Questions which deal with or solve for functions expressed in integral form, i.e., one or more unknown functions that appear under an integral sign.

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17
votes
1answer
3k views

How to solve a non-linear integral equation?

I have a non-linear integral equation that I'd like to solve with Mathematica: $$ \int_{0}^{1} \mathrm{d}x \frac{B(x) v}{(B(x) + B(v))^2} = 1$$ ...
15
votes
3answers
3k 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
2k 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) ...
8
votes
2answers
6k 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 ...
8
votes
0answers
2k views

Integro-differential equation [closed]

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|) ...
7
votes
3answers
566 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
2answers
327 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: ...
6
votes
2answers
133 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 ...
5
votes
1answer
133 views

Solve a differential equation with an integral inside?

I am trying to solve this differential equation in Mathematica: y'[t]+integral from 0 to t of y[x]dx =e^(-t) where y[0]=0. Is this possible?
5
votes
3answers
374 views

Different results of a definite integral $\int_0^{\cosh ^{-1}(a)} \frac{1}{\sqrt{a^2 \text{sech}^2(x)-1}}\, dx$

fixed in 10.0.2 Update I have tried like these. I think there is a bug. ...
5
votes
1answer
568 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 ...
4
votes
2answers
1k 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
1answer
715 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
171 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: ...
4
votes
1answer
278 views

How to solve this Integral equation

D[x[t] - x[t - 1]/(2 E), {t, 3}] + Integrate[E^(-δ)*x[t - δ]/5^t, {δ, 2, 2.5}] == 0 I found solve this problem is hard with Mathematica. I also find a article ...
4
votes
2answers
162 views

Compute the average distance from the base of a rectangular pyramid to its apex

How can I compute the average distance from the base of a rectangular pyramid to its apex? For example, if the base of the pyramid is 30 feet by 8 feet, and the height of the pyramid is 12 feet, then ...
4
votes
1answer
305 views

Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

I need to solve this Fredholm integral equation of the second kind: f[s]+integrate[f[t] K[s,t],{t,0,1}]=s where ...
4
votes
3answers
326 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?
4
votes
1answer
326 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 ...
4
votes
2answers
619 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 ...
4
votes
2answers
68 views

DSolve results not what expected

I have the following formula: s'[t] == 230.94 Tan[0.914743 - 0.138564 t] When I solve this by hand I get: ...
3
votes
1answer
1k views

Can Mathematica solve integro-differential equations?

I have integro-differential equations like this: ...
3
votes
0answers
80 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } ...
3
votes
0answers
119 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
161 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
1answer
150 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
280 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
1answer
165 views

Complex Integration

There are some sequential functions: ...
2
votes
3answers
288 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 ...
2
votes
1answer
113 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 ...
2
votes
2answers
347 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
510 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: ...
2
votes
0answers
40 views

NDSolve with NIntegrate [duplicate]

I'm new to Mathematica and I'm trying to use it to solve some equation like: NDSolve[x'[t]+NIntegrate[x[tau],{tau,0,t}]==20,x,{t,0,10}] But it keeps giving me ...
2
votes
1answer
134 views

Integral equation

I'm trying to solve the following integral equation: Integrate[-k/2 Exp[-k Abs[x-z]] f[x], {x,0,d}] == a f[z] Differentiating twice with respect to z should ...
2
votes
0answers
162 views

Volterra integral equation

I have to find an approximate numerical solution for the equation $$ F(x) - \lambda \int\limits_1^{x} \text{d}s \;s^2 F(s) Z(x-s) = G(x) $$ $$Z(s) = (\psi''(1-2\ h\ i\ s)- 0.5 \psi''(1-2\ h\ i\ s))$$ ...
2
votes
0answers
270 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: ...
1
vote
2answers
136 views

Solving for the limits of an integral

I am trying to get Mathematica to solve for the symmetric limits of an integral of a Square Wave. ...
1
vote
1answer
107 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
102 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
2answers
105 views

Output of NIntegrate depends on MaxRecursion

I have an integral in this form: ...
1
vote
1answer
116 views

Is this integral solvable?

I am trying to solve the below mentioned integral. Is it possible to solve this with mathematica. If not, can I get an approximation for this by introducting the variable not to take certain values ? ...
1
vote
2answers
79 views

Integral too slow and with conditionals

When I try to solve this integral: $$\int_{-a}^a \frac{x}{\left(x^2+y^2\right)^{3/2}} \, dy$$ it's return ...
1
vote
1answer
114 views

Quantum Physics question [closed]

I'm not sure if anyone here would be able to help me. But if you know a more appropriate place where I could ask for help, please let me know. So here's the question: And here's my attempt at ...
1
vote
1answer
83 views

symbolic integration of product of hankel function and trignometric function

I want to perform the following integration. my function is : r*cos(r)hankel(0,kr) i want to integrate from 0->pi , and want to get the expression in terms of k. I used this command ...
1
vote
1answer
338 views

Solving Fredholm Equation of the first kind [duplicate]

I want to numerically solve Fredholm integral equations of the first kind, equations of the form $$g(t)=\int_a^b K(t,s)f(s)\,\mathrm{d}s$$ where we know the functions $K(t,s)$ and $g(t)$ and seek to ...
1
vote
1answer
115 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
0answers
128 views

Numerical solution to integral equation

I have an integral equation where one factor $f(t)$ in the integrand is defined in terms of an integral equation where it is again a factor. $\quad \quad y[t]=\int _0^tf[\tau ]g[t-\tau ]d\tau + ...
1
vote
0answers
114 views

PDE with Integral constraint

I am trying to solve the Non-linear Schrodinger equation $-\Delta \psi(r) + \psi(r) - |\psi(r)|^2\psi(r) = 0$ where $r \in \Omega$ In a square domain ($(x,y) \in \Omega$ where $\Omega=[0,1]\times ...
1
vote
0answers
91 views

Long running time

I gave this data as input: ...
1
vote
0answers
85 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. ...