Questions on the use of numerical functions NIntegrate and NDSolve.

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8
votes
2answers
577 views

How to solve the differential equation with Duhamel's integral?

How do I solve a differential equation with Duhamel's integral? I tried to solve it with NDSolve, but failed: ...
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: ...
4
votes
0answers
113 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 ...
1
vote
1answer
80 views

Assymptotically Solving ODE with free parameters

I have a first order ODE, $F(y[x], y'[x], A, B) = 0$. I want to solve this numerically (with the boundary value $y[1] = i$ ) but also with the following requirements: A and B are free parameters which ...
0
votes
0answers
84 views

NIntegrate fails to converge around a value out of integration range

This is the function that I am trying to integrate, I have interpolated it for best results (would rather not): There is a 'singularity' around 0, but I get the warnings and bad results even when ...
0
votes
0answers
54 views

Integration problem: lrgexp

I want to calculate the following integration but it gives the error PolynomialGCD::lrgexp: Exponent is out of bounds for function PolynomialGCD. The code is: ...
2
votes
1answer
567 views

Plotting NIntegrate

Plot[E^(-0.5 x) NIntegrate[Cos[t] E^(Cos[t] + 0.5 t), {t, 0, x}], {x, 0, 40}] Mathematica evaluates this integral for each point, which takes a long time. It is ...
3
votes
1answer
301 views

Approximate $h$ in $F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$

Consider $$F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$$ $$|z|\le L$$ $$0 \le \theta \le \pi$$ By having knowledge of $F(\theta)$, how can one approximate $h(z)$? In ...
2
votes
1answer
848 views

Use Euler method to solve differential equation

Use Euler's Method or the Modified Euler's to solve the differential equation ${dy/dt=y^2+t^2-1, y(-2)=-2}$ on $[- 2, 2]$. Take h = 0.2 (...
2
votes
2answers
541 views

Plotting several numerical solutions plus the analytic solution of ODE in one plot

I want to be able to plot several numerical solutions of an ODE, plus its analytical solution in one plot, in order to see how the numerical solutions converge towards the analytical one with respect ...
1
vote
1answer
1k views

Euler's method for system of differential equation

I need to program Euler's method to solve a system of two diffferential equations of first order. Fist, I have programmed the Euler's method for just one differential equation: ...
6
votes
1answer
807 views

NestList and Euler's method

I am new to mathematica and so just experimenting with various programming constructs. Recently have been looking at NestList and how I could use this to implement ...
3
votes
1answer
237 views

Precision of NIntegrate

At the moment I am considering a "difficult", highly-oscillatory integral in Mathematica. It calculates the integral without any complaints. However, I am also trying out a numerical method with which ...
1
vote
2answers
149 views

Plot3D and NIntegrate issues

f[x_, y_] := 2*x - y Plot3D[f[x, y], {x, -1*Sqrt[4 - y^2], Sqrt[4 - y^2]}, {y, -2, 2}] NIntegrate[f[x, y], {x, -1*Sqrt[4 - y^2], Sqrt[4 - y^2]}, {y, -2, 2}] I ...
2
votes
1answer
219 views

Evaluation of the second argument to NIntegrate

The expression Integrate[x^2, Flatten[{{x},{1,2}}]] evaluates properly, to $\frac{7}{3}$. However, ...
1
vote
1answer
150 views

Integrate and NIntegrate yield different results for double integral

Evaluating a double integral with bivariate normal distribution yileds widely different results depending on the method used. I define a bivariate normal distribution with ${10, 3}$ and ${8, 1.5}$ as ...
1
vote
2answers
321 views

How to numerically integrate this integral?

I want to integrate a function (spherical coordinates): $$\int _0^{2 \pi }\int _0^{\pi }\frac{r^2 \sin (\theta ) e^{-\lambda \sqrt{\rho ^2+r^2-2 \rho r \cos (\theta )}-2 r}}{\pi \epsilon \sqrt{\...
17
votes
5answers
527 views

Mismatch between numerical and analytic evaluation of an integral

I evaluated the following integral $$\int_0^1 \sqrt{r} \left | \cos \left(\left(k+\frac{1}{2}\right) \pi r\right)\right | dr$$ ...
2
votes
2answers
349 views

Perturbation theory with Mathematica: Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

I would like to ask also Mathematica users about my question from the math forum. To expand, I'm adding the code which calculates the full double integral for $n=0$ and $\mu=0$ (the second in the post)...
19
votes
4answers
928 views

A bug in Integrate

Integrate[(1 + 16 Tan[2 x - y]^2)/(1 + 4 Tan[2 x - y]^2), {x, 0, 2 π}] Mathematica (wrong) output is (tested under versions 8 and 10.0, took ~ 1 minute of CPU ...
0
votes
1answer
99 views

Numerical integral speed

I have the following code to calculate a numerical integral for any given a, however it takes a very long time, even with adaptivemontecarlo, which is not accurate enough: ...
9
votes
2answers
284 views

Why does Mathematica say $\int_0^1\int_0^1\int_0^1\frac{1.0}{xyz}\,dz\,dy\,dx=0$?

Mathematica 9 says that $\int_0^1\int_0^1\int_0^1\frac{1.0}{xyz}\,dz\,dy\,dx=0$ and $\int_0^1\int_0^1\int_0^1\frac{1}{xyz}\,dz\,dy\,dx=0$. ...
2
votes
1answer
178 views

Solving an integral equation numerically

my problem is: I get the result of definite integral and now I need to find the upper limit for the same integral but with opposite sign value so f2=-f1. ...
2
votes
1answer
177 views

NIntegrate Warning / Error Messages

I am doing: NIntegrate[Sin[Exp[(x^4)]], {x, 2, Infinity}, PrecisionGoal -> 12] It prints out a host of warnings, but also shows the results as: $$-0....
1
vote
2answers
132 views

Integral too oscillatory

Is there any way top make this integral less oscillatory? ...
0
votes
0answers
70 views
2
votes
0answers
170 views

The idea behind Stiffness switching method with NDsolve [closed]

Does the Stiffness switching method with NDsolve switch just between multiple variants of 4th order Runge Kutta method or it uses also other methods?
2
votes
1answer
68 views

How to NDSolve a set of equations, one of which itself contains NIntegrate of a desired function?

How to NDSolve a set of equations, one of which itself contains NIntegrate of a desired function waited to be solved by NDSolve first? For example, ...
2
votes
1answer
111 views

Why isn't Table iterator value inserted in failed NIntegrate arguments?

Consider this simplest example: Table[{z, NIntegrate[f[x], {x, 0, z}]}, {z, {1}}] Here f is not defined, so ...
6
votes
2answers
513 views

Starting NDSolve from intermediate time step?

I always wondered if I could start NDSolve from an intermediate time step. What I mean is, in the code sample below, if I were to run my solution from ...
0
votes
2answers
95 views

Fix my code to return a table of values

Here is a “procedural” program that we wrote in my class, implementing the rectangle rule of numerical integration: ...
3
votes
3answers
306 views

NIntegrate-ing a compiled function

I'm trying to integrate numerically in 6 dimensions a very long expression and I read about the option to NIntegrate a compiled function which should be faster. ...
3
votes
0answers
361 views

Problem solving a nonlinear partial differential diffusion equation [closed]

EDIT: actualy the nonlinear partial differential equations for interacting density distributions, including boundary conditions, should be given as $$ \frac{\partial\phi}{\partial t} = D \frac{\...
5
votes
1answer
151 views

Finding minimum fly-by radius between Mars and spacecraft from interpolating function

I've written an interplanetary trajectory solver/plotter that plots the path taken by a spacecraft on an Earth-Mars mission, but have run into a little trouble when the spacecraft actually reaches ...
4
votes
1answer
127 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 ...
1
vote
1answer
305 views

Locating Periodic Orbits

Here is the code for the numerical integration of an orbit. First the module for the definition of the equations of motion. ...
2
votes
0answers
200 views

Volterra integral equation

I have to find an approximate numerical solution for the equation $$ F(x) - \lambda \int\limits_1^{x} \text{d}s \;s^2 F(s) Z(x-s) = G(x) $$ $$Z(s) = (\psi''(1-2\ h\ i\ s)- 0.5 \psi''(1-2\ h\ i\ s))$$ ...
1
vote
1answer
301 views

Runge-Kutta Butcher tables

I like to have the Butcher's table for Explicit (or implicit as well) Runge-Kutta method of a fixed order. I do not understand reading http://blog.wolframalpha.com/2013/09/10/numerical-methods-runge-...
2
votes
2answers
378 views

NIntegrate over a list of functions

This question is the result of these other two questions. Question 1 and 2. I thought it would be better to ask a new question rather than deleting previous one. I think When ...
3
votes
2answers
346 views

NIntegrate piecewise vector function

Is there a way to numerically integrate a vector function defined via Piecewise? Example: ...
0
votes
1answer
82 views

NIntegrate producing SetDelayed::write message [closed]

I used NIntegrate to calculate an integral with the final limit as a variable, which later will be listed in a table, ...
0
votes
1answer
222 views

Plotting results of NIntegrate with variable integration limit

I have tried to use NIntegrate with variable limits and compute the following ...
7
votes
1answer
202 views

Difficulty in getting correct Gaussian curve for diffusion of point source

I want to solve diffusion of a point source numerically and check it against analytical solution. first I define initial profile, ...
5
votes
1answer
413 views

Numerical solution of IVP for linear ODE with variable coefficient runs wild soon

Cross posted in scicomp.SE. A friend of mine showed me this initial value problem (IVP) for a linear ordinary differential equation (ODE) with variable coefficient: $$y''(x)=\left(x^2-1\right) y(x)$$...
0
votes
0answers
109 views

Volume by NIntegrate gives zero

I want to Integrate the following expressions for different p < n and 2 < n with p and n both natural numbers: ...
5
votes
1answer
597 views

How to plot the solution of a Partial Differential Equation?

My attempt. I need to solve numerically the Complex Ginzburg-Laudau Equation (CGLE): $$ \frac{\partial A}{\partial t}=\epsilon A-(1+i\beta)|A|^2A+(1+i\alpha)\nabla^2A $$ I'm using a uniform initial ...
1
vote
1answer
264 views

Plot a numerical integration as a function of a variable

Suppose we have a function that is hard to evaluate analytically but a numerical estimate suffices. For example consider the $f(x,\Lambda) = \Lambda\cdot \sin(x)$ where $\Lambda$ is some parameter I ...
7
votes
2answers
566 views

Integrating a function over a surface integral

From a first principles bandstructure calculation I get an energy scalar field in three dimensions $E(x,y,z)$. It's now easy to plot a constant energy (contour)-surface for dedicated values $\epsilon=...
4
votes
2answers
149 views

Coarse-graining in numerical integrations

I have been working recently in a coarse-graining problem I found when using NIntegrate: I am trying to evaluate the function $$f(a)=\int_0^{\infty}x\,e^{-(a^2+b^2)x^2}\text{d}x$$ numerically as a ...