Questions on the use of numerical functions NIntegrate and NDSolve.

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4
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1answer
204 views

How to evaluate differential entropy from raw data?

I want to evaluate the differential entropy according to here $$h(X) = - \int f(x) \log{f(x)} dx$$ where $f$ is the probability density function. Lets create some test data (normal distribution): ...
4
votes
1answer
215 views

Efficient Dyson series implementation

I'm trying to implement a Dyson series \begin{array}{lcl} U(x,x_0) & = & 1 + \int_{x_0}^{x}{dy_1V(y_1)}+\int_{x_0}^x{dy_1\int_{x_0}^{y_1}{dy_2V(y_1)V(y_2)}}+\cdots \\ & &{} + ...
4
votes
1answer
94 views

System of equations is solved by NDSolve over just a tiny domain

I'm solving numerically a system of differential equations with the use of NDSolve. The numerical integration works only a very small interval of the argument. I'm ...
4
votes
1answer
104 views

Parametric numerical integration [closed]

I would like to solve the integral: $$ \iiint\frac{e^{-\sqrt{x^2 + k^2}}x^2 \sin{\theta}}{\sqrt{x^2 + k^2}} dx d\theta d\phi$$ defined in $x \in [0,+\infty]$, $\theta \in [0,\pi]$, $\phi \in [0,2 ...
4
votes
2answers
98 views

How should I evaluate the numerical integral of a function with divergent part substracted?

I want to evaluate the following integral numerically in Mathematica, $$\int_0^{\infty}(\frac{1}{(1 - e^t)^{10}} - (\frac{1}{t^{10}} - \frac{5}{t^9} + \frac{145}{12 t^8} - \frac{75}{ 4 t^7} + ...
4
votes
1answer
436 views

Understanding of method for NDSolve

I used automatic method for NDSolve. Then I asked myself - which method Mathemathica prefered? I got an answer for this question on this forum, that I need to use: ...
4
votes
1answer
251 views

Find lengths of contours in a ContourPlot

I am trying to find the lengths of different contours in the following plot: It is a complicated piecewise function evaluated on the unit disk. I am hoping there is an easy, generalized way to ...
4
votes
1answer
221 views

How to do this complex integration on the real line?

$m, r$ are parameters in the following integral: Integrate[z Exp[I z r]/Sqrt[z^2 + m^2], {z, -∞, ∞}] How to do this integration directly? The result should be ...
4
votes
1answer
73 views

Issues with modeling pulses in a very simple system of DAEs

Some time ago I had asked this question about evaluation difficulties using Euler Integration to solve a system of ODE where discrete pulses occur. While I have now abandoned Euler integration and ...
4
votes
1answer
52 views

NIntegrate crashes without error message when using high precision integrand with non-zero tailing digits

I have a numerical integral that evaluates fine for floating point arguments with low precision (e.g., 4 digits), with argument where only zeros follow the first, e.g., 4 non-zero digits, or with ...
4
votes
1answer
1k views

Combining Gravity Turn and Orbit Models

I have a mathematical model for the motion of an orbiting spacecraft about Earth: ...
4
votes
1answer
123 views

Possible bug / numerical issues with HypergeometricU — any suggestions for a fast workaround?

I've encountered some problematic behaviour with HypergeometricU. I have a probability distribution on the positive integers that takes the following form after ...
4
votes
1answer
924 views

Integrating a functional of an InterpolatingFunction

It is straightforward to Integrate an InterpolatingFunction. However, even for a simple functional of an ...
4
votes
2answers
1k views

Why does this integral have a complex component?

I wanted to find the probability of my normally-distributed random variable being at least 15, so I set up this integral: ...
4
votes
1answer
296 views

How to demonstrate lack of stability with advection equation

I am trying to that using a coarse grid with an explicit method for, say, the advection equation leads to an unstable solution. The trouble is Mathematica avoids unstable solutions with good ...
4
votes
1answer
562 views

Multi-dimensional integral in the complex plane with poles and essential singularity

I've passed the last week searching a way to numerically integrate this multi-dimensional integral in the complex plane at the poles and avoiding the singularity at z=0: $$ \oint_{C}\oint_{C\ auound\ ...
4
votes
1answer
413 views

Error Interpretation in NIntegrate

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code: ...
4
votes
2answers
974 views

How to Solve this ODE with Mixed Boundary condition

I have an ODE equation which is sort of y''[x] + 2 y'[x]/x + .0001 (y[x])^3 ==0 subject to the boundary conditions ...
4
votes
1answer
204 views

Construct DifferentialMatrices and Kernel for LevinRule for this integral and ODE set

I've made a lot of progress on my problem the last few days thanks to all the help I've received on here. I think I'm upto the final step of greatly improving the performance of NIntegrate[..] on my ...
4
votes
1answer
478 views

Efficient way to perform elementary integration step with NDSolve internal method

I'm trying to tweak the NDSolve function to perform one elementary integration step (using some explicitly selected stepping algorithm via ...
4
votes
2answers
2k views

PDE Boundary Conditions

I am solving a PDE using Mathematica and I would like to know how to implement the condition that the two-variable function y[t,s] is zero whenever ...
4
votes
1answer
137 views

Using `N` gives strange result

Consider these two functions which are almost the same: ...
4
votes
1answer
363 views

How to choose MaxStepFraction for optimal speed of NDSolve

I'm trying to use NDSolve to solve a 1D Schrodinger's equation, and it seems that MaxStepFraction has huge effect on the ...
4
votes
1answer
655 views

Multiple simultaneous events in EventLocator method for NDSolve

I'm using NDSolve to integrate a system of ODEs, and EventLocator to stop the integration when it leaves a certain region in phase space. This works perfectly as it should. However, I've also added ...
4
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0answers
60 views

Numerical instabilities of a convection-(non-)diffusion equation when shrinking from a square to a triangular domain

I am trying to evaluate a parameter-dependent indefinite integral using a PDE-based scheme I described here, and I'm having some trouble when I try and cut down the domain from a square to a triangle. ...
4
votes
0answers
67 views

Avoid Evaluation of Function at NDSolve

I have a huge "black-box" f function, which I want to integrate. Let's define it: f[x_,y_,a_]:=a*Exp[-(a*10000)(x^3+y^3)] as ...
4
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0answers
101 views

Using indexed array elements as integration dummy variables with EvaluationMonitor. Bug?

As part of a routine that must cope with integration of varying numbers of dimensions, I would like to use indexed variable names (e.g., x[0], x[1],...) as dummy integration variables. However, it ...
4
votes
1answer
488 views

How can I invoke the solution of NDSolve to determine a parameter in my equation just inside NDSlove?

I am trying to solve a differential equation by NDSlove for $h(x,t)$. It reads $$h_t=h_{xx}-V_h-\lambda(t)$$ where $V_h$ is a given function of $h(x,t)$ denoted by ...
4
votes
0answers
87 views

Integrate yields complex value, while after variable transformation the result is real. Bug?

I have the follwoing integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, Infinity}, Assumptions -> z \[Element] Reals] >> -3.36354 - 3.85013 I the ...
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0answers
411 views

Solve integral equation for upper bound

I need to find the upper bound of an integral knowing the value of the lower bound and the result of the integral. Here is my function: ...
4
votes
0answers
69 views

NIntegrate::ncvbr: How should we interpret and handle this error not mentioned in any documentation?

I have some user-defined module describing my integrand which has to be computed numerically (it's much more complicated than this but bear with me): ...
4
votes
0answers
159 views

How to specify the time variable for NDSolve

I recall that it is possible to specify which independent variable is the "time" variable in NDSolve, but I can't find it documented anywhere. Does anyone recall ...
4
votes
0answers
101 views

What are good/best practices to take the Fourier transform of an InterpolatingFunction?

I have a function which I have obtained from numerical integration of a differential equation, and I would like to take its Fourier transform. What are good practices for doing this? To make things ...
3
votes
2answers
713 views

Unexpected result of summation

I wrote a small module that gives me an incorrect output-set. It should be a single number! I don't understand what went wrong. This is the form of summation used: $$\frac{1}{2} (b-a) \sum_{i=1}^n ...
3
votes
3answers
173 views

Inefficient NIntegrate and Which

In version 10 a simple numerical integraton of piecewise function is highly inefficient: ...
3
votes
4answers
270 views

Using NIntegrate to integrate functions with sharp peaks (Lorentzian-like)

I have a problem with NIntegrate that I do not understand. I want to integrate a function f[x], which has a complicated analytic ...
3
votes
2answers
195 views

Error in NIntegrate command

I am trying to evaluate (numerically) the integral $$\int_0^1\int_0^1\int_0^1\int_0^1\int_0^1\int_0^1\frac{dx\,dy\,dz\,ds\,dt\,du}{(x - s)^2 + (y - t)^2 + (z - u)^2}$$ with Mathematica using the ...
3
votes
1answer
2k views

Methods to speed up numerical NDSolve, NIntegrate,

I am not very used to do numerical simulations on Mathematica. Do you have any ideas how to improve i.e. speed up my code? ...
3
votes
2answers
587 views

How to numerically integrate this integral

I am unable to do this definite integral in Mathematica 9. Is there any command so that I can get the numerical value of the above integration? Code: ...
3
votes
2answers
855 views

How to avoid this kind of numerical error caused by extreme parameters when using NDSolve?

Here I use a one-dimensional heat conduction equation as the example. I found that when the thermal diffusion coefficient is small enough, Mathematica will give a result against the second law of ...
3
votes
2answers
326 views

Integration and Fourier transform

I would like to calculate the distribution function from the characteristic function. There is a formula given as $$F_X(x)=\frac{1}{2}+\frac{1}{2\pi}\int_{0}^\infty \frac{e^{i w x}\phi(-w)-e^{-i w ...
3
votes
2answers
145 views

Help with Integration

I am new to Mathematica, and I need help with integration of the following Reliability expression (from a well-known Reliability Model). $$R(t)=e^{-Ne^{-bt_i}(1-e^{-bt})}, t \geq 0$$ I have ...
3
votes
3answers
339 views
3
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2answers
166 views

Integrate and NIntegrate not the same

Why am I getting two different results here? ...
3
votes
4answers
407 views
3
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2answers
331 views

NIntegrate piecewise vector function

Is there a way to numerically integrate a vector function defined via Piecewise? Example: ...
3
votes
2answers
650 views

Speed of convergence for NIntegrate

I'm trying to optimise numerically a function that entails computing the expected value of a truncated trivariate normal distribution and this is taking extremely long -I also get warned about ...
3
votes
3answers
1k views

How to integrate a function over a 3D planar polygon?

I am trying to integrate a function over a planar polygon in 3D. In 2D, this is fairly straightforward, using either answer from this question (I use the second answer). If we use an equilateral ...
3
votes
2answers
113 views

Reaction-diffusion PDE with NDSolve: either very slow or very inaccurate

I am struggling to have Mathematica 10.3 solve a system of PDE's (with periodic boundary conditions and random initial conditions), but either it produces a set of very noisy InterpolatingFunction ...
3
votes
4answers
261 views

NIntegrate Trapezoid rule with even subdivision producing poor results

NIntegrate[x^2, {x, 5, 9}, Method -> {"TrapezoidalRule", "Points" -> 2, "RombergQuadrature" -> False}, MaxRecursion -> 1] returns the ...