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16 votes
Accepted

Integrating over $x$ in numerically solving a partial integrodifferential equation

NDSolve is not capable of solving this sort of problem as a PDE. Thus, it is necessary to perform the computation by discretizing the PDE in ...
bbgodfrey's user avatar
  • 61.6k
13 votes

Solving the Lotka-McKendrick model with NDSolve

I'm not an expert on age-structured populations (particularly this continuous-time model) and I know better numerical methods exist, but why not just discretize in age ...
Chris K's user avatar
  • 20.3k
13 votes
Accepted

Why can't Mathematica solve this definite integral?

Likely this is due to the fact that antiderivative are troublesome in computer algebra systems. Do this: ...
march's user avatar
  • 23.8k
12 votes
Accepted

Solving integral equation

With some effort you can find numerical solution. Let us discretize function f at num points, e.g. ...
Andrzej Odrzywolek's user avatar
12 votes

Fredholm Integral Equation of the 2nd Kind with a Singular Difference Kernel

This is an eigenvalue problem. Let's apply a Galerkin scheme: We fix a finite dimensional space of functions, pick a basis $u_0, u_2,\dotsc,u_{n}$ and define the matrices $$A_{ij} = \int_0^1 \!\!\!\!...
Henrik Schumacher's user avatar
11 votes
Accepted

Numerically solve an integro-differential equation

This integro-differential equation can be solved with the method mentioned in this answer i.e. differentiate the equation to make it a pure ODE. First, interprete the equations to Mathematica code. (...
xzczd's user avatar
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11 votes

Solving Fredholm Equation of the second kind

Use DSolve: ...
Ulrich Neumann's user avatar
11 votes

Solving partial differential equation involving Hilbert transform

I used the method of solving integro-differential equations proposed by Michael E2 on Solving an integro-differential equation with Mathematica I added new options to his code to solve this problem. ...
Alex Trounev's user avatar
  • 45.4k
11 votes

Numerically solving a non-linear integro-differential equation

This equation can be solved numerically using method described in our paper. First note, that integral $\int_{0}^{\tau} \mathrm{d} s \, \big[ y(\tau -s) \big]^{2} y(s) $ can be mapped on (-1,1) by ...
Alex Trounev's user avatar
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10 votes

NIntegrate into NDSolve with variable integrand

Numerical solution: solution = NDSolve[{D[y[x], x] == x + f0[x], y[0] == 1, f0'[x] == y[x], f0[1] == 0}, y[x], {x, 0, 1}]; Symbolic solution from @rewi (Works ...
Mariusz Iwaniuk's user avatar
10 votes

Solving an integro-differential equation with Mathematica

Based on the hacky way I used in my answer here; I had to split up the NDSolve process, so as not to redefine MapThread too soon:...
Michael E2's user avatar
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9 votes
Accepted

Vanishing of gravitational force inside hollow spheres imply the Newtonian law of gravity

Solutions to integral equations are equivalence classes of functions, i.e. two functions are in the same class if they are different on a (Lebesgue) measure zero subsetset of their domains. Having ...
Artes's user avatar
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9 votes

Fredholm Integral Equation of the 2nd Kind with a Singular Difference Kernel

As mentioned by Henrik, this is an eigenvalue problem. Since Mathematica doesn't have a built-in eigenvalue problem solver for integral equation, we need to discretize the equation to matrix form by ...
xzczd's user avatar
  • 66.8k
9 votes

Find area between three curves

To get an overview: ContourPlot[{y == 6/x, y == x + 4, y == x - 4}, {x, -10, 10}, {y, -10, 10}] Define it as a region through logical combinations, the signs of ...
Roman's user avatar
  • 48.1k
9 votes

How to find the area locked between two curves and a line

...
Nasser's user avatar
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8 votes
Accepted

Numerically solving an ODE whose right-hand side involves an integral

Being a mathematician, I resist fudging by cutting off the singularity by some small eps = 10^-12. But if you're an engineer, you should be satisfied @Nasser's ...
Michael E2's user avatar
  • 237k
8 votes

NDSolve Integro-differential equation

Seems, an analytical solution is possible. ...
Akku14's user avatar
  • 17.3k
8 votes
Accepted

Numerical solution of Fredholm Equation

Perhaps NestList gives an iterated solution (Picard iteration) ...
Ulrich Neumann's user avatar
8 votes
Accepted

Inverting integral transform $f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x$

In the math.stackexchange post I have shown that $$\left\{\theta(y-a)-\theta(y-b)\right\} \left(g^{-1}\right)^\prime(y)\, y=\mathcal{L}^{-1}[f](y)$$ where $g^{-1}$ is the inverse function $g^{-1}(g(x))...
yarchik's user avatar
  • 18.7k
8 votes

Why is Mathematica unable to solve this integral equation?

Mathematica can solve your equation, but it requires some understanding of mathematics and human intervention. This is always the case for nontrivial problems, isn't it? Additive solution was ...
yarchik's user avatar
  • 18.7k
8 votes

Shape of a boomerang

I do not clearly understand your question. However, if you want to plot the shape of a boomerang use the following code. ...
user444's user avatar
  • 2,456
7 votes

Numerically solve an integro-differential equation

If you can convert the integro-differential equation into an IVP or BVP ODE, then that would be the best approach. If you can't do so, then you can try the following iterative approach. The basic idea ...
Carl Woll's user avatar
  • 131k
7 votes

How to solve a system of integral equations

...
Dr. belisarius's user avatar
7 votes

Functional fixed points (ie fixed point of mapping from function space C[0,1] to itself)

Just as an addition to @Okkes and @Ulrich's answer following the idea lined out by @LukasLang, we can also start with a symbolic solution of the integral for every p...
Thies Heidecke's user avatar
7 votes

Numerically solving a multivariable integro-differential equation

We need a functional to evaluate the right-hand-side from a given function $f$: ...
Roman's user avatar
  • 48.1k
7 votes

Solving the Lotka-McKendrick model with NDSolve

There is no unique solution for data provided by @Pillsy, since boundary and initial conditions are inconsistent. To show it we just use exact solution in a form: ...
Alex Trounev's user avatar
  • 45.4k
7 votes
Accepted

How to increase the speed of Collocation method to solve System of differential equations

First of all, the NDSolve solution can be further improved: ...
xzczd's user avatar
  • 66.8k
7 votes
Accepted

Multiple Integration

Try this 1D integration ...
yarchik's user avatar
  • 18.7k
7 votes

How to find the area locked between two curves and a line

Thanks @Nasser work out all the formulas. ...
cvgmt's user avatar
  • 75.1k
7 votes

Galactic rotation speed

We can solve this problem using colocation method with Euler wavelets ang Gauss quadrature rule. First we define interval $0\le R\le Rmax$ and map interval $(0,R)$, on ...
Alex Trounev's user avatar
  • 45.4k

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