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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0answers
84 views

How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
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1answer
52 views
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0answers
86 views

Complicated Integral output with Unfamiliar Regularized Hypergeometric Function

I need the solution for following integral and it has output in MATHEMATICA as: ...
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0answers
26 views

Help with NIntegrate with singularity and variable integration limit

I have a function of the type: $$ F(x,z)= \int_{-\infty}^{\infty} dx' \int^{z}_{-\infty} dz' f(z,z',x,x')\frac{\partial g(z',x')}{\partial z'}$$ where the integration order may be interchanged. I ...
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1answer
57 views

Definite Integral [closed]

I am not able to do the following integration. \begin{equation} \int_{0}^{a} x\sin^2 \left(\frac{n \pi x}{a} \right)dx \end{equation} It shows an error that "more input is needed". The mathematica ...
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2answers
48 views

Can't solve for y in a fairly simple work done by magnetic field equation

The equations below are essentially calculations for a railgun-type mechanism (here is a good image description). I also drew an image using my own variables as definitions: I assumed that ...
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0answers
53 views

Help with unsolved indefinite integral ! $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ [migrated]

I really need to solve this indefinite integral: $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ It seems much more complicated than it looks. I have found a integral table ...
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2answers
99 views

Trouble second-order ordinary differential equations with integral terms

I am a beginner of mma. During the learning of differential equations, I encountered a very difficult problem, a second-order system of ordinary differential equations with integral terms. This ...
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2answers
82 views

Problem with FindRoot function

I have set up a code to solve simultaneous integral equations. These result essentially from taking integrals of the Fermi-Dirac distribution. Although there are no obvious problems with the code it ...
1
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1answer
53 views

Regression to fit data with an integral equation

I have a set data and I know that this data are best fitted by the following equation, assuming that f($\tau$) is a log normal distribution and $M_{eq}$ = 1.5: $M(t,\tau)= M_{eq} \int (1-e^{-t/\tau}) ...
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1answer
91 views

Solving IDE system using FDM based on NDSolve`FiniteDifferenceDerivative

I wanted to solve the following system of integro-differential equations (IDEs) using finite difference method (FDM). $y_1''(t)+t^2y_1(t)-y_2''(t)+\int\limits_0^t[(t-x)y_1(x)+y_2(x)]\mathrm{d}x=(2+t^...
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2answers
175 views

solve the iterative integral equatio [duplicate]

The following code is incorrect, please specify my mistakes ...
0
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1answer
58 views

Solve integral equation by iteration

For[{n = 1, y[0][x_] = 1}, n < 4, n++, y[n][x_] = 1 + Integrate[y[n - 1][t]^2 + t^2, {t, 0, x}]; Print[{n, y[n][t]}]] here how can I get table of values ...
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1answer
180 views

How to solve this differential equation analytically?

How can I DSolve the following differential equation? Is there an analytical answer to this equation? ...
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2answers
47 views

problem with constant variables

Variable x is supposed to be constant, but the program calculates for each variable x, 100 numbers of x. I want all my x variables to have the same definition, but the program constantly changes each ...
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5answers
279 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 ...
3
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2answers
218 views

Solution or artifact?

I am trying to increase the precision of the code ...
4
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1answer
237 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, ...
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1answer
161 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{\...
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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. ...
2
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2answers
103 views

Unexpected limit result (was: How to fix the limit problem?)

MATHEMATICA fails in proving that Limit[ Integrate[ x^(x - 1) , {x, t, 2*t} ] , t -> 0 , Direction -> "FromAbove" ] == Log[2] How to fix ...
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0answers
95 views

How to integrate this expression one time? [duplicate]

How can I integrate this expression? I want to have the second derivative of V i.e. V''[r]. Note that, you can multiply it be ...
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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] ...
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1answer
84 views

How can I create loop for this iteration integration [closed]

I have definite integral of the form $$f(x)=\int_{-\pi}^\pi\frac{x^2-1}{(x^2-1)^2+x^2+y^2+f(x)}dxdy$$ I want to make a code to calculate $f(x)$ numerically, starting from $f(x)=0$ and stopping when ...
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1answer
79 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}] ...
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1answer
40 views

Convolve doesn't yeild expected answer, nor does integrate

I am convolving, or trying to, a Gaussian and a Lorentzian as follows: ...
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1answer
54 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
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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 ...
1
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1answer
82 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 ...
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1answer
55 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
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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 ...
0
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1answer
36 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
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1answer
62 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 ...
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0answers
38 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>...
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1answer
93 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 ...
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0answers
27 views

Using `Integrate` and or `FourierTransform`

I have some general questions about Integrate and FourierTransform. Firstly with ...
0
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1answer
47 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 ...
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0answers
61 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 ...
6
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2answers
229 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}...
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0answers
96 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 ...
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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 ...
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3answers
352 views

NDSolve Integro-differential equation

I have hard time to solve numerically the following integro-differential equation: ...
0
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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 ...
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0answers
47 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 ...
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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 <...
8
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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 ...
2
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1answer
523 views

NDSolve: Couple PDE and ODE involving integral

I am trying to numerically solve an equation with NDSolve, where there is a ODE coupled to a PDE, like the following: ...
3
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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)\...