# Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations. By a Boundary value problem, we mean a system of differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.

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### Techniques for solving the Navier-Cauchy equations of linear elastostatics

I'm trying to get an exact solution to a specific boundary value problem in linear elasticity. Standard techniques one would use to solve the heat or wave equations, such as separation of variables ...
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### Proving a regularity estimate for $−εu′′ + bu′ = f$

I've been given: $$−εu′′ + bu′ = f ,x ∈ (0, 1), u(0) = u(1) = 0$$ and have been asked to prove the regularity estimate $$∥u′′∥_{L^2(0,1)} ≤ C_R∥f∥_{L^2(0,1)}$$ I normally try to provide some working ...
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### Method of characteristics - inhomogeneous pde

The boundary conditions problem: $2\frac{\partial u}{\partial x}-14\frac{\partial u}{\partial y}=3(x+y)$ Where the boundary conditions are: $u(x,-6x)=Cos((2x)^2) \quad \forall x \in \mathbb{R}$ How ...
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### General compatibility condition for pure Neumann problem

It is known than for a second order elliptic boundary-value problem with pure Neumann conditions, a certain compatibility condition between the data must be satisfied. For example, in the case of the ...
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### Transformation of boundary conditions at Laplace Equation in polar Coordinates

It is given the following BVP \begin{align} \Delta u &=0 , \quad 0\leq r <R,\quad 0<\phi<\alpha\leq 2\pi \tag1\\ u(r,0)&=0,\quad 0\leq r <R \tag 2\\ u(r,\alpha)&=\alpha,\quad 0\...
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### FEM code for PDE subdomain interface condition

I have a simple problem to solve. I have two pde, one for $u_{1}$ and one for $u_{2}$, where the domain in space for $u_{1}$ is $[-L_{0}, 0]$ , and for $u_{2}$ the space domain is $[0,L_{0}]$. I ...
1 vote
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I am trying to solve the following ODE with boundary conditions in the set $C^1[-2,1]$, with $u(-2)=-1/2$ and $u(1)=1$. that is I must tell if there exist a solution $u$ in $C^1[-2,1]$ $$0= \frac{d}{... • 2,103 1 vote 1 answer 38 views ### Finding the value of a constant in a Boundary Value Problem Let t \to y(t) be a real-valued smooth function on any open interval containing [0,1]. Suppose that y(t) satisfies the differential equation:$$y''(t) + w(t)y(t) = \lambda y(t)$$where \lambda ... • 395 1 vote 1 answer 85 views ### Invert Laplace Transform with Heaviside function I'm solving the following boundary value problem$$ y \frac{\partial u}{\partial y}+\frac{\partial u}{\partial x}=1, \quad u(x, 1)=1=u(0, y) . $$I've derived that \bar{u}(p, y)=p^{-2}+p^{-1}-p^{-2} ... • 219 0 votes 0 answers 41 views ### How can I solve u_t = u_x + u_{xx} on an isolated half-rod (Robin conditions)? This question seems so simple, yet the answer eludes me.. I'm trying to solve the advection-diffusion equation on the half-line x > 0:$$ u_t = u_x + u_{xx} $$The advection is in the direction of ... • 163 2 votes 0 answers 91 views ### The solution of a set of equations is about to appear but still elusive. A set of equation is derived from an elastic problem with axisymmetry. The solution is assumed in a serial form, whose coeffecients are almost found but still remain elusive because of the expansion ... • 33 2 votes 0 answers 61 views ### Initial boundary value problem for the 1-D Schrödinger equation Consider the following initial boundary value problem for the 1-D Schrödinger equation for a function u(t,x) in the domain \Omega=[0,T]\times[0,L]:$$ \begin{cases} iu_{t}(t,x)+\Delta u(t,x)=0,\...
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I am reading Gazzola et al's book and there for a space $H^{m}(\Omega)$ they seem to allow boundary conditions that are of degree as high as $2m-1$. How is this reconciled with the fact that ...