Articles of greens function

Understanding the use of continuity and 'derivative change' in finding Green functions

Below is a university question and the corresponding solution for which I do not understand small parts of: Question: Show that the Green’s function for the range $x \ge 0$, satisfying $$\frac{\partial^2 G(x,z)}{\partial x^2}+G(x,z)=\delta(x-z)$$ with boundary conditions:$$G(x,z)=\frac{\partial G(x,z)}{\partial x}=0\quad\text{at}\quad x=0$$ is $$G(x,z)=\begin{cases}\cos z\sin x-\sin z\cos x, &\text{for} & x\gt z \\ 0 &\text{for} & x\lt […]

How to find the change in a piecewise derivative at a given point?

If given some function (in this case it’s a Green function but that’s not important here as my question is much more simple) and you are given that it’s derivative is $$\bbox[yellow]{\frac{\mathrm{d}}{\mathrm{d}x}G(x,x^{\prime}) = \begin{cases} A(x^{\prime})\cos x, & \text{for}\quad x\lt x^{\prime} \\ -B(x^{\prime})\sin x, & \text{for}\quad x\gt x^{\prime} \end{cases}}$$ Now you ask yourself $$\fbox{$\color{blue}{\text{What is the […]

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

If $\Delta u(x)=\delta(x)$ then $u(x)=C |x|^{2-d}$ via Fourier transform?

Can the behaviour $$ \text{constant}\times|x|^{2-d} $$ be obtained for the solution of the distributional equation in $\mathbb R^d$, for $d\ge 3$, $$ \Delta u(x)=\delta(x) $$ via Fourier transform method?

Fundamental solution to the Poisson equation by Fourier transform

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{cases} \dfrac{1}{(2-d)\Omega_d}|x|^{2-d}\text{ for } d=1,3,4,5,\ldots\\ \dfrac{1}{2\pi}\log|x| \ \ \ \ \ \ \ \ \ \ \ \ \text{for } d=2, \end{cases} $$ where $\Omega_d$ is the $d$-dimensional solid angle […]