## New answers tagged differential-equations

1
vote

### Motivation and physical interpretation of the Laplace transform

The following simple motivation for the formula was given in my undergrad class, which I hope I'm not misremembering:
Typically the way the Laplace transform arises in applications is when solving ...

4
votes

### The Fourier transform of the Liouville function?

Your first formula for $\lambda(x)$ is equivalent to
$$\lambda(x)=\underset{N\to \infty}{\text{lim}}\left(\sum\limits_{z=-N}^N \cos(\pi(\Omega(z)+x-z))\, \text{sinc}(\pi(x-z))\right)\tag{1}.$$
...

0
votes

### Green's function on sphere

You could watch this video Here, whose title is 'Green’s function for the Laplace-Beltrami operator on a toroidal surface', at the beginning, he introduced the Green function on sphere,
as Prof.Hardy ...

9
votes

Accepted

### Nicer expression for 2.1369288...?

Apply Lagrange reversion to @TheSimpliFire’s equation:
$$\frac{k}{(k-1)^2}+1=e^k\iff k=1+\sqrt{\frac k{e^k-1}}\\\implies k=1+\sum_{n=1}^\infty\frac1{n!}\left.\frac{d^{n-1}}{dx^{n-1}}\left(\frac x{e^x-...

12
votes

### Nicer expression for 2.1369288...?

With a few substitutions, we find that $$c=\frac{k^3}{k^2-k+1}\quad\text{where}\quad1+\frac k{(1-k)^2}=e^k.$$ The solution for $k$ requires a more advanced function than Lambert $W$.

4
votes

### Motivation and physical interpretation of the Laplace transform

Re "I have not yet encountered any good explanation of how the Laplace transform formula arises." I think it's useful to look at early origins of the LPT, relations to other transforms, and ...

4
votes

### Motivation and physical interpretation of the Laplace transform

From R. N. Bracewell, "The Fourier Transform and Applications", McGraw Hill 3rd ed., pp.381:
"Advantages of the Laplace transform over the Fourier transform for handling electrical ...

19
votes

### Motivation and physical interpretation of the Laplace transform

Interest is continuously compounded at rate $r,$ so that if you deposit $\\\$1$ now it will be worth $\\\$e^{rx}$ at time $x.$ How much do you need to deposit now in order to withdraw at rate $\\\$f(x)...

6
votes

### Motivation and physical interpretation of the Laplace transform

I will not discuss the uses of the Laplace transform.
Myself I think of the analogy with sequences. A sequence $(a_n)_{n\geq 0}$ is determined by its generating function (convergence issues aside)
...

15
votes

### Motivation and physical interpretation of the Laplace transform

The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\varepsilon )$ (a non-statistical concept) into the ...

23
votes

### Motivation and physical interpretation of the Laplace transform

Besides the important physical motivation pointed out by Carlo Beenakker, there is another one, purely mathematical. Laplace transform is a generalization of a power series (and Dirichlet series).
In $...

26
votes

### Motivation and physical interpretation of the Laplace transform

The physical motivation for the Laplace transform is causality.
Consider the linear input-output relation
$$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$
Causality dictates ...

3
votes

Accepted

### Bound on $L^1$ norm of solution of two-point boundary value problem

First of all, there is clearly no bound if $0$ is an eigenvalue of $Lu=(pu')'+qu$, $u(0)=u(1)=0$. (This will not happen if $p(x)<0, q(x)\ge 0$ because then the operator is positive, but if $p,q$ ...

3
votes

### Fréchet-valued symbols

Check if this follows with appropriate adjustments from the arguments for Thm.44.1 and Exr.44.6 in
Trèves, François, Topological vector spaces, distributions and kernels, Pure and Applied Mathematics ...

3
votes

Accepted

### Existence of solution to nonlinear first order PDE with C^1 bounds

I don't think in the level of generality you are looking at you can say anything useful. Let me give two examples with very contrasting behaviors. Here I am, per your comment, allowing myself to think ...

1
vote

Accepted

### Same occupation measure $\Rightarrow$ same trajectory

Under your hypotheses, you'll have (for each bounded continuous function $\varphi$)
$$
\int_0^T \varphi(x^f(s))\,ds = \int_0^T \varphi(x^g(s))\,ds,\qquad\forall T\ge 0,
$$
where $x^f$ and $x^g$ are ...

Top 50 recent answers are included

#### Related Tags

differential-equations × 1634ca.classical-analysis-and-odes × 295

ap.analysis-of-pdes × 249

dg.differential-geometry × 221

ds.dynamical-systems × 214

fa.functional-analysis × 159

reference-request × 130

real-analysis × 114

differential-operators × 78

mp.mathematical-physics × 76

elliptic-pde × 57

na.numerical-analysis × 55

riemannian-geometry × 54

ag.algebraic-geometry × 52

cv.complex-variables × 47

pr.probability × 34

oc.optimization-and-control × 31

functional-equations × 27

stability × 27

linear-algebra × 26

lie-groups × 26

differential-topology × 26

inequalities × 26

fourier-analysis × 26

integration × 26