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seen Nov 18 at 14:38

Apr
14
comment Show Green's function solves Poisson's equation
$H$ is called a fundamental solution. You might want to take a look at Fritz John book. To prove it classically, you'll have to separate the singularity at $x_0$ from the rest of the integral (using a ball of radius $\epsilon$), and then investigate the behavior in the limit. I've used the argument here
Apr
13
comment Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$
$\text{}\text{}dp$?
Apr
10
comment Method of characteristics inhomogeneous nonlinear wave equation
Studying the Burger's equation, as @user88595 suggest, will not only help you to derive the solution, but also to understand how it behaves.
Dec
10
comment Stability of the System of Differential Equations
For 1) you need initial conditions, since $A' + B' + C' =0$ only imply that $A + B + C = k$, where $k$ is a constant. To show stability, you'll have to linearize around the stationary solution and see what's happening with the eigenvalues.
Nov
25
comment Separating Partial Differential Eq
let us continue this discussion in chat
Nov
25
comment Separating Partial Differential Eq
The separation I've shown you is correct. What is $a$ and $b$? As I told you, the relationship between $C$ and the separation constant comes from the boundary conditions. Why don't you edit your question and show us how did you derived the relation $a^2 + b^2 = C^2$? Here is a quick guide on how to typeset equations.
Nov
25
comment Separating Partial Differential Eq
Well, if by a relationship you mean $C = C(\lambda)$, then no. The solution of the system is \begin{align}\Theta(\theta) &= A e^{\sqrt{\lambda} x} + B e^{-\sqrt{\lambda} x} \\ R(r) &= C J_\sqrt{\lambda}(C x) + D Y_\sqrt{\lambda}(C x)\end{align} where $J_\sqrt{\lambda}$ and $Y_\sqrt{\lambda}$ are Bessel and Neumann functions of the first kind, of order $\sqrt{\lambda}$. If no boundary conditions are given, then you have an infinite collection of solutions spanned by both parameters.
Nov
25
comment Separating Partial Differential Eq
I'm guessing you're getting Helmoltz equation from separating time and space from a cylindrical wave equation. If you need to quantize $\lambda$ and $C$, then boundary conditions are needed.
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy Indeed.
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy It gets absorbed in the definition of $\hat{c}_1(k)$. I abused the notation, sorry!
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy When $t=0$, $\sin(\omega t) = 0$, and the equality follows.
Jul
22
comment Perturbation Theory on Finite Domains
You should read about the WKB approximation. A great treatise on the asymptotics of $y'' + p(x) y = 0$ can be found in chapter 6 of Olver's Asymptotics and Special Functions.
Jul
14
comment Nonlinear equation (oscillon) comparison
@ComplexGuy en.wikipedia.org/wiki/Virtual_work
Jul
12
comment Solve the following differential equations by converting to Clairaut's form through suitable substitutions.
@Vish.Math See my last edit but be warned that there is never a satisfactory explanation for intuitive steps. Math is experience.
Jul
11
comment Black Scholes Merton PDE with a time variant boundary condition
A few questions: Is there a relationship between $r$, $\delta$ and $\sigma$? Are all constants positive? One bigger than the other, etc? What's the domain for $V$? What's the behavior you're expecting for $S$ as the price $V$ goes to infinity?
Jul
9
comment Solution of a differentiation in integral form
@ComplexGuy Nope, the definition of Spherical Bessel functions is \begin{align}j_n(x) &= \sqrt{\frac{\pi}{2 x}} J_{n + \frac{1}{2}}(x) \\ \\ y_n(x) &= \sqrt{\frac{\pi}{2 x}} Y_{n + \frac{1}{2}}(x)\end{align}
Jul
8
comment How to find out where a solution to a differential equation is defined?
The answer to your question is that theorem. Any ODE book will have a detailed proof, along with the relation between existence, uniqueness and continuation of solutions depending on parameters and initial conditions.
Jul
8
comment Ordinary Differential Equations - Sturm Liouville
Hint: Sturm separation theorem
Jul
8
comment How to find out where a solution to a differential equation is defined?
Have you read the Picard–Lindelöf theorem?
Jul
8
comment Solve the following differential equations by converting to Clairaut's form through suitable substitutions.
@Vish.Math See the edit. If you find it appropriate, please consider accepting the answer.