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.

3
votes
4answers
266 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] ...
0
votes
1answer
33 views

Convolve doesn't yeild expected answer, nor does integrate

I am convolving, or trying to, a Gaussian and a Lorentzian as follows: ...
2
votes
1answer
89 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{\...
-1
votes
0answers
35 views

Сontour integral [migrated]

everyone. Can you help with the task?
1
vote
1answer
51 views

Area of a parametric plot with interpolating functions

I am looking for the area of a parametric plot defined by two solutions (defined as: VVLt & PVLt) of Numerically solved differential equations. The problem is when i'm trying to integrate the two, ...
4
votes
2answers
281 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 ...
1
vote
1answer
70 views

How to speed up the calculation of a multi-dimension matrix involving symbolic integral?

The following program succeeds in getting matrix CC, but it takes time badly, especially in the case varNumber becomes larger just as the following ...
0
votes
1answer
43 views

Delayed differential equation with convolution

So I have a differential equation : $\left\{\begin{split} & c'(t) = \alpha q(t) c(t)^{2/3} - \beta(1 - q(t))c(t)\\ &\chi'(t) = \kappa c(t) q(t) \\ & q(t) = \int_{-\infty}^t \Big( ...
4
votes
2answers
141 views

Solution of a singular integral equation using interpolation and truncated methods

I am trying to code the following integral equation to find the solution numerically using Mathematica. In fact the exact solution is x^2 (1 - x). First we define ...
4
votes
3answers
171 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 ...
0
votes
1answer
28 views

Inverse stochastic problem solution

I am trying to determine a probability density $p(\mu)$ such that, when $\mu$ is inputted into a forward simulation equation: $d \sim \mathcal{N}(\mu,0.1)$, I obtain a distribution on $d\sim \mathcal{...
1
vote
1answer
59 views

How can I solve this integration or does it have a closed form solution?

The integral I am dealing with is below. I need to find the closed-form expression of this integral. $\int_0^\infty \ln(1+\frac{A}{1+B+Cx})\frac{e^{-x/M}}{M}dx$ Here, $A$, $B$, $C$ and $M$ are ...
0
votes
0answers
36 views

Calculate a definite integral containing an ExpIntegral

I need to calculate the following integral Integrate[Exp[-(y/bn)] (-y)^c ExpIntegralE[d, p y], {y, 0, z/x}] Where $z<0,z\in \mathbb{R}$, $x\in\mathbb{R},x>...
0
votes
1answer
84 views

Substitution into integral [duplicate]

I have to do a substitution (u = pi/2-x) into an definite integral, which I have defined as f[u]. When I evaluate my code, no ...
0
votes
0answers
25 views

Using `Integrate` and or `FourierTransform`

I have some general questions about Integrate and FourierTransform. Firstly with ...
0
votes
1answer
43 views

My integral doesn't evaluate

i'm doing something called Sommerfeld expansion i got somehelp online source i will show after code. Sommerfeld expansion to integrate Fermi-Dirac equation to find total number of particles N for ...
1
vote
0answers
54 views

Solving a finite Hilbert transform from 2D thin wing theory

I have a problem where I know $w(x) =1$ and want to find $u(x)$. It's a well-known problem I am recreating, from a 1972 paper. Note the paper doesn't solve the problem, the integral equation, it just ...
2
votes
0answers
91 views

Can this integral equation problem $\int_\Gamma \frac{e^{ik|x-y|}}{4\pi|x-y|}\varphi(y) \, \mathrm{d} y = u_{x_0}^{in}(x)$ be solved?

I am not sure if Mathematica is capable of solving integral equations in 2D/3D. I found this page in the documentation, but this is just for 1D. The following is what I would like to solve, it can ...
6
votes
2answers
198 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}...
10
votes
3answers
438 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. ...
-3
votes
1answer
91 views

Solving an integral equation analytically [closed]

I have an inegral equation like this $\qquad n(\phi)=\int_0^\sqrt{\phi} f(w)\sqrt{2w+\phi}dw$. I need to find $f(w)$ analytically. Here $n(\phi)$ is known. Here $n(\phi) = -\frac{2}{\delta ^2 \phi ...
0
votes
1answer
76 views

It's wrong the solution to definite integral from Mathematica 11.3? [closed]

I'm solving the exercise 23 from 4.8 section from "The Calculus 7th Leithold" (I use the Spanish edition "El Cálculo 7"), I write the solution in (physical) notebook, was 8*sqrt(2)/3, but when run in ...
0
votes
0answers
34 views

Integral of intersection of a spiral and a circle

I need to prove that for a full turn of a logarithmic spiral (independent from the constant tangent of it) and a straight line on the z axis, the logarithmic spiral always covers more space-time in ...
3
votes
1answer
126 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
172 views

Solution or artifact?

I am trying to increase the precision of the code ...
0
votes
0answers
51 views

Numerical solution to approximate the singular integration using collocation method

I am working to solve "numerically" the following integral equation IE: u[x]=f[x]+Integrate[(1/(x-t)^(1/4))*u[t],{t,0,x}]+Integrate[(1/(x-t)^(1/4))*u[t],{t,0,1}] ...
4
votes
2answers
667 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:...
1
vote
1answer
133 views

How Can We Solve The Eigenvalues of partial-integral equation?

Here, my problem is that $$ \left(\int_{-L_0}^{L_0} \left(\int_{-L_0}^{L_0}\mathrm e^{-(x-x_1)^2-(y-y_1)^2} ({\bf u_{\lambda}}(x_1, y_1) + {\bf v_{\lambda}}(x_1, y_1)) \, \mathrm dx_1\right) \, \...
1
vote
0answers
52 views

Strange Integral Calculation With Traditional Form

I must be doing something wrong. I define the following integral: The code: ...
0
votes
0answers
48 views

Get Integral Result without Assumptions

I have the following two functions: ...
2
votes
1answer
113 views

Symbolic integration of Exponential

How can I solve this integration? I want to solve this integration with c, B, H, Y, b, w surviving as constants in the result: ...
0
votes
2answers
146 views

An integral problem with Sin and Cos

I have a difficulty in calculating the following integral with Mathematica: $$ \int^{\pi}_{0}\rm d\theta_1\frac{ \sin^2\theta_1[2(2-\cos\theta_1\cos\theta)^2+(\sin\theta\sin\theta_1)^2]}{[(2-\cos\...
1
vote
1answer
85 views

Unable to obtain Analytical expression for a long integral

I have the following parameter c1 that I wish to integrate analytically (I need a close form expression in order to study them in detail): ...
2
votes
1answer
255 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')}(...
1
vote
1answer
63 views

Evaluation of an integral using Mathematica

I am trying to evaluate the following integral, but the result I got is imaginary part. Do you know if there is a way to get a better evaluation of difficult integrals? ...
0
votes
0answers
67 views

How to write code for replacing result in integral

I have about 100 integral term, the general form of them is as follows: $$\int \int dx~dy~N_\mu(x)~N_\nu(x)~F(x,y)~A_\nu(y)$$ For each $\mu$ and $\nu$, the definition of functions are different. I ...
1
vote
1answer
227 views

Solving Integro-differential equation numerically with shooting method

This question is related a question I previously asked here Solving integro-differential equation with boundary condition at infinity and for which a solution was found . Now I am dealing with a ...
0
votes
1answer
38 views

fisrt 5th iterations of a sequence of functions: integro-differential equations [closed]

Please I am struggling to compute with mathematica the first 5th iterations of the sequence of functions $u[n,t]$ satisfying the integro-differential equation Please help me with the code. Patrick
2
votes
2answers
180 views

How to Integrate the Derivative of a function with Assumptions on the function?

I have a monotonic increasing function g and constant αsuch that ...
1
vote
0answers
116 views

I need to solve integro–differential equation with derivative

There are 2 equations a1 and a2, the difference between them is the red words part(integral term). Boundary x(0)=0,x'(0)=0 x'(t-u) means Bring t-u into the derivative of x ,for example, if x(t)=t^3, ...
3
votes
2answers
290 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 ...
2
votes
1answer
159 views

How to solve the intergo-differential equation with derivative [duplicate]

I do not know how to handle the integro-term. ...
1
vote
1answer
81 views

How to plot a 2 variable function of an integral?

I have this function: $$E_p(x_0,y_0,t)=E_0e^{-i\omega t}\int^{\frac{a}2}_{\frac{-a}2}\int^{\frac{b}2}_{\frac{-b}2}\frac{e^{ik\sqrt{(x-x_0)^2+(y-y_0)^2+z^2}}}{\sqrt{(x-x_0)^2+(y-y_0)^2+z^2}}dxdy$$ ...
10
votes
2answers
435 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 ...
0
votes
1answer
120 views

Can I make Wolfram Alpha use a specific substitution when integrating? [closed]

Can I define the '$u = \dots$' for an integral? For example when I integrate $\frac{x^3}{\sqrt{6 - x^2}}$ with respect to $x$, the program automatically sets $u = x^2$, how can I change this so it ...
1
vote
3answers
293 views

NDSolve Integro-differential equation

I have hard time to solve numerically the following integro-differential equation: ...
2
votes
1answer
172 views

How do I solve a kind of integral equation with Mathematica?

I would like to prove that only the functions of the form $f(R)\propto 1/R^2$ satisfy the following integral equation (assume $r>r_0>0$): $$ \int\limits_{r-r_0}^{r+r_0}\left({r^2\over{2r_0^2}}-{...
1
vote
3answers
161 views

integral equation iteration

So I am trying to recreate the numerical treatment in this papir (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.29.130), equation 3.7. The main point is to try to solve the following equation ...
2
votes
2answers
314 views

How to solve this differential equation with integral?

How to solve this differential equation with integral? $\omega = 1.2;$ $i1 = 25640;$ $ia = 4*10^6;$ $c2 = 1.4513*10^5;$ $t5 = 16000*\cos({\omega*t});$ $tpto = 2.8*10^6*{\theta'}(t);$ $cr(\omega)...