# Tagged Questions

202 views

### Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
30 views

48 views

### A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
65 views

### Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
45 views

### A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
31 views

314 views

43 views

### Finding Fourier transform of initial condition

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
33 views

### Checking if a function is in the Schwartz space of rapidly decreasing functions.

Is there any neat bi-implication other than the definition that I can use to check this? This question was motivated by a question that asked if $f(x) = e^{-|x|^3}$ was in S. It isn't infinitely ...
35 views

### Stationary phase approximation for dominant frequencies

http://en.wikipedia.org/wiki/Stationary_phase_approximation I am studying the method of stationary phase, and I was thinking about one of the examples on the given webpage. ...
195 views

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
120 views

### Fourier Series and Solving Differential Equations

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where $$E(t)=200t(\pi^2-t^2),$$ for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), ...
34 views

### The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$\frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
735 views

### Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...