What is Boundary value problem: Definition and 78 Discussions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.
I want to solve this using difference equation. So I set up the general equation to be
Pi = 0.5 Pi+1 + 0.5 i-1
I changed it to euler's form pi = z
0.5z2-z+0.5 = 0
z = 1
since z is a repeated real root
I set up general formula
Pn = A(1)n+B(1)n
then
P0 = A = 1
PN = A+BN = 0 -> A= -BN...
Homework Statement
I try to integral as picture 1.
The result that is found by me, it doesn't satisfy Green's function for boundary value problem.
Homework EquationsThe Attempt at a Solution
show in picture 2 & picture 3.
Hello! (Wave)
I want to find the solution of the following initial and boundary value problem:
$$u_t(x,t)-u_{xx}(x,t)=0, x>0, t>0 \\ u_x(0,t)=0, t>0, \\ u(x,0)=x^2, x>0.$$I have done the following so far:
$$u(x,t)=X(x) T(t)$$
$$u_t(x,t)=u_{xx}(x,t) \Rightarrow...
Hello! (Wave)
Let $a,b>0$ and $D$ the rectangle $(0,a) \times (0,b)$. We consider the boundary value problem in $D$ for the Laplace equation, with Dirichlet boundary conditions,
$\left\{\begin{matrix}
u_{xx}+u_{yy}=0 & \text{ in } D,\\
u=h & \text{ in } \partial{D},
\end{matrix}\right.$...
Homework Statement
Suppose we have the standard rectangular potential barrier in 1D, with
$$
V =
\left\{
\!
\begin{aligned}
0 & \,\text{ if } x<0, x>d\\
V_0 & \,\text{ if } x>0,x<d\\
\end{aligned}
\right.
$$
The standard approach to solve for tunneling through the barrier is to match the...
Hi, I am interested in simulating the vacuum field equations, but solving a full boundary value problem rather than the initial value problem. i.e. I might have boundary conditions in all spatial and temporal extents/extremes, rather than just an initial 3D surface.
Does anyone know any free...
The solution of 1D diffusion equation on a half line (semi infinite) can be found with the help of Fourier Cosine Transform. Equation 3 is the https://ibb.co/ctF8Fw figure is the solution of 1D diffusion equation (eq:1). I want to write a code for this equation in MATLAB/Python but I don't...
a microchannel of length 2L and width h in the thermal cycling region. the temperature profile ...(1)
the cyclic temperature profile leads to a time dependent density ...(2)
using the mass conservation equation i.e. ...(3)
and momentum balance equation i.e. ...(4)
we have to find the exact...
The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time.
xsize = 10; % Model size, m
xnum = 10; % Number of nodes
xstp =...
What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask.
As part of an engineering project I've implemented...
Hello! (Wave)
I want to check if the following boundary value problem has a solution
$\left\{\begin{matrix}
-u_{xx}-4u=\sin {2x}, x \in (0,\pi)\\
u(0)=u(\pi)=0
\end{matrix}\right.$
I have thought the following:
We consider the corresponding homogeneous equation $-u_{xx}-4u=0$.
The...
I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0
where f(x) and g(x) are positive functions.
I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this.
I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...
I can't seem to find an explicit or analytical solution to a boundary value problem and thought I might ask those more knowledgeable on the subject than me. If t is an independent variable and m(t) and n(t) are two dependent variables with the following 8 constraints:
a) m' =0 @T=0 and...
Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...
So here I have Laplace's equation with non-homogeneous, mixed boundary conditions in both x and y.
1. Homework Statement
Solve Laplace's equation \begin{equation}\label{eq:Laplace}\nabla^2\phi(x,y)=0\end{equation} for the following boundary conditions:
\phi(0, y)=2;
\phi(1, y)=0;
\phi(x...
Hello! (Wave)Consider the boundary value problem
$\left\{\begin{matrix}
- \epsilon u''+u'=1 &, x \in [0,1] \\
u(0)=u(1)=0 &
\end{matrix}\right.$
where $\epsilon$ is a positive given constant.
I have to express a finite difference method for its numerical solution.
How can we know whether it...
Hey! :o
Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\partial{u}}{\partial{y}}\right ) -(1+x^2+y^4)u=f(x,y), x^2+y^2<1 \\ u(x, y)=g(x,y)...
Solve the boundary value problem:
$\left\{
\begin{array}{lcl}
y''&=&0,\hspace{1.0mm} 1<x<2\\
y(1)&=&0\\
y(3)+y'(3)&=&0
\end{array}
\right.
$
For the problem, I first calculate the eigenvalues and after check the roots and finally find the eigenvectors. Is correct this? Help me please :).
1. The problem statement, all variables a
nd given/known data
A rectangular trough extends infinitely along the z direction, and has a cross section as shown in the figure. All the faces are grounded, except for the top one, which is held at a potential V(x) = V_0 sin(7pix/b). Find the...
Hello, can anyone give me the general instructions of solving shooting method problem:
dy1/dx=-y1^2*y2
dy2/dx=y1*y2^2
with the boundary conditions: y1(0)=1, y2(1)=2
Suppose we have this rectangle that is stretched equally on both sides with some force, F.
Neglect shear force or moments and assuming transverse waves,
is the solution still
ε = Ae^(i(wt-kx))+Be^(i(wt+kx))
With boundary conditions:
X = +L/2, ∂ε/∂x = 0
and
X = -L/2, ∂ε/∂x =...
Hey! :o
I have to solve the following initial and boundary value problem:
$$u_t=u_{xx}, 0<x<L, t>0 (1)$$
$$u_x(0,t)=u_x(L,t)=0, t>0$$
$$u(x,0)=H(x - \frac{L}{2} ), 0<x<L, \text{ where } H(x)=1 \text{ for } x>0 \text{ and } H(x)=0 \text{ for } x<0$$
I have done the following:
Using the method...
Homework Statement
A unit sphere at the origin contains no free charge or conductors in its interior or on its boundary. It is, however, embedded in a dielectric medium. The dielectric is linear, but the permitivity varies by angle about the origin. It is constant along any radial direction...
Hi guys, so I'm stuck on quite an interesting problem, and have been for a few days now. If anybody can take the time to have a look at it that would be the most incredible thing ever, because I have reached a point where I am at a loss.
Solve the following 4th order differential equation...
Considering the classic problem in Electrodynamics "Conducting sphere with Hemispheres at different potentials"
How does one think in order to attack this problem? I didn't get it. What potential was considered in solving this problem? Was it the +V or the -V? Or both? Why is θ' considered...
Homework Statement
A rectangular plate extends to infinity along the y-axis and has a width of 20 cm. At all faces except y=0, T= 0°C. Solve the semi-infinite plate problem if the bottom edge is held at
T = {0°C when, 0 < x < 10,
T = {100°C when, 10 < x < 20.
Homework Equations
∇2T=0...
This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem.
I have to solve the following equation:
\frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t)
where \sigma > 0
The initial condition is:
\psi (x,0) =...
Hello,
I need help in solving the problem:
" find the lowest order uniform approximation to the boundary value problem εy''+y'sinx+ysin(2x)=0. y(0)=(pi), y(pi)=0. "
what I did:
y(out)=Ʃ(ε^n)y(n)
εy''(out)+y'(out)*sinx+y(out)*sin(2x)=0
for order 0: y'(out)*sinx+y(out)*sin(2x)=0...
Homework Statement
Solve the given BVP or show that it has no solution. (It does have a solution)
y"+2y = x, y(0)=y(\pi)=0
Homework Equations
Characteristic polynomial is r^2 + 2 = 0. μ = √2
The Attempt at a Solution
The solution to the complementary homogeneous equation is y_h...
Homework Statement
L[y] = \frac{d^2y}{dx^2}
Show that the Green's function for the boundary value problem with y(-1) = 0 and y(1) = 0 is given by
G(x,y) = \frac{1}{2}(1-x)(1+y) for
-1\leq y \leq x \leq 1\
G(x,y) = \frac{1}{2}(1+x)(1-y) for
-1\leq x \leq y \leq...
Let's say I have a set of nonlinear differential equations of the form.
x' = f(x,y) \\
y' = g(x,y)
Where f and g contain some parameter 'a' that is constrained to within certain values.
Let's say I know x(0), y(0) and x(T), y(T) where T isn't a set value. What methods can I use to...
Homework Statement
y'' +λy=0
y(1)+y'(1)=0
Show that y=Acos(αx)+Bsin(αx) satisfies the endpoint conditions if and only if B=0 and α is a positive root of the equation tan(z)=1/z. These roots
(a_{n})^{∞}_{1} are the abscissas of the points of intersection of the curves y=tan(x) and...
Homework Statement
Show that the boundary-value problem $$u_{tt}=u_{xx}\qquad u(x,0)=2f(x)\qquad u_t(x,0)=2g(x)$$ has the solution $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$ where ##G## is an antiderivative/indefinite integral of ##g##. Here, we assume that ##-\infty<x<\infty## and ##t\geq 0##...
Hi, I'm not sure if this is on the right thread but here goes. It's a perturbation type problem.
Consider the boundry value problem
$$\epsilon y'' + y' + y = 0$$
Show that if $$\epsilon = 0$$ the first order constant coefficient equation has
the solution
$$y_{outer} (x) = e^{1-x} $$...
Hi, I'm not sure if this is on the right thread but here goes. It's a perturbation type problem.
Consider the boundry value problem
$$\epsilon y'' + y' + y = 0$$
Show that if $$\epsilon = 0$$ the first order constant coefficient equation has
the solution
$$y_{outer} (x) = e^{1-x} $$
I have...
Homework Statement
Find the eigenvalues and eigenfunction for the BVP:
y'''+\lambda^2y'=0
y(0)=0, y'(0)=0, y'(L)=0
Homework Equations
m^3+\lambdam=0, auxiliary equation
The Attempt at a Solution
3 cases \lambda=0, \lambda<0, \lambda>0
this first 2 give y=0 always, as the only...
Hello,
I am trying to solve a vibration problem analytically but I don't understand how to implement the non-homogeneous boundary conditions.
The problem is defined as below:
y_{t}_{t}(x,t) = a^{2}y_{x}_{x}(x,t)
With
Boundary conditions:
y(0,t) = 0 [ fixed...
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I'm having difficulty with the boundary conditions on this problem. I don't need a solution or a step by step. I've just never solved a boundary condition like this.
Its the u(pi,t) = cos(t) that is giving me...
Homework Statement
Determine all the solutions, if any, to the given boundary value problem by first finding a general solution to the differential equation:
y" + y = 0 ; 0<x<2π
y(0)=0 , y(2π)=1
The attempt at a solution
So the general solution is given by: y = c1sin(x) +...
Homework Statement
y^{(4)}+\lambda y=0
y(0)=y'(0)=0
y(L)=y'(L)=0
Homework Equations
The hint says...
let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution
Listening to the hint, I got
r=\pm\mu With multiplicity 2 of each. So that means..
y=c_1 e^{\mu t}+c_2te^{\mu...
Can't seem to work this out,
any solutions would be greatly appreciated!
Thanks in advance!
Solve the boundary-value problem
Uxx + Uyy + U = 0 , 0<x<1,0<y<1
U(0,y) = 0 , Ux(a,y)= f(y)
U(x,0) = 0 , Uy(x,1)= sin(3*pi*x)
Homework Statement
Solve the given boundary value problem or else show that it has no solutions: y'' + 4y = cos x, y'(0) = 0, y'(pi) = 0.
Homework Equations
N/A
The Attempt at a Solution
So I made it all the way through the problem I think, but I am not getting the correct answer...
Homework Statement
A cantilever beam has uniform load w over a length of L as described by the eq.
EI y'''' = -w y(0) = y'(0) = 0 y''(L) = y'''(L) = 0
EI are constants
find y(x)
Homework Equations
L[y^4] = S^4*Y(s) - S^3*Y(0) - S^2*Y'(0) - s*Y''(0) - Y'''(0)
The...
Homework Statement
Use a Green's function to solve:
u" + 2u' + u = e-x
with u(0) = 0 and u(1) = 1 on 0\leqx\leq1
Homework Equations
This from the lecture notes in my course:
The Attempt at a Solution
Solving for the homogeneous equation first:
u" + 2u' + u = 0...