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I want to solve the non-linear Caputo-type fractional equation of the form ($0 < \alpha < 1$)

$$ ^cD^{\alpha}_0 f(t) = af(t)^4 + bf(t) + c$$

I have found, the equation $^cD^{\alpha}_0 f(t) = af(t) + g(t)$ is the Cauchy-type equation and is solved. But, setting $g(t) = af(t)^4 + c$ doesn't work.

Just for clarification, I want something like $f(t) =$ something in $t$. (Numerical solutions / Hints is also fine).Thanks in advance!

PS: If Riemann-Liouville or any other fractional derivatives will make it easier, that's also fine.

PS2: I am a newbie to fractional calculus, so I have no clue other than this book.

Ref:

Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204. Amsterdam: Elsevier (ISBN 0-444-51832-0/hbk). xv, 523 p. (2006). ZBL1092.45003.

gciriani
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Oleg Farenskiy
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    here is a similar question which is IMO a bit simpler than your equation. Please note that pursuing an analytic solution of even the simplest forms of fractional differential equations is usually a waste of time, because there is none. So your best bet is a numerical solution. To find a numerical solution of a FDE, it is often converted to its discrete-time equivalent. Among the discretization methods for FDEs, the Grunwald-Letnikov definition is the most famous one. – polfosol Jan 28 '23 at 14:45

1 Answers1

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I have successfully solved numerically a different fractional differential equation (FDE) by using the Matlab code to solve FDEs by Prof. R. Garrappa. At the link, see the answer with the plot for the code and paper. I hope this helps.

gciriani
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