Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.