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İ am just stuck on this step : If we have the following volterra fractional integral equation : $$ x(t)=x_0+\frac 1 {\Gamma(\alpha)} \int_0^t ({t}-\tau)^{\alpha-1} f(\tau,x(\tau)) \, d\tau, \qquad t\in\left[0,{T}\right] $$

And then let : $$ \tau=t-(t^\alpha-\upsilon\Gamma(\alpha+1))^{1/\alpha} $$

So Volterra fractional integral equation can be written as : $$ x(t) =x_0 + \int_0^{t^\alpha/\Gamma(\alpha+1)} f(t-(t^\alpha-\upsilon\Gamma(\alpha+1))^{1/\alpha},x(t-(t^\alpha-\upsilon\Gamma(\alpha+1))^{1/\alpha})) \, d\upsilon $$ Is there any explanation how did we write it in a the last form Thanks a lot in advance .

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This is a simple substitution or re-parametrization of the integration variable.

Set $$ \tau=g(υ)=t-(t^\alpha-\upsilon\Gamma(\alpha+1))^{1/\alpha} $$ then you get $τ=0$ for $υ=0$ and $τ=t$ for $t^α−υΓ(α+1)=0$ or $ υ=\frac{t^α}{Γ(α+1)}$. Furthermore, $dτ=g'(υ)\,dυ$ with $$ g'(υ)=\frac1αΓ(α+1)(t^α−υΓ(α+1))^{\frac1α-1}. $$ Note that $Γ(α+1)=αΓ(α)$ by definition of the Gamma function.

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