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.

Filter by
Sorted by
Tagged with
26
votes
1answer
7k 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 ...
20
votes
2answers
15k 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]...
15
votes
2answers
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} \mathrm{d}t' {\exp[2 (t^\prime - t)] E(t^\prime) f(t^\prime)} + \int_{t}^{0}{\mathrm{d}t' \exp(t^\...
12
votes
1answer
173 views

High precision numerical solution of the nonlinear Volterra integral equation

Let consider nonlinear Volterra integral equation 5.1 from the paper An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order $$...
11
votes
3answers
813 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
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: ...
10
votes
2answers
705 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
2answers
358 views

Testing Turbulence Models

In the 90s, several turbulence models were proposed. Using Mathematica vv 10-12.1 we tested two models: 1) Spalart-Allmaras turbulence model (SA) for aerodynamic applications, published in AIAA ...
8
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: ...
8
votes
5answers
360 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
245 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
124 views

Solving the Lotka-McKendrick model with NDSolve

The Lotka-McKendrick model is a demographic model that represents the way a population changes over time due to fertility and mortality. For an age-specific population density $ u(a, t) $, and a total ...
8
votes
2answers
368 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 ...
8
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(...
7
votes
2answers
683 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{\...
7
votes
2answers
414 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}...
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 <...
6
votes
2answers
526 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 ...
6
votes
2answers
3k 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, \...
6
votes
3answers
316 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
158 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
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...
6
votes
1answer
690 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
1answer
227 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
597 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
622 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
1k 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
148 views

Numerically solving a multivariable integro-differential equation

I have an integro-differential equation of the form, $$\small\frac{\partial f(x,t)}{\partial t}=\int_{-5}^5 |x-y|\,f\left(-\frac{x}{3}+\frac{4y}{3},t\right)\,f\left(\frac{2x}{3}+\frac{y}{3},t\right)\,...
5
votes
1answer
233 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
74 views
5
votes
1answer
260 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 ...
5
votes
0answers
864 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
2answers
236 views

Solution or artifact?

I am trying to increase the precision of the code ...
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
679 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
361 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
223 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
2answers
351 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 ...
4
votes
1answer
279 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
276 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
2k 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
288 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
2answers
279 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
334 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
509 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
1answer
234 views

Inverse Laplace transform of this complicated function

I have been solving a coupled PDE system analytically and I need to find the inverse Laplace transform of $(1)$ and get $T(x,y)$. $s$ is the Laplace domain variable and $\alpha, \beta, \gamma, T_{fi}, ...
3
votes
1answer
217 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)\...

1
2 3 4 5