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12

Use Plancherel's theorem to evaluate this integral. The basic idea is that for two functions $f,g$, $$I=\int_{-\infty}^{\infty} f(x) g^*(x)dx = \int_{-\infty}^{\infty} F(k) G^*(k) dk$$ where $F,G$ are the Fourier transforms of $f,g$. Your functions both have relatively small support in the spectral domain. Here, $\sin x / x \rightarrow \text{rect}(k)$ ...

7

The key to the evaluation of oscillatory integrals is to truncate integral at the right point. For this example you need to choose upper limit of the form $$\pi\mathbb{N}+\frac{\pi}{2}$$ Before explaining why it should work, let me at first show that it actually produces good results. Asymptotics It is easy to guess that asymptotic series has the ...

6

Ooura's method for Fourier sine integrals works here, see: Ooura, Takuya, and Masatake Mori, A robust double exponential formula for Fourier-type integrals. Journal of computational and applied mathematics 112.1-2 (1999): 229-241. I wrote an implementation of this algorithm but never put in the work to get it fast (by, say caching nodes/weights), but ...

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I want to mention another idea in case it helps. The truncation error of replacing $\int_{-\infty}^{\infty}$ with $\int_{-L}^L$ is on the order of \begin{aligned} \int_L^\infty \cos(tx)g(x)\,\mathrm{d}x &= \frac{1}{t}\sin(tx)g(x)\big|_{L}^{\infty} - \int_L^\infty\frac{\sin tx}{t}g'(x)\,\mathrm{d}x \\&= -\frac1L g(L)\sin(tL) + \text{asymptotically ... 5 There is something very basic that you should know about hyperbolic problems. Consider the most basic example \partial_tu+a\partial_xu=0 with a numerical marching scheme of the formu_j^{n+1}=\sum_kc_ku_{j+k}^n$$. This covers all explicit schemes, and all implicit schemes like Crank Nicolson also, if you begin by solving the tridiagonal system. It ... 4 This is a Hamilton-Jacobi equation. You can read about how to apply WENO to such equations in Section 4 of Chi-Wang Shu's 2009 WENO review paper, and references therein. 4 This is part of the (complex-valued) Fourier transform for which there is (provably) no more efficient way than the Fast Fourier Transform (FFT) if you want to compute the integrals for at least a significant fraction of all frequencies n between zero and the largest frequency you care about. In other words, while your intuition may tell you that ... 4 Is the sign accurate? If so, you may have an issue since your var form should be (\nabla u, \nabla v) - \langle n\cdot \nabla u, v\rangle_{\Gamma_{\rm rest}} = 0 substituting the Robin condition in gives (\nabla u, \nabla v) - \alpha\langle u, v\rangle_{\Gamma_{\rm rest}} = -\langle 1, v\rangle_{\Gamma_{\rm rest}} which can mess with your ... 3 For debugging the code, there is a set of analytic solutions here for several reduced models corresponding to subsets of terms on the right-hand side. These analytic solutions have to be reproduced by the code. Verification testing of this kind is a standard practice for debugging simulation models. Reduced model 1:  m \ddot{x} = - \gamma \dot{x}  Solution:... 3 The point important to understand when thinking about multigrid is that the lower levels of the hierarchy do not actually have to solve the problem accurately. Rather, the operators at the lower levels just need to provide good approximations of a part of the spectrum (eigenvalues) of the operator on the finest level -- specifically, they need to well ... 2 If f is very nice, then you can approximate f with a sequence of piecewise polynomials and integrate over the resulting intervals exactly. This may be much, much cheaper than using an FFT. This is also true for approximations of f by any functions where you can easily know or determine the exact antiderivative in each interval. If f is a black box, ... 2 Isn't this the same as the real part of the Fourier transform of f(x)$$ \begin{align} I(n) &= \int^1_0dxf(x)\cos(n\pi x) \\ &=\Re{{\int^1_0dx f(x) e^{-j n \pi x}}} \\ &= \Re\int^1_0dx f(x) (cos(n\pi x) - j\sin(n \pi x)) \\ &= \int^1_0dxf(x)\cos(n\pi x) \end{align} $$2 Most of this was already discussed in the comments, but I would like to elaborate and put a detailed answer. There are no elementary characteristics (definiteness, symmetry, bandwidth) which can tell you whether the underlying (mixed or not) FEM/FVM is stable to solve the continuous problem. You can not tell anything about that just by looking at those ... 1 If I understand your problem correctly, you just need to do parameter identification from a known model and non-noisy data. The standard way to do this is via a nonlinear least-squares framework. To do this, after solving the ODE for your given choice of \rho and f at some number of time points, you create a cost function that is a function of \rho and ... 1 I finally found the problem and corrected in the code. The issue was that the scheme was not able to handle left-traveling waves due to an incorrect implementation of the eigendecomposition. The projection on the characteristic space was being done only on the point-values of the vector of conserved variables, that is:$$ V_i = L(u_{i+1/2})U_i G_i = L(...

1

That is a Volterra integral equation in principle, and there are several methods to solve it. I would first differentiate the whole expression to obtain $$\tag 1 \partial_t \mathbf{a}^{(n+1)}(t) = e^{i\mathbf{H}t} \mathbf{D}(t)e^{-i\mathbf{H}t}\mathbf{a}^{(n)}(\tau)$$ Define $\mathbf b^{(i)}(t) = e^{-i\mathbf{H}t} \mathbf a^{(i)}$, then  \tag 2 i\...

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You can use some scheme where value at a point is some type of average of its neighboring values. You will have to decide whether this kind of smoothing is appropriate in your case or not. In MATLAB, smooth3 function is used to smooth data in 3D. Using same principle, you can perform Gaussian smoothing or box smoothing in 1D. This is how smoothing is done. ...

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The simplest scheme you could employ is an upwind scheme. It's first-order and introduces artificial viscosity/diffusion, but doesn't require limiters. Probably the next simplest class would be Lax-Wendroff schemes, for which you can find a comprehensive explanation in LeVeque's book on Finite Volume schemes.

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One and two do not lead to the same flow. One is like a piston that actually compresses the flow, two will behave differently. The details are dependent on the inlet flow characteristics, but you should not expect the same behavior. If the flow is slow ($M \ll 1$), you might be able to get away with a short-time simulation of the starting conditions in a ...

1

If there's some model for the signal for $t < t_0$ and $t > t_f$, perhaps it would be possible to try to fit the full signal using a Metropolis-Hastings or Goodman-Weare algorithm. This could then also yield some information about the probability distributions for the parameters involved.

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