15 votes
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How to solve the differential equation with Duhamel's integral?

NDSolve currently can't handle this kind of differential equation, LaplaceTransform is your friend. Since in this case inverse ...
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15 votes
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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 ...
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13 votes
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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: ...
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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 \!\!\!\!...
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12 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 ...
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11 votes
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Solving integral equation

With some effort you can find numerical solution. Let us discretize function f at num points, e.g. ...
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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. ...
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10 votes
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PDE of real-world system, integral boundary condition

Because NDSolve cannot accommodate the x=0 boundary condition, it is necessary to perform this computation by discretizing the ...
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10 votes

How to solve the differential equation with Duhamel's integral?

One can use Picard-type iteration to get the solution: Using an approximation to x'[t] (in the integral), we can integrate the ODE to obtain a new approximation. ...
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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 ...
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10 votes

Solving Fredholm Equation of the second kind

Use DSolve: ...
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9 votes
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Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

There exist few typos in your kernel definition. This is how your kernel looks assuming a=2 (we denote it as A while defining ...
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9 votes
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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. (...
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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 ...
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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:...
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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 ...
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8 votes
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How to solve this Integral equation

Large t Approximation Because this is a neutral delay integral-differential equation, solving it may seem very difficult at first glance. However, the term 1/5^t ...
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8 votes
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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 ...
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8 votes
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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 ...
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8 votes
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Numerical solution of Fredholm Equation

Perhaps NestList gives an iterated solution (Picard iteration) ...
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8 votes
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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))...
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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 ...
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7 votes

Solve a differential equation with an integral inside?

Method 1, using Laplace transform ...
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7 votes

How to solve a system of integral equations

...
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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 ...
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7 votes
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How do I solve an integral equation related to the Newtonian 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 ...
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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...
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7 votes

Numerically solving a multivariable integro-differential equation

We need a functional to evaluate the right-hand-side from a given function $f$: ...
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  • 36.2k
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: ...
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7 votes
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How to increase the speed of Collocation method to solve System of differential equations

First of all, the NDSolve solution can be further improved: ...
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