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Questions tagged [boundary-value-problem]

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

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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}{...
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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
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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. $e_k, k=1,2,3$ ...
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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 $V_i \subset\Omega$, $i=1,\dots,N$...
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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}\...
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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 ...
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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
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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 $x=0$: $\...
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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\...
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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^...
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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 ...
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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 ...
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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 \...
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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{\...
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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 ...
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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|\...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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_{...
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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, ...
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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}\|...
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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
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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
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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
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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 ...
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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
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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
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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 $$f(y)=sin(k_ny), \quad \lambda=k_n^2 $$ $$f(a)=0 \Longrightarrow k_na = n\pi, \quad n=0,1,2,...$$...
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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 - \...