Questions tagged [boundary-value-problem]
For questions concerning the properties and solutions to the boundary-value problem for differential equations.
2
votes
0answers
90 views
Eigenvalues keep giving trivial solutions everytime.
I am trying to find the eigenvalues of this Eigen BVP. $\mu$ is the eigenvalue parameter
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \...
3
votes
1answer
52 views
Newtons law of cooling applied to spherical region
I'm having trouble solving the following problem:
Formulate a mathmatical model for a stationary (steady) temperature distribution inside the spherical volume
$$
R^2\leq x^2+y^2+z^2\leq (2R)^2,
$$
...
0
votes
0answers
46 views
How to proceed further in this Eigen Boundary value problem
I have the following eigenvalue BVP
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F = 0
$$ wit BC(s) $F(0)=0,\frac{F''(0)}{F'(0)}=\...
0
votes
0answers
30 views
Converting BVP into standard Eigenfunction Eigenvalue form
I have a eigen boundary value problem
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2F=0
$$
$\mu$ is the separation variable or the ...
0
votes
1answer
29 views
Understanding a proof exercise Evans PDE mean value formula
I am trying to understand a solution to exercise 2.5 (3) in Evans' PDE pag 85-86, which tates the following:
The part in the Oval is the one I have trouble understanding. Thanks in advance for any ...
0
votes
1answer
67 views
solve 2nd order PDE
I have a PDE:
$$
\frac{\partial^2 u}{\partial x^2} + a\frac{\partial^2 u}{\partial y^2} +bu = f(x,y)
$$
where $a$ and $b$ are constants and $b>a>0$.
Also $\space u(x,0) = g(x)$, and $\space u(0,...
0
votes
0answers
16 views
Initial guessing to bvp4c MATLAB
I am working on a 4th order non-linear variable coefficient homogeneous ODE bvp.
I am having issues getting a solution using bvp4c. This could be one of many things. Not having a solution within the ...
1
vote
1answer
62 views
Laplace transform on a Boundary value problem (ODE)with . Am i attempting correctly?
From a system of PDEs where i used the following ansatz: $$\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$$. $F(x) := \int f(x) \, \mathrm{d}x$ and $G(y) := \int g(y) \, \mathrm{d}y$
So, $$\...
1
vote
1answer
47 views
Analytical solution of heat equation with non-homogenous boundary conditions
I am trying to get analytical solution of heat equation with non-homogenous boundary conditions, which i can code in MATLAB and compare with my numerical results.
In short, i am unable to reach the ...
2
votes
1answer
37 views
Integro-differential equation including a convolution of the first derivative.
I am having difficulty finding the right approach to solving the following differential equation,
$$
y''(t)+\int_t^Tg(s-t)y'(s)\,ds=f(t),
$$
with the boundary conditions,
$$y(0)=y_0\,,\quad y(T)=0.$$
...
2
votes
0answers
27 views
Can this Helmholtz PDE with Robin boundary conditions be solved analytically?
Consider the following Helmholtz problem in the infinite triangle $y>0,\;x>y$ with parameters $Q<0$, $P\ge0$, $P<|Q|$.
$$\left\{\begin{align}
&\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\...
1
vote
1answer
48 views
Tipping ladder equation
I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $\phi$...
4
votes
0answers
93 views
Partial differential equation with a nowhere differentiable boundary
Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. ...
0
votes
1answer
28 views
What numerical method does Matlab's bvp4c use?
Can anyone shed some light on how matlab's bvp4c function works? I've looked online but I haven't found any specifics on the method it uses.
With that question asked, what are some different ways on ...
1
vote
0answers
23 views
Underdetermined IBVP's: constructing boundary condition during evolution
For $(t,x)\in[0,T]\times[0,X]$ let there be a partial differential equation (PDE) of the generic form
$$\partial_ty=f(t,x,y,\partial_xy)$$
along with some initial condition (i.c.)
$$y_i(x):=y(0,x)$$
...
0
votes
1answer
27 views
2 Layer Finite Difference Scheme PDE
I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.
$\frac{∂^2}{∂x^2}(k(x,t) \frac{∂^2U(x,t)}{∂t^2})=0$
k is just a parameter, which is dependent on x and t. The problem ...
2
votes
0answers
44 views
Weird eigensystem of ODE with regular singularity
Consider the eigenvalue problem of the following 2nd-order ODE $$(x/2+a)^2y(x)-xy'(x)-x^2y''(x)=\lambda^2y(x),$$ in which $y\in(-\infty,+\infty)$ and parameter $a>0$. It has a regular singularity $...
0
votes
2answers
65 views
Analytical solution for PDE-system's IBVP to validate method of lines
I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific ...
1
vote
0answers
31 views
best software package to numerically solve high order nonlinear ODE boundary value problem
I need to numerically solve a 4th and 5th order nonlinear ODE BVP and I was hoping I could get some advice on the best software package to solve these types of problems. I've used MATLABS bvp4c for ...
2
votes
1answer
62 views
First-order quasilinear PDE system analysis
For $(t,r)\in[0,\infty)\times[0,1]$ let be the following PDE's system
$$\dot{\vartheta}= w' +w\vartheta' $$
$$\dot{w}= \vartheta'+ww' $$
along with initial conditions (i.c.)
$$w(0,r)=w_i(r)\qquad \...
1
vote
1answer
19 views
How to build linear system of approximation of solution to boundary value problem
We want to numerically approximate the following problem using finite differences.
$$y''=f(t,y,y') \ \ \ \ \ a \leq t \leq b$$
$$y(a) = \alpha \ \ \ \ \ y(b) = \beta$$
We can divide the interval $[a,...
2
votes
1answer
28 views
Inequality for the $L^2$ norm of a derivative in a Dirichlet problem
I found some trouble in solving this problem:
given $f$ a continuous function in $[0,1]$ and $u\in C^2([0,1])$, such that $u''(x)=f(x)$ on $(0,1)$ and $u(0)=u(1)=0$, prove, $\forall\epsilon > 0$, ...
0
votes
1answer
42 views
Solutions to the heat equation with a ring of coolant
I'm interested in solutions to the heat equation for the following problem
$ \dfrac{\partial u}{\partial t} = c^2 \nabla^2 u $
on $ \left\{ (x,y) \vert x^2+y^2 \leq R^2 \right\} $
such ...
0
votes
0answers
14 views
Systems of Partial differential equations: initial-boundary value problems: in search of numerical code test.
I am in search of constrained evolution problems defined in some region $$(t,r)\in[0,\infty)\times[0,R]$$
concerning PDE's systems and boundary conditions of the general form
$$ x_1' = f_1(\vec{x},\...
1
vote
0answers
34 views
Numerical solution for boundary value problem
I need to solve a 4th order non-linear boundary value problem in the following form:
$$
\left\{\begin{matrix}
x'_1 = f_1(t, x_1, x_2, x_3, x_4) & x_1(0) = a_1 \\
x'_2 = f_2(t, x_1, x_2, x_3, ...
0
votes
1answer
49 views
Symmetric Boundary Conditions/Eigenvalues (PDEs)
Consider the following eigenvalue problem for the Laplacian
$-\Delta u = \lambda u$ in $U$
$u + a \left(\frac{\partial u}{\partial v}\right)$ on $\partial U$
where $v$ is the outward unit normal to ...
0
votes
2answers
38 views
Laplace equation with the Robin's boundary problem
$\textbf{Problem}$ Let $\Omega$ be an open, bounded and connected subset of $\mathbb{R}^n$. Suppose that $\partial \Omega$ is $C^{\infty}$. Consider an eigenvalue problem
\begin{align*}
\begin{...
0
votes
0answers
31 views
Boundary conditions and decomposition on spherical harmonics
Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or ...
3
votes
1answer
93 views
Estimate of a weak solution in a nonhomogeneous equation
$\textbf{Problem}$ Let $\Omega \subset \mathbb{R}^n$ be open, bounded and connected with $\partial \Omega\in C^1$. For each $i,j=1,\cdots,n$, assume that $a_{ij},b_i,c \in L^{\infty}(\Omega)$ (real ...
0
votes
0answers
22 views
Mixed boundary condition for the heat equation
Would someone help me understand the way the solution obtained in this question:
Heat Equation Mixed Boundaries Case: Fourier Coefficients
I did not understand why in the final solution, he took $...
0
votes
0answers
16 views
Solving an initial boundary value problem for the wave equation in one spatial dimension
The question is as follows:
Solve the following initial boundary value problem for the wave equation in one spatial dimension:
$$
\begin{cases}
\displaystyle \frac{\partial^2 u}{\partial t^2} - \...
0
votes
0answers
22 views
Transform an inhomogeneous dirichlet problem into a homogeneous one? i.e. Burgers equation
I need to transform an inhomogeneous dirichlet problem into a homogeneous one. For exeample the burgers equation:
$$\begin{cases} \partial_t u(x,t) + u(x,t) \partial_x u(x,t)=0\\
u(0,t)=0,& \...
0
votes
0answers
13 views
PDE with Robin Boundary Condition at alpha Solving with Poisson's Equation
Using a code like this, I am having a hard time applying a Robin Boundary condition for a instead of a dirichlet for the following problem:
IMG OF QUESTION FOR POISSON ROBIN BC
Nx = ...
-3
votes
1answer
67 views
How to solve this Boundary Value problem? [closed]
$$ \left(\frac{d^2 u}{d x^2} \right)+\frac{2}{x+x_0} \frac{d u}{d x}=\frac{\lambda}{k} u; \quad x \in [0,L], k,x_0>0 $$
$$u(0)=0$$
$$u(L)=0$$
0
votes
0answers
15 views
Burgers equation with zero boundaries
I am curious how I can avoid non-zero boundary values for the burgers equation, my idea is something like this:
$$\begin{cases} \partial_t u(x,t)+ u(x,t)\partial_x u(x,t) = f(x,t)&\\
u(x,0)=0,&...
0
votes
1answer
26 views
Finite intersection property for a locally finite open cover $\{U_j\}$ seems to contradict itself and imply that $\{U_j\}$ isn't actually a cover?
On page 84 of Adam's book on Sobolev spaces he states the 'uniform $C^m$-regularity condition' for a domain $\Omega$. This condition states that when we have a locally finite open cover $\{U_j\}$ of ...
1
vote
1answer
35 views
Solve a Sturm-Liouville Boundary Value Problem
Solve the Wave Equation
I've been trying to solve the above wave equation
where $u = u(x, t)$ and $c ∈ \mathbb{R}$ is a constant, subject to
$$
u(x, 0) = 0,\;\; 0 < x < 1, \\
u_t(x, 0) = U_0x,\...
0
votes
0answers
34 views
Obtaining boundary conditions
I'm trying to numerically solve the three coupled PDEs;
$\frac{\partial\theta}{\partial t} = w + \nabla^2 \theta, \ \ \ \ \ (1)$
$\frac{\partial Q}{\partial t} = -RaPr\nabla^2_H\theta + \nabla^2 Q, ...
0
votes
0answers
10 views
Systems of BVP with unbalanced BCs
I have a system of ODEs forming a BVP. I was thinking of using Finite Difference to solve it but for some functions I do not have the BCs on both sides (though in total they are enough to make the ...
0
votes
0answers
27 views
boundary problem: use main theorem of monotone operators
I am trying to investigate for which $\alpha \in \mathbb R$ the boundary problem
$$-u''(x)+\alpha sin(u(x))u'(x)=f(x)$$
$$u(a) = u(b) = 0$$
is weakly solvable using the main theorem of monotone ...
0
votes
0answers
22 views
Finite difference replacement of a PDE
I'm having trouble with this question. I think that of of the partial derivatives should be looking for finite difference approximation of these two derivatives using the Taylor series expansion but ...
2
votes
2answers
45 views
Well-posedness of heat-equation PDE with only one initial condition
Consider the PDE given by $u_t = \alpha u_{xx}$ with initial condition $u(x, 0) = f(x)$.
Now suppose we discretize the problem in the time variable, so we approximate $u_t(x, t)$ by a finite ...
0
votes
0answers
8 views
Showing that a square wave of unity amplitude can be bound by the subtraction of its fundamental component and a third order harmonic.
Consider a normal square wave that has a switching angle of $\alpha = \cos^{-1}\left(\frac{\pi}{4} m \right)$. (Note that these values are found from Fourier Analysis of a square wave with varying (...
0
votes
0answers
63 views
Applying neumann boundary conditions to diffusion equation solution in python
For the diffusion equation
$$\frac{\partial u(x,t)}{\partial t}=D\frac{\partial^2 u(x,t)}{\partial x^2}+Cu(x,t)$$
with the boundary conditions $u(−\frac{L}{2},t)=u(\frac{L}{2},t)=0$ I've programmed ...
0
votes
1answer
46 views
Uniform Cone Condition with Unit Normal Vector
I was going through this article from M. BRAMSON, K. BURDZY AND W. KENDALL
https://projecteuclid.org/download/pdfview_1/euclid.aop/1362750941
where the uniform interior cone condition is given this ...
1
vote
1answer
58 views
Theoretical Solution for this Poisson Equation Problem
$\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + 1 = 0$
with the following boundary conditions:
$\phi(\pm 1,y)=0 \ and \ \phi(x,\pm 1) = 0$
I was able to solve this ...
0
votes
0answers
27 views
Mixed Cauchy and Dirichlet and unspecified boundary conditions for Laplace equation on $I^2$
I am looking for a reference where the following problem is discussed:
$u \in C^{\infty}(I^2)$
so that
$\Delta u = 0$
$u(0,y) = f(y)$, $u(1,y) = g(y)$, $u(x,0) = h(x)$
$\nabla u(x,0) \cdot \hat{n} (...
1
vote
0answers
44 views
How to solve one variable given two variables and one inequality?
So I have a function:
$\phi(P) = \lvert (1-\alpha_2)^P - (1-\alpha_1)^P\rvert$
It is easy to show that $\phi$ is increasing between $0$ and $P_M$, where
$P_{M} = \lvert \ln\bigg(\frac{\ln(1-\...
0
votes
2answers
98 views
Solution of the parabolic equation
The problem is set as follows:
$$\frac{\partial W(\tau,z)}{\partial \tau}=-\frac{1}{\varepsilon} \frac{\partial(z_1 W(\tau,z))}{\partial z_1}-\frac{\partial(z_2 W(\tau,z))}{\partial z_2}+h^2\frac{\...
1
vote
0answers
41 views
Finding the Green Function
I'm having some trouble finding the Green function of the following differential equation:
$$
\frac{d[x y'(x)]}{dx} = f(x)\\
0 \leq x \leq 1\\
y(1) = 0
$$
$y(0)$ is finite.
