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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1answer
38 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
65 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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0answers
25 views

Nonlinear integral equation — numerical solution

I am trying to solve numerically the following nonlinear integral equation: $\qquad \int_{0}^{x}\left( \frac{y}{x}\right)^{f(y)-1}g(y)\,dy=h(x)$, where $f(y)$ is the unknown function, while $g(y)$ ...
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2answers
161 views

solve the iterative integral equatio [duplicate]

The following code is incorrect, please specify my mistakes ...
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1answer
51 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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2answers
44 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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2answers
99 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 ...
4
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1answer
226 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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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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1answer
175 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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1answer
76 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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4answers
442 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
39 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
152 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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1answer
53 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
315 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 ...
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1answer
78 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
51 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( ...
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3answers
189 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 ...
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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
61 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
37 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
87 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
26 views

Using `Integrate` and or `FourierTransform`

I have some general questions about Integrate and FourierTransform. Firstly with ...
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1answer
45 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
57 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
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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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2answers
215 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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3answers
512 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. ...
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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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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
38 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
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1answer
146 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)\...
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2answers
215 views

Solution or artifact?

I am trying to increase the precision of the code ...
1
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1answer
71 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
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2answers
668 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:...
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5answers
272 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 ...
1
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1answer
138 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) \, \...
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0answers
52 views

Strange Integral Calculation With Traditional Form

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

Get Integral Result without Assumptions

I have the following two functions: ...
2
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1answer
115 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: ...
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2answers
149 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\...
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1answer
87 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): ...
3
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1answer
279 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
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1answer
65 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? ...
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0answers
70 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
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1answer
236 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
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1answer
40 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
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2answers
182 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 ...