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 ...
- 60.8k
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 ...
- 19.6k
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:
...
- 22.9k
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 \!\!\!\!...
- 104k
11
votes
Accepted
Solving integral equation
With some effort you can find numerical solution. Let us discretize function f at num points, e.g. ...
11
votes
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. ...
- 42k
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 ...
- 42k
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 ...
- 13.5k
10
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. (...
- 63.8k
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 ...
- 56.8k
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 ...
- 63.8k
9
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:...
- 234k
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 ...
- 46.2k
9
votes
8
votes
Accepted
Solving PDE involving Hilbert transform numerically
As I pointed out in a comment above, this problem can be solved by performing a Fourier Transform in x, solving the resulting ODE, and transforming back. The ...
- 60.8k
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 ...
- 234k
8
votes
8
votes
Accepted
Numerical solution of Fredholm Equation
Perhaps NestList gives an iterated solution (Picard iteration)
...
- 49.6k
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))...
- 17.8k
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 ...
- 17.8k
7
votes
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 ...
- 130k
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...
- 8,724
7
votes
Numerically solving a multivariable integro-differential equation
We need a functional to evaluate the right-hand-side from a given function $f$:
...
- 46.2k
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:
...
- 42k
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:
...
- 63.8k
7
votes
Accepted
7
votes
How to find the area locked between two curves and a line
Thanks @Nasser work out all the formulas.
...
- 63.9k
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 ...
- 42k
Only top scored, non community-wiki answers of a minimum length are eligible