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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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Difference between “essential boundary conditions” and “natural boundary conditions”?

In a boundary value problem, what's the difference between "essential boundary conditions" and "natural boundary conditions"?
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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 ...
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1answer
682 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta (x-x');...
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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 ...
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1answer
345 views

Problem with Heat Equation and Laplace Transform

So, my problem is this: I have to solve the heat equation with this boundary condition: $$ \begin{cases} \dfrac{\partial}{\partial t} u(x, t) - D\ \dfrac{\partial^2}{\partial x^2} u(x, t) = S_0\delta(...
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1answer
61 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-...
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1answer
55 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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1answer
73 views

Entropy solution of the Burgers' equation 2

I'm trying to construct a solution to the following problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $0<a<b$ I try to find a shock curve starting from point $(t,x)=(...
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2answers
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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\...
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1answer
320 views

Fourth order PDE solution for vibrating beam rigidly fastened at one end and simply fastened at other end.

I have Partial Differential equation in the form: $$ \frac{\partial^2 y}{\partial t^2} + \frac{\partial^4 y}{\partial x^4} =0, \quad 0<x<1 $$ Vibrating Beam Boundary Conditions: $$ y(0,t) = \...
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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 $...
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0answers
117 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D heat equation (Laplace equation) $$\nabla^{(3)}T_s=0$$ where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
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1answer
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Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
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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,...
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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 $y(x) = \sin \pi x$ $f(x)=(e^x- \pi^2)\sin \pi x + \pi \cos \pi x$...
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2answers
707 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 u}{\...
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3answers
815 views

Uniqueness of a holomorphic function with certain boundary values on an arc

Is it true that if a holomorphic function in the unit disk converges uniformly to the $0$ function some connected arc of the unit circle, this function is globally null? If that is true, this would ...
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1answer
818 views

Green's function using method of images

I'm working on the exercises 1.42 in the textbook "Green's functions and boundary value problems" by Stakgold and Holst, used in a graduate class "Applied Mathematics". Show that the electrostatic ...
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2answers
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Localized center modes with exponential decay tails, solved from non-linear differential equations

Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$: $$-a\big(\partial_r^2+\frac{\partial_r}{r}\big) U(r)+ B(r) \partial_r V(r)=0, $$ $$ -B(...
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1answer
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Solving Wave Equations with different Boundary Conditions

Right now I'm studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)...$) I know how to solve it when the boundary ...
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1answer
386 views

Reaction diffusion equation solution

This has been driving me spare for the last week, and I feel pretty bad for not being able to get a solution because (at least on the face of it) it's a pretty simple equation. I have the following ...
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0answers
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Oscillating modes with quickly exponential decay tails, solved from non-linear differential equations

Inspired by a previous post, I foresee a more generic set of real-function PDE may be solved analytically, with the property I am going to describe. Two coupled non-linear differential equations in ...
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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 ...
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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, ...
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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 ...
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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/...
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1answer
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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 ...
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2answers
363 views

Simplify Laplace equation in rectangle geometry

Consider Laplace's equation in a rectangle as shown in the following figure. The boundary conditions are shown in the figure. The problem is solved in the case of a1 =a2=1. Is there a way to ...
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0answers
340 views

second order differential equation with Green's function

I need to solve following differential equation \begin{eqnarray} y''(x) - k = \delta(x-x_0) \end{eqnarray} subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} Is it ...
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1answer
374 views

Counter example for uniqueness in Poisson equation with Robin Boundary conditions.

My question is related to the following. Prove the uniqueness of poisson equation with robin boundary condition I was thinking about the use of $a$ being positive. So I tried to find an example ...
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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^{-\...
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1answer
1k views

Biharmonic Equation on a square (fourier series solution needed)

$\nabla^{4}f=1$ for $f$ defined in a square from $x\in[-1,1]$ and $y\in[-1,1]$ The boundary conditions are: $f=0,f_{xx}=0$ on $x=\pm 1$ $f=0,f_{yy}=\mp 1$ on $y=\pm 1$ I intend to solve this ...
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0answers
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On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\...
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1answer
141 views

Effective Boundary Condition for a Heat Equation with Variable Conductivity

Consider a heat equation in one space dimension $$\frac{\partial u(t,x)}{\partial t} = \frac12\Theta(x)\frac{\partial^2u(t,x)}{\partial x^2} \tag{1}$$ where Heavyside function $$ \Theta(x) = \begin{...
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1answer
86 views

Solution of $y''(x) -k = \delta(x-x_0)y(x)$

I need to solve following differential equation $y''(x) -k = y\delta(x-x_0)$ subject to boundary conditions \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} I am not sure if it is possible ...
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1answer
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Why is $u(x, y) = A(x - y)$ the general solution for this PDE?

My Professor's notes had this problem: Solve the PDE $u_x + u_y = 0$ in the domain $y > \phi(x)$, $x \in \mathbb{R}$ given that $u = g(x)$ on the curve $y = \phi(x)$, where $\phi(x) = \frac{x}{1 ...
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0answers
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mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
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1answer
349 views

One dimensional wave equation on a finite string with Robin boundary condition

Show that the initial-boundary value problem \begin{align} & {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\ & u\left( x,0 \...
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1answer
548 views

Initial Boundary Value Problem for Wave Equation

I'm trying to solve this initial boundary value problem. But I'm getting stuck, some help would be appreciated. $$u_{tt}-u_{xx}=0, x>0, t>0$$ $$u(x,0)=0, u_t(x,0)=0, x\ge0$$ $$u(0,t)=1-\cos(t), ...
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1answer
106 views

Non zero solutions of second order equation by solving for $a$

Im stuck on this problem: Let a be a real constant. Consider the equation $y''+5y'+ay=0$ with boundary conditions $y(0)=0$ and $y(3)=0$ For certain discrete values of $a$, this equation can have non-...
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0answers
84 views

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
155 views

Finite difference method, boundary conditions

I'm not gonna show every approximation here since I know many of you are familiar with it, but my start equation is: $$-u''(x)+u(x)=f(x)$$ $$u(0)=0$$ $$u(1)=1$$ I can approximate this using finite ...
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1answer
41 views

Hyperbolic system with nonhomogeneous boundary conditions

I want to solve this problem but I'm stuck in the last step. I have followed all the steps below, but I don't know how to finish. Any Ideas? We consider the standard wave equation with ...
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2answers
183 views

Minimum value of the solution of the ODE.

So the given problem is $$y''-y=e^{-x},~y(0)=y'(0)=0$$ and $$y:\mathbb{R}\to\mathbb{R}$$ is a solution. They ask whether $y(x)$ is bounded and does it attain minimum in $\mathbb{R}$? So I solved the ...
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1answer
236 views

Trouble with derivatives using Newton-Raphson in MatLab

I'm finding it very difficult to get my head around how best to express the following system of equations in MatLab in order to solve it. The equations come from Von Karman's similarity solution to ...
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0answers
127 views

Solving an elliptic equation.

First, my question is : $U$ : bounded, open w/ $\partial U$ : $C^{1}$. Let $Lu=-\sum_{i,j=1}^{n}\left(a^{ij}u_{x_{i}}\right)_{x_{j}}$ where $a^{ij}\in L^{\infty}\left(U\right)$ : symmetric ...
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0answers
106 views

$w_x(L,t)w_t(L,t) - w_x(0,t)w_t(0,t) \le 0$?

Is $w_x(L,t)w_t(L,t) - w_x(0,t)w_t(0,t) $ equal to zero or nonpositive assuming any of the following? $$w_{tt} (x,t) = w_{xx} (x,t), \ x \in (0,L), \ t \in (0,T) \ (1)$$ $$w_{tx} (x,t) = w_{xt} (x,t)...
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2answers
41 views

Why $\sum_nA_n(X_n,X_m)=A_m(X_m,X_m)$?

Why in the following proof $$\sum_nA_n(X_n,X_m)=A_m(X_m,X_m)$$ ? The author says it's because orthogonality but orthogonality means $(f,g)=\int_a^bfgdx=0$. So how come orthogonality helps to prove it ...
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0answers
58 views

Can I use separation of variables?

I have the PDE $u_t-u_{xx}=-u^2$ with boundary conditions $u(0,t)=0$ , $u(a,t)=0$ and $u(x,0)=0$ for $0<x<a$ and $0<t<T$ I have tried to use separation of variables and Fourier's method ...
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1answer
3k views

Laplace equation in 1D with MATLAB - Dirichlet boundary condition

Here is a Matlab code to solve Laplace 's equation in 1D with Dirichlet's boundary condition u(0)=u(1)=0 using finite difference method ...