# Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations.

997 questions
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### 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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### Discretization Dirichlet boundary condition for Elliptic PDE with finite volume method

I want to discretize the following equation using FMV: $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ To this end, let $V_i \subset\Omega$, $i=1,\dots,N$...
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### Differential equation with different initial condition

Given an vector field $v:U (\subset R^n)\to{R^n}$. Consider the differential equation $\dot{x} = v(x)$. It's given that this differential equation with an intial condition $x(0) = x_0$ has unique ...
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### How to solve wave equations with boundary condition $u_x(0,t)=h(t)$?

\begin{align*} u_{tt}-c^2u_{xx}=0, x>0\\ u(x,0)=u_t(x,0)=0\\ u_x(0,t)=\frac{t}{1+t^2},t>0 \end{align*} According to the textbook, I should look for solutions in the form $u(x,t)=F(x-ct)$ and ...
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### Converting between Solution forms using Green's Functions in Linear Differential Equation

EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ...
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### 1D Wave equation mixed boundary conditions and I.C.

I have been searching for a solution online, but cannot find one that fits the B.C. and I.C. for this wave equation. I read through this PDF, page 7; although I had similar conditions I just obtained ...