Articles of boundary value problem

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla u)=(\gamma u_x)_x+(\gamma u_y)_y=0,\\ u(x+2\pi,y)=\mu u(x,y),\\ u(x,1)=0,\\ u(x,0)=0 \mbox{ or } u_y(x,0)=0. \end{cases}$$ Let’s denote the set of possible values of the eigenvalue parameter $\mu$, for which the system has a solution $\mu_\gamma$. Does there exist a conductivity $\alpha(y)$ on […]

second order differential equation with Green's function

I need to solve following differential equation \begin{eqnarray} y”(x) – k = \delta(x-x_0) \end{eqnarray} subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} Is it possible to solve the equation using Green’s function? I tried as follows: solution of the homogeneous equation \begin{eqnarray} y”(x) – k = 0 \end{eqnarray} is $y(x) = […]

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the initial/boundary-value problem \begin{cases}u_t-u_{xx}=0 & \text{in }\mathbb{R}_+ \times (0,\infty) \\ \qquad \quad \, u=0 & \text{on } \mathbb{R}_+ \times \{t=0\} \\ \qquad \quad \, u= g & \text{on }\{x=0\} \times [0,\infty). \end{cases} (Hint: Let $v(x,t):=u(x,t)-g(t)$ and extend […]

Developing solution for electrodynamics problem

Although it is a question related to physics, since the point it really matters is its mathematical aspect, I post this question on MSE. There’s an additional exercise from Introduction to Electrodynamics by Griffith. Problem 4.34 A point dipole p is imbedded at the center of a sphere of linear dielectric material (with radius $R$ […]

Solving Wave Equations with different Boundary Conditions

Right now I’m studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)…$) I know how to solve it when the boundary conditions are $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ with d’Alambert’s formula. But I don’t know how to solve it in any other case. Does anyone know of any book, notes, etc. […]

Solve Boundary Value Problem for $y''+ y' + e^xy = f(x)$

Consider to solve Boundary Value Problem : $y”+ y’ + e^xy = f(x)$ with $0 < x < 1$ and $y(0)=y(1)=0$ with exact solution $y(x) = \sin \pi x$ $f(x)=(e^x- \pi^2)\sin \pi x + \pi \cos \pi x$. How to plot two curves for exact solution and numerical solution by MATLAB?

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y” + xy’ + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green’s Funtion to Solve this problem. I am facing issues with equations and the number of variables. Please help me solving this

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\in B_r, \\ u(x) = g(x), & \qquad x\in \partial B_r, \end{align} for some $f \in C^1(\overline{B}_r)$ and $g \in C(\partial B_r)$ I have to show […]

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green’s function. I have to solve the given differential equation using Green’s function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta (x-x’);y(0)=y(L)=0$

Reaction diffusion equation solution

This has been driving me spare for the last week, and I feel pretty bad for not being able to get a solution because (at least on the face of it) it’s a pretty simple equation. I have the following reaction diffusion equation: $$\frac{\partial M}{\partial t}=d\frac{\partial^2 M}{\partial x^2}-gM$$ With: $$\frac{\partial M}{\partial x}(0,t)=-h, \quad M(1,t)=0, \quad […]