Pragabhava
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 Apr14 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 Apr13 comment Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$ $\text{}\text{}dp$? Apr10 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. Nov25 comment Separating Partial Differential Eq Nov25 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. Nov25 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. Nov25 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. Sep10 comment Solution of a differentiation in integral form @ComplexGuy Indeed. Sep10 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! Sep10 comment Solution of a differentiation in integral form @ComplexGuy When $t=0$, $\sin(\omega t) = 0$, and the equality follows. Jul22 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. Jul14 comment Nonlinear equation (oscillon) comparison @ComplexGuy en.wikipedia.org/wiki/Virtual_work Jul12 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. Jul11 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? Jul9 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} Jul8 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. Jul8 comment Ordinary Differential Equations - Sturm Liouville Jul8 comment How to find out where a solution to a differential equation is defined? Have you read the Picardâ€“Lindelöf theorem? Jul8 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. Jul8 comment Solution of a differentiation in integral form No offence, but it is my impression that you might be reading something more complex than you can handle right now. You really need to step up your knowledge on special functions, separation of variables, eigenfunctions and eigenvalues, Fourier and Laplace transforms, etc. if you want to fully understand the maths that are being used in the articles you are trying to read.