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To continue the questions:

https://math.stackexchange.com/questions/3743374/approximation-of-the-convolution-operator?noredirect=1#comment7697800_3743374

https://math.stackexchange.com/questions/3744687/convolution-using-integral-transforms?noredirect=1#comment7702418_3744687

Convolution using the Laplace integral transform of certain functions

Approximate solution of a nonlinear ODE in the form of a Fourier series containing the coefficients of the initial ODE

https://math.stackexchange.com/questions/3745783/find-an-approximate-formula-of-one-of-the-components-of-the-solution-of-the-diff

I have a differential equation, for example:

$\frac{dx}{dt}= \mu \cdot e^{-x^2} \cdot a_1 \cdot \sin(\omega_1 \cdot t) + a_1 \cdot \cos(\omega_1 \cdot t)$

{\[Alpha] = 0.1, \[Omega] = 2 Pi 0.1, \[Mu] = 5

His numerical solution is as follows:

enter image description here

The figure shows that one of the components of the solution can be:

$x(t) ≈ e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t}$

enter image description here

If we represent the solution of the equation in the form:

$x(t) = e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t} + \Delta(t)$

And substitute in the original equation, we get:

$\frac{e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t} + \Delta(t)}{dt}= \mu \cdot e^{-(e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t} + \Delta(t))^2} \cdot a_1 \cdot \sin(\omega_1 \cdot t) + a_1 \cdot \cos(\omega_1 \cdot t)$

Is it possible to somehow analytically and very approximately get a solution for $\Delta(t)$

EDIT:

I thought about this for a long time and came to the following assumption.

If we take the integral from both sides and transfer the $- e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t}$ to the right side, then we can get the following equation:

$\Delta(t) - $$\int_{}^{} \mu \cdot e^{-(e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t} + \Delta(t))^2} \cdot a_1 \cdot \sin(\omega_1 \cdot t) + a_1 \cdot \cos(\omega_1 \cdot t) dt = - e^{-\mu \cdot \frac{\alpha^2}{2} \cdot t}$

But is it possible to solve this equation, like the Fredholm equation, and find $\Delta(t)$?

I will be glad to be advised and helped.

dtn
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    This is Volterra equation. It can be solve numerically. – Alex Trounev Jul 05 '20 at 11:19
  • Take your time to close the question. Numerically solved, I agree. An approximate analytical solution is needed. – dtn Jul 05 '20 at 11:20
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    It is better to use differential equation for approximate solution. Are any restrictions for parameters $\mu , a_1, \omega_1 $? – Alex Trounev Jul 05 '20 at 11:59
  • There are only one restrictions on the parameters - they must be greater than zero. However, the system may become unstable. At what values is unknown. For simplicity, let's limit ourselves to a range of parameters like this: $\mu = [0,10]$,$\alpha_1 = [0,3]$,$\omega_1 = [0,5]$ – dtn Jul 05 '20 at 12:03

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