# Questions tagged [boundary-value-problem]

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

1,013 questions
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### Weak form of steady Navier-Stokes equations with special boundary condition

Suppose we want to solve the steady low-Mach-number Navier-Stokes equations coupled with a passive scalar $\xi$, which read: \begin{align} \nabla \cdot \rho \mathbf{v} & = 0,\\ \rho \mathbf{v} \...
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### 1D wave equation with Boundary Conditions: Fourier Transform solution

I am considering the 1D wave equation with $c=1$ for the sake of simplicity: $$u_{tt}-u_{xx}=0,\quad \forall x\in\mathbb R,\; \forall t\in\mathbb R\tag{1}\label{eq:1}$$ with the following boundary ...
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### Green's function for homogeneous PDE

I was looking for some Green's function method to solve a homogeneous PDE with nonhomogeneous boundary conditions (i.e., $Lu=0$ in $D$ with $u=f(\mathbf{x})$ in $\partial D$), but most of the ...
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### Can I solve the following problem in integration along with constraints? [closed]

max $\int (x-f(x)dx$ such that $f(0) =a$ $f(1) = b$ $f'(0) = c$ $f'(1) =d$ $f''(0.5) = 0$ $f'(x) >0$ $\forall x \in[0,1]$
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### Green Solution to Laplace Equation with Robin Boundary Conditions

Let's say that I know a solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=0\quad \mathbf{x}\in\mathbb{R}^2$$ And I need a solution for the laplace equation in the ...
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Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(... 1answer 55 views ### 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.$e_k, k=1,2,3$... 0answers 16 views ### 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$V_i \subset\Omega$,$i=1,\dots,N$... 0answers 27 views ### Curious Question on Inhomogeneous Boundary Conditions of a PDE I am given the PDE,$\ u_t=u_{xx},$with boundary conditions$u(0,t)=A, \ u(1,t)=B$and$u(x,0)=f(x)$. I have found the solution of this PDE is $$u(x,t)=A+(B-A)x+\sum_{n=1}^{\infty}B_ne^{n^2\pi^2 t}\... 0answers 58 views ### Solving definite integral in two variables. Solving a PDE with the following boundary problem with arbitrary constant b:$$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$Hint given in the question is as follows: Split this integral in two ... 2answers 86 views ### Solve PDE using method of characteristics with non-local boundary conditions. Given the population model by the following linear first order PDE in u(a,t) with constants b and \mu :$$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0u(a,0)=u_0(a)\,\,\,a≥0u(0,t)=F(t)=b\... 0answers 12 views ### REFEREENCE REQUEST for Non-Local Boundary Value problems It would be really helpful if someone could suggest me any reference (Books or Papers) where I would find worked-out examples of Elliptic Boundary value problems (especially Laplace equation) with non-... 1answer 725 views ### Converting Dirichlet Boundary Conditions to Neumann Boundary Conditions for the Heat Equation I'm solving the heat equation on a two dimensional square domain. The problem is defined as: $$u_{xx}+u_{yy} = 0 \hskip{0.5cm}\text{for} \hskip{0.5cm} 0 \leq x \leq 1, 0 \leq y \leq 1$$ with the ... 1answer 710 views ### 2d laplace equation with neumman boundary condition $$\Delta u(x,y)=0$$ $$x,y\in(0,1),$$ $$\frac{\partial u(0,y)}{\partial x}=0,\quad \frac{\partial u(1,y)}{\partial x}=0,\quad\frac{\partial u(x,0)}{\partial y}=0,\quad\frac{\partial u(x,1)}{\partial y}=... 2answers 43 views ### Solving the BVP u_{xx}+a^2u=\sin(\pi x) with u(0)=1 and u(1)=-2 I am trying to solve the BVP$$u_{xx}+a^2u=\sin(\pi x), \ \ \text{for} \ \ 0<x<1$$with u(0)=1 and u(1)=-2, \forall a\in\mathbb{R}. I begin by solving the homogeneous equation u_{xx}+a^... 1answer 24 views ### Laplace Equation with Inhomogeneous Boundary Condition I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous. I'm having trouble deciding how to handle one of the boundary ... 1answer 28 views ### Separating variables in a PDE with multiple constants My question is: How do you use separation of variables on a PDE that has more than one constant in it? All the examples I can find in my book/online only have one constant in it, like$$ \frac{\... 0answers 23 views ### Matrix representation of a finite difference with Neumann boundary conditions Given 1D data$[c_1,c_2,c_3,\cdots,c_N]$I can represent the derivative operation as a matrix product. For example, using the central difference $$\left.\frac{\partial c}{\partial x}\right|_k \... 2answers 707 views ### Laplace Equation on the Corners and Boundary of a Rectangle? Consider for some rectangle [a,b] \times [c,d] \in \mathbb{R}^2, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 u}{\... 0answers 22 views ### A sufficient condition for a Neumann problem to have solution Suppose we have the Boundary Value Problem (BVP): \Delta{u}=f , in the domain (topos) D \partial_{n}{u}=h, in \partial{D} It is easy to prove using 1st Green's identity that a necessary ... 0answers 27 views ### Poisson integral formula for boundary value problem I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is:$$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\... 1answer 32 views ### Helmholtz equation with robin boundary condition Consider the differential equation$(\nabla^2+\frac{1}{R^2})\psi(\bar{r}) = 0$in 2 dimensions, with the boundary condition$\partial_r\psi(R)+ \kappa \psi(R) = 0\$, on unit disk of radius R. What is ...
I have two equations like $$\frac{\partial Y}{\partial x}=A(L-2x)\;\text{ at }(x,0)$$ And $$\frac{\partial Y}{\partial t}=0\;\text{ at }(0,0)$$ Is it possible to find out the real equation? (That is ...
I am solving an exercise concerning the Airy eigenvalue problem $$-y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*)$$ which (among other things) asks me to prove that all eigenvalues are positive. I ...