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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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21 views

Solving an integral equation analytically

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. Is there any way for this using ...
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1answer
61 views

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

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
28 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
96 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
146 views

Solution or artifact?

I am trying to increase the precision of the code ...
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0answers
37 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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2answers
655 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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0answers
25 views

EDO with parameter, divergence and stop integration

I'm trying to solve an ODE which independent variable is time, where there is a parameter involved, let's say $\beta$. I need to plot the final solution in terms of $\beta$. The equations, initial ...
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1answer
107 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
47 views

Get Integral Result without Assumptions

I have the following two functions: ...
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1answer
103 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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0answers
89 views

Integrate under assumptions

I am trying to solve the following integral: $$\int_0^\infty u^{\sigma-1} \exp\left[-c u^\sigma\right] \mathrm{d}u$$ under the assumptions $\sigma \in (0,1)$ and $c>0$. I know the result is $$\...
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2answers
144 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
83 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
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1answer
224 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')}(...
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1answer
59 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
64 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 ...
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1answer
210 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 ...
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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
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2answers
174 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 ...
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0answers
100 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, ...
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2answers
271 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
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1answer
141 views

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

I do not know how to handle the integro-term. ...
7
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1answer
533 views

2d catenary in Mathematica

I'm working in Mathematica to plot a 2-d catenary in Euclidean space according to this Math Exchange Link, specified with a square Dirichlet boundary condition. Here's what I've coded using ...
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1answer
62 views

How to plot a 2 variable function of an integral?

I have this function: $$E_p(x_0,y_0,t)=E_0e^{-i\omega t}\int^{\frac{a}2}_{\frac{-a}2}\int^{\frac{b}2}_{\frac{-b}2}\frac{e^{ik\sqrt{(x-x_0)^2+(y-y_0)^2+z^2}}}{\sqrt{(x-x_0)^2+(y-y_0)^2+z^2}}dxdy$$ ...
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2answers
355 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
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1answer
75 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 ...
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3answers
221 views

NDSolve Integro-differential equation

I have hard time to solve numerically the following integro-differential equation: ...
2
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1answer
157 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}}-{...
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3answers
130 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
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2answers
282 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)...
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1answer
50 views

Solving linear equation with integrals

I have a simple equation with respect to constants A and B, unfourtunately it's written using definite integrals with unknown ...
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0answers
82 views

Nonlinear vibration

I am trying to solve this equation: with the following constants: alpha1=3.7 alpha2=3.84 Nnon=0.0009 Cnon=0 VDC=2 VAC=0.6 The solution should be as shown in the ...
5
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2answers
284 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{\...
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0answers
128 views

How can I solve a nonlinear fractional order integro-differential equation?

How do I solve the nonlinear fractional order integro-differential equation shown above in Mathematica?
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0answers
81 views

Multiple variable integration of a function of multiple variables

I'm having trouble writing code in Mathematica for this multi variable integration: $\int_0^1\int_0^1...\int_0^1F[x_1,x_2,...,x_n]dx_1dx_2...dx_n$ If I already know the number of elements of A it is ...
4
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2answers
228 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 ...
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0answers
79 views

Can Mathematica do a Double Path Integral?

I've obtained some analytical results that I'd like to verify numerically by doing a double path integral. I haven't done this before with Mathematica and am unsure if it's possible, I'd like to do ...
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0answers
112 views

Solving (numerically) integro-differential equation

I am absolutely new to Mathematica, and I installed it to solve a specific equation describing dopamine diffusion in cerebral tissue. I get a number of mistakes (boundaries for a start), but I was ...
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1answer
58 views

Finding a function which is defined by a function of it's integral

How can I find $f(x)$ in below relation? $f(x)= e^{I}$ $I = c_1 e^{-x^2/x0^2} - \int_0^\inf (\frac{c_2 (x_2 - x)}{(c_3^2+(x2-x)^2)^{\frac{5}{2}}}+ \frac{c_4(x2-x1)}{(c_3^2+(x2-x1)^2)^{\frac{7}{2}}})...
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0answers
101 views

Solving a cumulative distribution from a first order condition

I could use some guidance in using Mathematica's NDSolve. I'm attempting to numerically uncover a cumulative distribution function (CDF), $F(p)$, with density $f(p)$ from a first order condition, ...
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1answer
192 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 ...
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2answers
135 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
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0answers
44 views

Efficiently Plotting NIntegrate with Variable Integration Limit [duplicate]

Consider $g(x)$ which is mathematically defined as $g(x) = \int^{x}_{x_0} f(x') dx'$ such that $x\in\{x_0,x_{max}\}$. If $f(x)$ can only be integrated numerically, one could define the following ...
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1answer
90 views

PDE with integral

I am trying to solve PDE containing integral. Its form is showed below: D[Pa[t],t] == Integrate[-k[x]*Pa[t],{x,0,1}] k[x_] := 2*x with initial conditions: ...
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1answer
175 views

Triple integral with dependent parameters

The form of my problem is as follows: $\Psi=C\int_{-\infty}^\infty\int_{-\infty}^\infty\left(f(x,y)\times \left(\int_0^t g(x,t)dt\right)dxdy\right)$ I have already attempted numerical solutions ...
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2answers
239 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
237 views

How to handle this constant-interval integral in ODE?

The problem is to solve a system of nonlinear integro-differential equation with a Fredholm integral. $Q_i(x), i=1,2,3, x \in [-\text{max},\text{max}]$ \begin{align*} m_i{Q_i}''(x)-V_i(\vec{Q})-d_i\...