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Questions tagged [integral-equations]

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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25
votes
1answer
6k views

How to solve a non-linear integral equation?

I have a non-linear integral equation that I'd like to solve with Mathematica: $$ \int_{0}^{1} \mathrm{d}x \frac{B(x) v}{(B(x) + B(v))^2} = 1$$ ...
23
votes
3answers
6k 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 ...
17
votes
2answers
14k 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 f[x]...
12
votes
1answer
3k 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) f(t^\...
11
votes
3answers
544 views

Solving an integro-differential equation with Mathematica

I try to solve a nonlinear integro-differential equation with this code. Here i used a periodic condition. ...
10
votes
2answers
512 views

Fredholm Integral Equation of the 2nd Kind with a Singular Difference Kernel

After reviewing the literature, I could not find an analytic solution to the equation $$\int^{1}_{0}dx\frac{f(x)}{|x-y|^{2/3}}=cf(y)$$ for $f(y)$, where $c$ is a constant and $y\in[0,1]$. I'm ...
9
votes
3answers
2k 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: ...
8
votes
5answers
280 views

Functional fixed points (ie fixed point of mapping from function space C[0,1] to itself)

I am looking for some tips or guidance as to what machinery in mathematica can help me get at this problem numerically. I am looking for fixed points of a mapping, but the objects in question are ...
8
votes
3answers
193 views

Numerically solving an ODE whose right-hand side involves an integral

In a 1955 paper on surfaces of constant negative Gaussian curvature, Marc-Henri Amsler defines a function $w(x)$ by giving the ordinary differential equation $$w'(x)=\frac{1}{x}\int_0^x \sin(w(t))dt,$$...
8
votes
2answers
298 views

Is it possible to take a numerical (integral) average of the dependent variable, within NDSolve, at each iteration?

tl;dr: want to integrate (average) the dependent variable within NDSolve. I am currently trying to implement a basic diffusion-advection equation for a reactant, A. The species is converted between A ...
7
votes
1answer
1k 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 <...
7
votes
0answers
2k 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: $\dot{x}=\big|y(t)-x(t)\big|^{1/n}\left[\text{Sign}[y(...
6
votes
2answers
4k 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: ...
6
votes
3answers
289 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: $$x_1+x_2+\cdots+x_k=m\quad\...
6
votes
2answers
231 views

Solving Fredholm Equation of the second kind

Consider the Fredholm Equation of the second kind, $$\phi(x) = 3 + \lambda \int_{0}^{\pi} \text{cos}(x-s) \, \phi(s) \,ds$$ Where the analytical solution is found as, $$\phi(x) = 3 + \frac{6\lambda}...
6
votes
2answers
153 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 ...
6
votes
1answer
630 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?
6
votes
1answer
1k 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
2k views

Numerically solve an integro-differential equation

So the problem is to numerically solve this Integro-Differential Equation, $$-v \frac{\:\mathrm{d}\kern0.1mm x(\zeta)}{\:\mathrm{d}\kern0.1mm \zeta} + \psi(\zeta) x(\zeta) + \int_{0}^{\zeta} f(\zeta, \...
5
votes
2answers
420 views

NIntegrate into NDSolve with variable integrand

I need to solve an integral into an ordinary differential equation like this: NDSolve[{y'[x] == x + NIntegrate[y[r], {r, 1,x}], y[0] == 1}, y, {x, 0, 1}] Note ...
5
votes
2answers
441 views

Solving integral equation

How can I solve the following integral equation for $f(x)$ on $x \in [0,1]$? $$ (f(x) -x) \int_0^1 \left( \frac{\mathrm e^{2-2f(x)}}{\left(\mathrm e^{1-f(x)}+\mathrm e^{1-f(y)}\right)^2}- \frac{\...
5
votes
1answer
162 views

Solving partial differential equation involving Hilbert transform

While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it. $$\frac{\partial f}{\partial t}−(\mathcal{H}(f)\left(\frac{\...
5
votes
1answer
517 views

PDE of real-world system, integral boundary condition

I've stripped all the physical-significance for clarity, but I know that u[x,t] will be everywhere positive and continuous. here are the equations in Mathematica code: ...
5
votes
1answer
979 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 ...
5
votes
1answer
218 views

Numerical Solution of Fredholm Integral Equation Using Bernstein Polynomials

I'm trying to solve a Integral equation: $$y(x)-\int_{-1}^1 \left(x^4-t^4\right) y(t) \, dt=x$$ for n > 5 my code is very slow. How to speed up? Example 2 ...
5
votes
1answer
528 views

Solving PDE involving Hilbert transform numerically

I'm trying to solve the following equation numerically: $$u_{tt}-\mathcal{H}(u_x)=A^2_{xx},$$ where $\mathcal{H}$ is the Hilbert transform and $A$ is a prescribed forcing function which we assume ...
5
votes
1answer
52 views
5
votes
0answers
768 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: f[t_] = Sqrt[1 + E^(-2 t)] ...
4
votes
1answer
1k 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 $f(s)...
4
votes
2answers
669 views

Mathematica gives an unexpected answer for Integrate [closed]

I need to integrate the following: \begin{equation}\tag{1} \frac{\sqrt{C + (1 - C) x^3}}{x}, \end{equation} where $0 < C < 1$ and $x$ is a positive variable (then $x^3 \ge 0$). When I integrate:...
4
votes
2answers
326 views

Nonlinear boundary value problem of ODE involving principal value of integral

I have asked a question on a nonlinear eigenvalue problem (EVP). And I have worked on these for a week but I cannot solve it. I think I should first try to solve the related ODE boundary value problem ...
4
votes
3answers
195 views

Numerical solution of a singular integral equation

I am looking to approximate the solution u of the following equation using discretization method or any other idea. Is there any way on how to find a numerical ...
4
votes
1answer
221 views

Why doesn´t Mathematica solve the integral Integrate[Sqrt[a^2*(Sin[x])^2+1],{x,0,2*Pi}]?

I have to solve that integral and I am sure it is good written. It is an elliptical integral but Mathematica just doesn´t solve it. What is the result to that integral? ...
4
votes
2answers
259 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
3answers
1k 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
1k 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 0<=s<=1...
4
votes
1answer
239 views

How to speed up the integral in NDSolve?

On the MMA.SE, there have been several question on how to solve equations including integral using NDSolve, see example 1, example 2, and example 3. Many people, ...
4
votes
1answer
241 views

Solving integro-differential equation with DSolve in MMA 11.0

In this question DSolve(symbolic solution) gives wrong answer. I analyzed the 13 equations in which DSolve gives the same ...
4
votes
2answers
241 views

How can I solve a 2nd order ODE containing an integral with NDSolve?

I'm trying to solve a equation of gravitional wave propagation given by $$ \chi''(u) + \frac{2}{u} \chi'(u) + \chi(u) = -\frac{24 f_h (u)}{u^2}\int^u_0 \frac{j_2 (u - U)}{(u - U)^2}\chi'(U)\mathrm ...
4
votes
0answers
317 views

Non-linear integral equation [closed]

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 ...
3
votes
4answers
447 views

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$?

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ? Limit[Integrate[(1 - x/n)^n/E^x, {x, 0, n}], n -> Infinity] ...
3
votes
2answers
219 views

Solution or artifact?

I am trying to increase the precision of the code ...
3
votes
2answers
311 views

Solving integro-differential equation with boundary condition at infinity

I wish to solve a differential equation that contains a hard-to-evaluate integral and to plot the solution in a range at least $r\in(0,10)$. The equation comes from a Hartree equation (Schroedinger ...
3
votes
1answer
285 views

Solve integro-differential equations

I am trying to numerically solve the following integro-differential equation and get some plots for $n(s)$ vs $s$: $$ \dot{n}(s)-\frac{\dot{w}(s)}{2w(s)}\int_{-\infty}^s ds'\frac{\dot{w}(s')}{w(s')}(...
3
votes
1answer
164 views

Using DSolve to solve an integral equation

I've the following integral (or DE) that I need to solve (for $x(t)$ and all the constants are known, real and positive): $$k\cdot\theta\left(t-m\right)+(n-k)\cdot\theta\left(t-v\right)=$$ $$x(t)\...
3
votes
2answers
146 views

Package for numerically solving Linear Integro Differential equation

Are there any packages or readily available methods for solving linear integro differential equations of the form $\dot{f}(t)=-\int_0^t g(t-s)f(s)\mathrm{d}s$ Thanks already for the answer! Martin
3
votes
1answer
801 views

Solve integro differential equation numerically [duplicate]

I want to solve the following equation numerically: with initial condition y[0]=0 and y'[0]=0 I tried to do so by using the NDSolve function: ...
3
votes
1answer
125 views

Dog chases his tail ! - “parametric differential/Integral equation”..?

I have the following situation where I am interested in the function $m(t)$ $$ \frac{dm}{dt}=4T(t)^{3}+T(t)^{2} $$ $$ T(\tau)=T_{0}-\int_{0}^{\tau}(\frac{dm}{dt})dt*Q_{S} $$ Is there a way to solve ...
3
votes
1answer
264 views

exponential differential integral equation

I have a following equation, which has a singular kernel, ...