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12
votes
3answers
2k views

Solving a Volterra integral equation numerically

I would like to solve for $P(t)$, in Mathematica, a Volterra integral equation of the 2nd kind. It is: $$P(t) = R_0(t) + \int_0^t P(t') R_0(t-t')dt'$$ I know the function $R_0$ and would ...
9
votes
1answer
1k views

How to solve a non-linear integral equation?

I have a non-linear integral equation that I'd like to solve with Mathematica: Solve[Integrate[B[x] v / (B[x] + B[v])^2, {x, 0, 1}] == 1, B[v], {v}] This ...
8
votes
1answer
1k views

solve an integral equation numerically

I am trying to find a numerical solution for an equation of the form: $$ f(t)=\int_{t_{min}}^{t} {\exp[2 (t^\prime - t)] E(t^\prime) f(t^\prime)} + \int_{t}^{0}{ \exp(t^\prime - t) E(t^\prime) ...
7
votes
3answers
481 views

Any way I can solve this integral?

So, Mathematica can't solve it. Any workaround? \begin{equation*} \int_0^\pi \frac{1}{\sin\left(\frac{\theta}{2}\right)^\beta + 1} \mathrm{d}\theta \end{equation*} The code is: ...
7
votes
0answers
2k views

Integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation using Mathematica. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) ...
6
votes
2answers
113 views

Integrate perfomance

I have a notebook in Mathematica 4. I'm trying to convert it to use in Mathematica 9. One of the problem is the long computation of definite integral in the new version of Mathematica. Here's the ...
4
votes
1answer
422 views

Recursive Integration

I'd like to evaluate the following recursive integration using Mathematica $$ \ M(T) = \int_0^T\int_0^\infty e^{-\delta s}g(x,s)dxds\ +\int_0^T e^{-\delta s}f(s)M(T-s)ds\, $$ where $g(x,s)$ and ...
4
votes
3answers
149 views

How to solve an integral equation by iteration method? [duplicate]

How I can obtain $n^{th}$ approximation of the following equation $f(t)=t+\int_0^tds f(s)$ by iteration method?
3
votes
2answers
3k views

Integral equation numerical solution with NDSolve

I'm trying to solve something like: f[x] == Integrate[f[x]*g[x]] where g[x] is known and ...
3
votes
2answers
451 views

Evaluating the integral of $\cos (\theta )\cos\left[\tan ^{-1}\left(\frac{a^2 \tan (\phi )}{b^2}\right)\right]$

I am trying to evalute the following integral: Integrate[Cos[θ] Cos[ArcTan[a^2/b^2 Tan[φ]]], {φ, 0, 2π}] I know that for $a=b$ I should get 0 out of the integral ...
3
votes
0answers
111 views

What can I do to improve the performance of my calculations?

I'm trying to express a simple probability problem in Mathematica, but am having trouble getting my calculations to execute at a reasonable speed. I have an object whose unknown location is modeled ...
3
votes
0answers
90 views

Non-linear integral equation

I'm trying to solve with Mathematica an integral equation. I found this excellent answer (How to solve a non-linear integral equation?) solving with a collocation method a problem which can be ...
2
votes
2answers
175 views

Finding integral points on a surface

Suppose I have a dimension formula (for a Lie algebra representation) that is as follows: $$ d(a,b) = {(a+1)(b+1)(a+b+2) \over 2} $$ Now consider the surface $F(a,b,n) = 0 = d(a,b) -n$ where $n \in ...
2
votes
1answer
141 views

Problem with numeric integration

I am having trouble with the UnitStep function as in the title. My problem is very simple, but I am not able to get a numerical result. I have ...
2
votes
1answer
197 views

how can I solve integrals including recursion of some sequences of functions

I would like to solve the equations as given below in mathematica. I have written some codes before but I have no idea when there is sub indexing to indicate the series of functions. I will be very ...
2
votes
2answers
111 views

How do I generate a set of n-tuples containing integral solutions to a linear equation provided certain constraints?

Let $m,k,p$ be fixed positive integers. I want to create a table of k-tuples $(x_1,x_2,\ldots,x_k)$ comprised of solutions in positive integers to the equation below: ...
2
votes
1answer
137 views

Integral of GeneratingFunction

I know that GeneratingFunction can be used to compute the generating function $\sum_{n=0}^\infty a_n x^n$ of a sequence $(a_n)_n$ via GeneratingFunction[a[n],n,x] ...
2
votes
2answers
97 views

Finding Expectation of function of a Log-normal distribution

Say $Y=g(X)$ and $p_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ dy = ...
2
votes
1answer
147 views

Using Boole to integrate over region

I have a complex and long piecewise function to integrate, so I have written a Mathematica script that cuts the whole range of integration, evaluates the integrand in each region and then integrates ...
2
votes
1answer
379 views

Volterra integral equation : 'literally match the independent variables' message

I want to solve an Volterra type Integral equation, and as a practice I entered the following code to my mathematica: ...
1
vote
5answers
279 views

Compute Triple Integral on spherical coords

I need to compute: $\int \int \int z dxdydz$ over the domain: $\{x^2+y^2+z^2\leqslant 16,z\geqslant 0\}$ Im trying to use spherical coords as: $$\int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} ...
1
vote
1answer
72 views

Write an expression as a full derivative

Is there a command that tells Mathematica to try to write the terms (if possible) of an expression as total derivates? Example: I want to tell mathematica to write ...
1
vote
1answer
91 views

Difficulty finding Expectation of a special function

I have a special function given as: $${\rm f}\left(r\right) ={1 \over \beta\lambda}\,2^{r/\beta} \exp\left({\left[2^{r/\beta} - 1\right]K \over \lambda}\right)$$ I should find the Expectation of ...
1
vote
1answer
496 views

Definite integral using Riemann sums [closed]

I just started learning Mathematica. It seems like a good program to learn. I only know some basics though. I came across this problem online which I want to learn how to solve: Write a ...
1
vote
1answer
211 views

Need help evaluating definite integral to a function of Y

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. The iterated integral ...
1
vote
1answer
68 views

How to solve an operator form of an integral equation by iteration method?

How I can obtain the $n^{th}$ approximation of the following Operator form integral equation? $F(t)=A(t)+\int_0^tds B(s)F(s)$, where $A(t)=\bigg{(}\begin{matrix}t&0\\Cos(t)&1 ...
1
vote
1answer
67 views

How to obtain a useful answer from integration when Simplify is insufficient?

I have been trying to integrate a function AND get an output that is sufficiently simple as to be usable. Simply using the Simplify and ...
1
vote
1answer
735 views

Can Mathematica solve integro-differential equations?

I have integro-differential equations like this: ...
1
vote
0answers
80 views

Imaginary number appears in Integrate [closed]

I had a problem in the following code because when I use Integrate, there will always be imaginary number in the result. I put my code here. The satisfactory result should look like eq 9.138, which I ...
1
vote
0answers
74 views

Integrate boundaries defined as equations

Have you guys ever needed to define Integrate boundaries as equations? I tried to submit the equation as was written in original text but it seems that mathematica can't understand it. ...
1
vote
0answers
163 views

Problem with Integrate

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1
vote
0answers
309 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 ...
1
vote
0answers
344 views

Testing the convergence when solving an integral equation

I am trying to find the solution of an integral equation such that a certain convergence criterion is fulfilled. Now the problem is that the function doesn't look like what I am actually expecting, ...
0
votes
1answer
471 views

integration on real domain only!

i am trying to integrate a real valued function. For some reason, Mathematica gives an answer in the complex domain. I'd be grateful for your answers. Here is the integral: ...
0
votes
1answer
120 views

Differential Equation help

I have a differential equation that looks like this: ...
0
votes
1answer
365 views

Solving an integral equation numerically for an unknown within the integral

I'm trying to solve an integral equation of the form Constant == Integrate [g(x)f(x,Efermi), {x,-200,200}], for the parameter ...
0
votes
1answer
83 views

Mathematical Theory - Integral [closed]

Consider this integral f[g_,b_]=NIntegrate[1/Sqrt[r+g+b],{r,0,Infinity}]; I have to get a value for b , and this value i get when f[g,b]==0 ( or the minimum ...
0
votes
1answer
54 views

Compute integral symbolically

I want to compute the following integral: ...
0
votes
1answer
39 views

Definite Integral, Piecewise, Error

I am trying to evaluate the following: ...
0
votes
1answer
111 views

How to solve this integral equation?

I am trying to solve for x the integral equation : ...
0
votes
1answer
445 views
0
votes
1answer
1k views
0
votes
1answer
796 views

how to solve an implicit integral equation? (iterate to a functional fixed point?) [closed]

I reduced a (special case) of my problem to the following code. Even though in this special case all related functions are analytical, DSolve is not the tool for this, though I am indeed looking for a ...
0
votes
1answer
52 views

How to calculate definite integral when boundary is a limit?

Integrate[Log[x]/(1-x)^2,{x,eta,Infinity}] The conditions are eta>0, and eta->1, how ...
0
votes
1answer
57 views

Symbolic integral including Hermite polynomial does not evaluate

I am trying to evaluate the following integral: Integrate[HermiteH[n, Sqrt[a]* x] * Exp[- (c/2)* (x^2 + y^2) + b* x * y - (a * x^2)/2], {x, - Infinity, Infinity}] ...
0
votes
0answers
27 views

NIntegrate Issue: Changing integration limits makes calculation time extremely long

After a long time developing some code in mathematica and finally getting it to work, I have unfortunately encountered an odd problem: When I change the limits of integration (and shift the function ...
0
votes
0answers
57 views

Multivariable Delay Diff. Equ. with Ndsolve and Nintegrate

I'm trying to simulate the following equation in Mathematica for $u(x,t)$: $$\frac{1}{\alpha} \frac{\partial u(x,t)}{\partial t} = -u + \int_{-\infty}^\infty {\rm d} y w(x-y) f(u(y,t - |y|/v))$$ ...
0
votes
0answers
65 views

Solving for a variable within a definite integral using FindRoot

I want to solve for P in this equation: Integrate[1/((P/(M*V)) - r*g - c*A*(V^2)*p/(2 M)), {V, 0, 26.82}] = s , where all constants are given except P. I've ...
0
votes
0answers
63 views

Exponential function Integral

I have tried to solve this integral: Integrate[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2,a] Mathemathica is not able to solve it, I have tried the integration by ...
0
votes
0answers
279 views

Solving a linear fourth-order fractional integro-differential equation

Definition: A real function $f(x)$, $x>0$ is said to be in space $C_\mu$, $\mu\in \Bbb R$ if there exists areal number $p$ ($>\mu$), such that $f(x)=x^pf_1(x)$, where $f_1(x)\in C[0,\infty]$, ...