
Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$
$$C_1(0)=0, C_2(0)=A, C_3(0)=0$$
$$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$
where $A,B,\eta_k$ some known constant. $e_k, k=1,2,3$ ...
0
votes
0answers
11 views
Discretization Dirichlet boundary condition for Elliptic PDE with finite volume method
I want to discretize the following equation using FMV:
$$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2
\\u_{|\partial\Omega}=g$$
To this end, let $V_i \subset\Omega$, $i=1,\dots,N$...
3
votes
0answers
25 views
Curious Question on Inhomogeneous Boundary Conditions of a PDE
I am given the PDE, $\ u_t=u_{xx},$ with boundary conditions $u(0,t)=A, \ u(1,t)=B$ and $u(x,0)=f(x)$. I have found the solution of this PDE is $$u(x,t)=A+(B-A)x+\sum_{n=1}^{\infty}B_ne^{n^2\pi^2 t}\...
1
vote
0answers
39 views
Solving definite integral in two variables.
Solving a PDE with the following boundary problem with arbitrary constant $b$:
$$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$
Hint given in the question is as follows:
Split this integral in two ...
0
votes
0answers
11 views
REFEREENCE REQUEST for Non-Local Boundary Value problems
It would be really helpful if someone could suggest me any reference (Books or Papers) where I would find worked-out examples of Elliptic Boundary value problems (especially Laplace equation) with non-...
11
votes
1answer
233 views
Robin BC in the 1D wave equation
The problem of interest is as follows:
the quantity of interest: $u(x,t)$
the wave equation: $\partial_2^2u(x,t)-c^2\partial_1^2u(x,t)=0$ where $c>0$
one Robin boundary condition at $x=0$: $\...
1
vote
2answers
45 views
Solve PDE using method of characteristics with non-local boundary conditions.
Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ :
$$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$
$$u(a,0)=u_0(a)\,\,\,a≥0$$
$$u(0,t)=F(t)=b\...
2
votes
2answers
35 views
Solving the BVP $u_{xx}+a^2u=\sin(\pi x)$ with $u(0)=1$ and $u(1)=-2$
I am trying to solve the BVP $$u_{xx}+a^2u=\sin(\pi x), \ \ \text{for} \ \ 0<x<1$$ with $u(0)=1$ and $u(1)=-2$, $\forall a\in\mathbb{R}$.
I begin by solving the homogeneous equation $u_{xx}+a^...
0
votes
0answers
34 views
+50
Laplace equation in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]
I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
0
votes
1answer
19 views
Laplace Equation with Inhomogeneous Boundary Condition
I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous.
I'm having trouble deciding how to handle one of the boundary ...
0
votes
0answers
9 views
Matrix representation of a finite difference with Neumann boundary conditions
Given 1D data $[c_1,c_2,c_3,\cdots,c_N]$ I can represent the derivative operation as a matrix product. For example, using the central difference
$$
\left.\frac{\partial c}{\partial x}\right|_k
\...
0
votes
1answer
27 views
Separating variables in a PDE with multiple constants
My question is:
How do you use separation of variables on a PDE that has more than one constant in it?
All the examples I can find in my book/online only have one constant in it, like $$ \frac{\...
1
vote
0answers
19 views
A sufficient condition for a Neumann problem to have solution
Suppose we have the Boundary Value Problem (BVP):
$\Delta{u}=f$ , in the domain (topos) $D$
$\partial_{n}{u}=h$, in $\partial{D}$
It is easy to prove using $1$st Green's identity that a necessary ...
0
votes
0answers
22 views
Poisson integral formula for boundary value problem
I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\...
0
votes
1answer
30 views
Proving that the eigenvalues of the Airy problem are positive
I am solving an exercise concerning the Airy eigenvalue problem
$$ -y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*) $$
which (among other things) asks me to prove that all eigenvalues are positive.
I ...
2
votes
3answers
40 views
Method of Characteristics for $u_t + uu_x = -2u$
Consider the following quasi-linear PDE : $u_t + uu_x = -2u$, with the boundary condition $u(0,t) = e^{-t}$.
Show, using the method of characteristics, that the solution to this boundary value ...
0
votes
1answer
24 views
Helmholtz equation with robin boundary condition
Consider the differential equation $(\nabla^2+\frac{1}{R^2})\psi(\bar{r}) = 0$ in 2 dimensions, with the boundary condition $\partial_r\psi(R)+ \kappa \psi(R) = 0$, on unit disk of radius R. What is ...
1
vote
0answers
26 views
Finding the equation from boundary values [closed]
I have two equations like
$$\frac{\partial Y}{\partial x}=A(L-2x)\;\text{ at }(x,0)$$
And
$$\frac{\partial Y}{\partial t}=0\;\text{ at }(0,0)$$
Is it possible to find out the real equation?
(That is ...
0
votes
0answers
5 views
Boundary value problem for Hemlholtz equation with pseudospectral method.
I have an equation of the form $(1-0.1 \Delta)f=1$ with boundary condition $f=0$. I need to solve it for $f$ by pseudospectral method in python. Apparently, it should be $f=\frac{1}{1-0.1\Delta}$. I ...
0
votes
0answers
7 views
Inverse of differential operator and boundary conditions
I want to clear a point that "Why boundary conditions are important in taking inverse of any differential operator (lets say Laplace operator )?". What i understood is that any transformation is ...
1
vote
0answers
28 views
Non-Dimensionalizing Separation of Variables PDE
I have a problem where I'm considering a cylinder with a closed end at the bottom (x = 0) and open end at the top (x = L).
The problem is looking at pressure disturbances throughout the air column.
...
0
votes
0answers
18 views
MATLAB solving BVP using bvp4c, variable coefficient
Say I have an bvp that involve the equation below
h''' = f(r)*h'' + h'
If I was to have enough boundary conditions to satisfy h(r) ODE, would it be possible to solve the f(r) equation ...
1
vote
1answer
37 views
Solve $u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}=0.$
I have two BVP's and for both of them the solution is similar, however there is one thing I can't explain. First of all let me state the BBVP's. The first one is
$$(1)=
\begin{cases}
&u_{...
0
votes
0answers
6 views
The boundary value problem has a unique solution if and only if the corresponding homogeneous system has a unique solution
I want to prove that the following claims are equivalent:
1) The boundary value problem $Ly = f, By = c$ has a unique solution for all $f \in \mathcal{C}[a,b]$ and $c \in \mathbb{C}^2$,
2) $Ly = 0, ...
0
votes
0answers
39 views
What means norm $\|u'\|$ and $\|u\|$ in Sobolev space $W^{1,2}(I)$ and why is functional continuous?
We have functional $F: W^{1,2}(I),I\langle 0,1\rangle \rightarrow \mathbb{R}$, I proved that functional is linear, I also counted that is bounded and I get this state:
$$|F(u)|\leq {C_1} \|u\|+{C_2}\|...
0
votes
0answers
18 views
Two-dimensional Laplace equation with weird Robin BC
I need to solve the steady-state heat equation a.k.a. Laplace equation over a rectangle
For $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$ defined on $x \in [0,a]$ and $y ...
2
votes
2answers
39 views
Boundary value problem $u_t-u_{xx}=\cosh(x), \ 0<x<4, \ t>0.$ (PDE)
So I want to solve the problem:
\begin{align} u_t-u_{xx}&=\cosh(x), \quad 0<x<4, \quad
t>0\\u(x,0)&=v(x),\\u(0,t)&=0,\\u(4,t)&=0. \end{align}
The inhomogeneous part is ...
0
votes
1answer
34 views
Solution Poisson Equation by Homotopy Perturbation Method
I need to solve the following Boundary Value problem
$$\frac{\partial^2w }{\partial x^2}+\frac{\partial^2w }{\partial y^2}=c$$
Boundary conditions are
$$w(x,h)=0$$
$$w\left(\frac{\pm h}{\sqrt3},...
0
votes
0answers
17 views
Differential equation with different initial condition
Given an vector field $v:U (\subset R^n)\to{R^n}$. Consider the differential equation $\dot{x} = v(x)$. It's given that this differential equation with an intial condition $x(0) = x_0$ has unique ...
0
votes
2answers
43 views
Solve the boundary-value problem $u_t + u_{xx} = 0$
Consider the boundary-value problem
$$\frac{∂u}{∂t} - \frac{\partial^2u}{∂x^2} = 0, \quad x\in[0,2], t\in[0,\infty) $$
$$u(0,t) = u(2,t)=0$$
$$u(x,0)=x(x-1)(x-2)$$
Show that $u(x, t) = −u(2 − x, ...
1
vote
2answers
26 views
Differential equation: mixed boundary condition, how to solve numerically?
I have a differential equation (dot means derivative w.r.t time)
$$(\dot x, \dot y) = f(x,y)$$
and I am given the initial condition for $x$, but a final condition for $y$:
$$x(0),\qquad y(1)=g(x(1))$$ ...
0
votes
2answers
44 views
Solving 2nd order with boundary conditions
$$f''+\lambda f = 0, \quad f=f(y), \quad a \leq y \leq b, \quad f(a)=f(b)=0 $$
In this case, I tried
$$f(y)=sin(k_ny), \quad \lambda=k_n^2 $$
$$f(a)=0 \Longrightarrow k_na = n\pi, \quad n=0,1,2,...$$...
-2
votes
1answer
37 views
How to solve this differential equation for $\psi_n$?
How to solve this differential equation for $\psi_n$?:
$$\frac{1}{2}\frac{\partial^2}{\partial x^2}\psi_n=\lambda_n\psi_n$$
apparently this is a heat equation but I cannot find information on this. ...
0
votes
1answer
24 views
Finite difference method without fiction points
I need to use finite difference method without introducing fiction points to solve the following problem:
$−\mu u′′(x)+\eta u′(x)+\sigma u(x)=f(x)$, $a<x<b,$
subject to the boundary conditions $...
3
votes
0answers
34 views
How to solve wave equations with boundary condition $u_x(0,t)=h(t)$?
\begin{align*}
u_{tt}-c^2u_{xx}=0, x>0\\
u(x,0)=u_t(x,0)=0\\
u_x(0,t)=\frac{t}{1+t^2},t>0
\end{align*}
According to the textbook, I should look for solutions in the form $u(x,t)=F(x-ct)$ and ...
3
votes
0answers
72 views
Uniqueness of the potential flow past a cylinder
I have a question regarding the uniqueness of the potential flow past a cylinder.
Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \...
2
votes
1answer
60 views
Shock formation for conservation law with initial and boundary data
Suppose we have $$u_t + f(u) u_x = 0$$ where $t, x > 0$, and initial conditions $u(x,0) = C$, where $C>0$ is constant, and $u(0,t) = g(t)$, where $t>0$. We know the solution is $$u(x,t) = F(x-...
1
vote
2answers
22 views
non-trivial solution boundary problems
Given the boundary problem
$$X''=\mu X,\;X(0)=0,\;X'(L)=0$$
We need to find the non trivial solution $X(x)$ that satisfies the above equations. Let $\mu<0$, $\mu=-k^2$ for some $k>0$. The ...
1
vote
0answers
13 views
Redefining BC within heterogeneous BVP
I am trying to solve the following boundary value problem using an Artificial Neural Network, over the dimensions $x$ and $t$ for $x \in [0,1]$ and $t \in [0,1]$. It can be thought of a representing ...
4
votes
0answers
115 views
3D Homogenous Laplace equation with integral boundary conditions
I have the 3D heat equation (Laplace equation)
$$\nabla^{(3)}T_s=0$$
where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
2
votes
0answers
71 views
Laplace [Heat] type equation with source terms
$$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} =\beta_h e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \, \mathrm{d}x + \beta_c e^{-\...
3
votes
0answers
90 views
Converting between Solution forms using Green's Functions in Linear Differential Equation
EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ...
1
vote
1answer
50 views
Burgers' Equation with Initial and Boundary Conditions
Consider a first-order PDE:
$$u_t + (1 + 2u)u_x = 0$$
valid on
$$0 \leq x \leq \infty$$
$$0 \leq t \leq \infty$$
with
Initial condition:
$$u(x, 0) = 0$$
and boundary condition:
$$u(0,t) = \begin{...
0
votes
0answers
26 views
Laplace equaion with integral source terms
I have the following coupled PDEs:
\begin{eqnarray}
\frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\
\frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - ...
0
votes
1answer
61 views
Laplace's Equation in the semi infinite strip
The problem is $$\Delta u = 0$$ in the semi infinite strip $S = \{(x,y)|0<x<1,y>0\}$ subject to the boundary conditions $$
\begin{cases}
u(0,y) = 0,\ y\ge0\\
...
0
votes
1answer
57 views
1D Wave equation mixed boundary conditions and I.C.
I have been searching for a solution online, but cannot find one that fits the B.C. and I.C. for this wave equation. I read through this PDF, page 7; although I had similar conditions I just obtained ...
0
votes
1answer
53 views
Solving PDE using separation of variables (Heat diffusion)
I am trying to solve a standard PDE, but I got stuck on how to choose the separation constant such that I do not end up with a trivial, uninteresting solution. The system is presented below and my ...
0
votes
0answers
47 views
Suitable ansatz for Coupled system of PDEs
I have the following three PDEs
\begin{eqnarray}
\frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\
\frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \...
