Questions on the use of numerical functions NIntegrate and NDSolve.

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9
votes
1answer
1k views

Integration strategies for oscillatory multidimensional function

I am seeking to integrate a highly oscillatory, multidimensional function. I am currently using NIntegrate's QuasiMonteCarlo approach. However, this is time-consuming and, given my current resources, ...
8
votes
1answer
734 views

Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel

I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity: $$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m ...
0
votes
0answers
38 views

NIntegrate issue to do with unknown types [duplicate]

I have a numerical function zz: zz[s_ ? NumericQ] := so3toE3[Inverse[yn[s]].ND[yn[k], k, s]]; The other functions involved have 2 pages of code defining them so ...
3
votes
1answer
993 views

Monitoring the Evaluation of NDSolve: time to finish estimation

My problem is quite simple: I run a NDSolve with a system of many ODEs, a calculation that will run for many hours, and I would like to know the progress of the ...
11
votes
2answers
1k views

Determining which rule NIntegrate selects automatically

I need to numerically integrate a highly oscillatory function over the semi-infinite domain $(0,\infty)$: $$\int_0^\infty \frac{\sin^2(x) \sin^2(1000 x)}{x^{5/2}}\mathrm dx$$ Since the Levin rule ...
0
votes
0answers
228 views

Function not recognized by Mathematica?

I'm trying to do triple integration on a bivariate function where one of the upper integration limits is the variable of the outer-most integration. When I execute the following lines, Mathematica ...
1
vote
1answer
880 views

Integro-differential eqn with double integral

I am looking at the following variation of a integro-differential, with y[0]=1. The output is not great, any solutions to this? ...
1
vote
0answers
520 views

Cauchy principal value integral of a list of numbers. How?

I have a list of numbers that are numerical samples of a function for which I need to find the Cauchy principal value integral. I thought I should be able to combine Interpolation with Integrate to do ...
1
vote
0answers
2k views

NDSolve: methods and step size choosing

I am looking into the documentation of NDSolve[]; more precisely how this function chooses the StepSize and how it chooses which ...
1
vote
1answer
150 views

Double integral 0

I am trying to reproduce the results from a paper in Mathematica. This task involves $K$ double integrals of the form $$\int f(x,y)g(x)dx,$$ where $f(x,y)$ is a bivariate normal density with mean ...
5
votes
0answers
159 views

Numerical solution of Schrödinger-type equation in Mathematica [duplicate]

I want to solve the following differential equation numerically: \begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\Delta}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r,t)\right]\psi(r,t) \end{equation} ...
19
votes
4answers
1k views

How to calculate the volume of a convex hull?

Given a spatial curve represented by a parametric equation, is it possible in Mathematica 9 to calculate symbolically (or at least numerically) the volume of its convex hull?
2
votes
2answers
421 views

Strange Behavior of NDSolve

I am trying to evaluate the following ODE numerically: ...
3
votes
0answers
1k views

Using NDSolve for Integro-Differential Equations

I have a fairly complicated set of coupled non-linear integro-differential equations that I am trying to solve using NDSolve. The equations are: ...
4
votes
1answer
688 views

NDSolve with vector function

(Possible duplicate yet I still can't understand.) Basic 2D revolving around origin: ...
5
votes
1answer
662 views

Solving homogeneous Fredholm Equation of the second kind

I am trying to solve a homogeneous Fredholm integral equation of the second kind, i.e. $\lambda y(x) = \int\limits_a^b e^{i[\phi(t)+k(t-x/M)^2]} y(t)\,dt$ where $\lambda$ is the eigenvalue (to be ...
5
votes
2answers
1k views

How to deal with zero in NDSolve in mathematica?

I would like to solve the following ODEs $$\begin{cases} x'(t)&=y\\ y'(t)&=-y(t)/t-e^{x(t)},\\ x(0)&=1,\\y(0)&=0, \end{cases}$$ (EDIT : The second equation used to be $y'(t) = ...
5
votes
1answer
1k views

Getting MemoryAllocationFailure from NIntegrate [closed]

When numerically calculating a double integral using NIntegrate over an infinite domain (with all options at their default), Mathematica 7 calculates my integral ...
2
votes
0answers
51 views

NIntegrate/NSum with parameters [duplicate]

I'm trying to calculate a continuous integral within a discrete integral. Something similar to this (yet more complex): ...
1
vote
1answer
691 views

NDSolve diffusion equation over/underdetermined

I have a feeling the solution to my problem is very simple… but my knowledge of differential equations is pretty weak. I am trying to solve a scalar diffusion equation (used in NMR spectroscopy, but ...
0
votes
1answer
292 views

DAE - varying initial conditions

I want to solve a DAE-system and I want to vary more than one initial conditions and to manipulate them. I looked here: Putting NDSolve into ParametricPlot But it does not work: ...
20
votes
2answers
1k views

3D orbits and inaccuracy over time

I wrote a little program to use Newton's Law of Universal Gravitation to animate 3 planets orbiting a central star, but I have run into a problem. Here is the code that I used to create the program (I ...
3
votes
1answer
662 views

How to adjust parameters to experimental data on a NDSolve problem

I have 2 differential equations with 2 variables, x and y,which are a function of t and I have the parameters k1, k2 y k3. ...
12
votes
1answer
778 views

Numerically solve 2nd order differential equation with singularity

Consider a second order differential equation with a potential that diverges at some generic value in the variable. For example: $$-y^{\prime\prime}(s)+\frac1{\mathrm{cn}{(s\mid k^2)}}y(s)=0$$ where ...
5
votes
1answer
151 views
12
votes
1answer
723 views

Why can't I change the value of MaxRecursion in NIntegrate when integrating BesselJ?

Bug introduced in 8.0.4 or earlier and persists through 10.0.2. I am trying to evaluate this integral numerically $$ \int_0^{\infty } J_0(q R) \tanh(q) \, \mathrm{d}q $$ for large values of $R$. ...
5
votes
2answers
2k views

How to plot and solve the numerical solution of a integro-differential equation

I have a integro-differential equation of the form $y'(t) = - \int_0^t {y(t_1 )} e^{t_1 - t} dt_1, {\rm{ t}} \in {\rm{[0,10], y(0) = 1}}$ My code is: ...
2
votes
0answers
296 views

Is mathematica storing information it shouldn't store?

I'm seeking to find solutions to a numerical integration with a large set of parameter combinations (basically, I'm doing a brute parameter sampling). Yet, I believe the memory of the computer is ...
0
votes
1answer
171 views

Notation for numerical solutions to differential equations

Can somebody explain this notation to me? Using Mathematica's first example in the NDSolve documentation: ...
2
votes
0answers
373 views

Adapting NDSolve to circumvent NDSolve::bdord: error for 1-D Euler Equations

I attempted to use NDSolve for the 1-D isentropic unsteady flow equations with low subsonic inflow velocity and prescribed inflow total enthalpy; along with a ...
2
votes
1answer
731 views

Infinite Expression Error from NDSolve

I am trying to solve a differential equation numerically. So I have ...
7
votes
2answers
351 views

Problem with NIntegrate when WorkingPrecision is specified

I am trying to evaluate this integral numerically: $$ \int_0^{\infty } m \exp (-m) J_1(m){}^2 \, dm $$ Everything is OK when only the integration method is specified: ...
2
votes
1answer
429 views

How to integrate ParametricNDSolve solution with respect to a parameter?

I have just upgraded to the new version of Mathematica because of its new built-in ParametricNDSolve function. I need to solve a first-order non-linear ordinary ...
3
votes
1answer
926 views

NDSolve for a large system of simple ODEs

I am solving a system of many (more than 100) ODEs. It is the kind of standard rate equation encountered in semiconductor physics. Here is the system: ...
4
votes
1answer
323 views

Differentiating ParametricNDSolve solutions

Is there any way to differentiate a solution obtained by ParametricNDSolve? For instance, I have the position $\phi_\gamma(t)$ as a function of time, parametrized ...
0
votes
2answers
620 views

NDSolve solution for driven damped pendulum diverges

I want to solve numerically for the system of the driven damped pendulum using Mathematica. This is the second-order nonlinear equation \begin{equation} \ddot{x} + 2 \beta \dot{x}+ \omega_0^2 \sin ...
4
votes
2answers
715 views

Animating the Lorenz Equations

I am trying to use the Animate command to vary a parameter of the Lorenz Equations in 3-D phase space and I'm not having much luck. The equations are: ...
0
votes
1answer
171 views

Error messages when using NInverseFourierTransform

I have two functions that I need to inverse Fourier transform and I was trying to get Mathematica to help me. I tried simply using theInverseFourierTransform ...
3
votes
2answers
2k views

How does one specify Neumann conditions for NDSolve?

I have a series of functions defined in my notebook, and then want to use this to solve a diffusion-reaction type equation. At the moment, something like this works: ...
9
votes
3answers
2k views

How to speed up the plot of NIntegrate?

Here is a toy example: f[t_] := NIntegrate[Sin[x], {x, 0, t}]; Plot[f[t], {t, 0, 10}] // Timing Even such a simple example will take 2.8 seconds on my computer. ...
0
votes
2answers
402 views

Problems with NDSolve and stiffness

I am trying to solve an ODE in chemical kinetics: $$\begin{align*} \frac{\mathrm d[x]}{\mathrm dt} &= -k_1 [x][y]\\ \frac{\mathrm d[y]}{\mathrm dt} &= k_1 [x][y] - k_3[y] \end{align*}$$ My ...
1
vote
1answer
1k views

NIntegrate fails while Integrate works

I have a function $f(t)$ defined as $f(t)=\int\limits_0^t(t-\xi)^{\alpha-1}\ \cos(\xi)\ d\xi$ where $0<\alpha<1$. I now want to evaluate this integral at various values of time. Therefore, my ...
18
votes
1answer
2k views

Optimizing a Numerical Laplace Equation Solver

Laplace's Equation is an equation on a scalar in which, given the value of the scalar on the boundaries (the boundary conditions), one can determine the value of the scalar at any point in the region ...
8
votes
2answers
5k views

Solving a system of ODEs with the Runge-Kutta method

I´m trying to solve a system of ODEs using a fourth-order Runge-Kutta method. I have to recreate certain results to obtain my degree. But I'm a beginner at Mathematica programming and with the ...
2
votes
2answers
775 views

How to remove a singularity with WhenEvent

Bellow I have a differential equation which hits a singularity at low values of t. What I want to do is somehow utilize the WhenEvent command in order to replace ...
1
vote
0answers
803 views

How can I speed up numerical integration of multidimensional integral?

I am numerically solving an integral equation that contains a double integral. I have managed to get a solution but it takes forever. I am wondering if there is a way to speed up numerical integration ...
2
votes
1answer
594 views

Creating hexahedral finite elements in Mathematica

Is it possible to do FEM using hexahedral elements in Mathematica? If it possible, is there any help to do that?
6
votes
1answer
813 views

WhenEvent in NDSolve

How come this doesn't work as I intended? ...
0
votes
1answer
256 views

double integration

I want to evaluate a double integral, but the limits of one integral is a function of the second. Like this ...