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## Hot answers tagged integral-equations

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 ...
• 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 ...
• 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: ...
• 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. ...
12 votes

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 \!\!\!\!... 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. (... • 66.8k 11 votes ### Solving Fredholm Equation of the second kind Use DSolve: ... • 54.2k 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. ... • 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 ... • 45.4k 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 ... 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:... • 237k 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 ... • 57.5k 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 ... • 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 ... • 48.1k 9 votes ### How to find the area locked between two curves and a line ... • 145k 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 ... • 237k 8 votes ### NDSolve Integro-differential equation Seems, an analytical solution is possible. ... • 17.3k 8 votes Accepted ### Numerical solution of Fredholm Equation Perhaps NestList gives an iterated solution (Picard iteration) ... • 54.2k 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 $g^{-1}(g(x))... • 18.7k 8 votes ### Why is Mathematica unable to solve this integral equation? Mathematica can solve your equation, but it requires some understanding of mathematics and human intervention. This is always the case for nontrivial problems, isn't it? Additive solution was ... • 18.7k 8 votes ### Shape of a boomerang I do not clearly understand your question. However, if you want to plot the shape of a boomerang use the following code. ... • 2,456 7 votes ### Numerically solve an integro-differential equation If you can convert the integro-differential equation into an IVP or BVP ODE, then that would be the best approach. If you can't do so, then you can try the following iterative approach. The basic idea ... • 131k 7 votes ### How to solve a system of integral equations ... • 116k 7 votes ### Functional fixed points (ie fixed point of mapping from function space C[0,1] to itself) Just as an addition to @Okkes and @Ulrich's answer following the idea lined out by @LukasLang, we can also start with a symbolic solution of the integral for every p... • 8,834 7 votes ### Numerically solving a multivariable integro-differential equation We need a functional to evaluate the right-hand-side from a given function$f$: ... • 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: ... • 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: ... • 66.8k 7 votes Accepted ### Multiple Integration Try this 1D integration ... • 18.7k 7 votes ### How to find the area locked between two curves and a line Thanks @Nasser work out all the formulas. ... • 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 ...
• 45.4k

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