Can someone kindly revise these few lines

[sarrah1]

I have this Volterra integral equation

$X(t)=e^{At}+\int_{0}^{t} \,e^{A(t-\tau)}\epsilon BX(\tau) d\tau$ (1) ,
of matrix solution $X(t)$ and $X(0)=I$, $A,B\in{R}^{n\times n}$ , $t\in[a,b]$

By iteration:
Result 1 : the solution $X(t)$ of (1) can be written as a power series in $\epsilon$ in the form
$X(t)=e^{At}+\sum_{1}^{\infty}{\epsilon}^{n} {P}_{n}(t)$ (2)
where ${P}_{n}(t)$ satisfies the recurrence
${P}_{n}(t)= \int_{0}^{t} \,e^{A(t-\tau)}B {P}_{n-1}(\tau) d\tau$ , ${P}_{0}(t)={e}^{At}$ (3)
proof is by substituting (2) into (1) and equating equal powers of $\epsilon$

Result 2 : The series (2) is uniformly convergent in $[a,b]$ for sufficiently small $\epsilon$
Proof: $||{P}_{1}(t)||\le \int_{0}^{t} \,e^{||A||(t-\tau)}||B|| {e}^{||A||\tau} d\tau$
by Bonnet 2nd mean value theorem $=||B||{e}^{||A||t}\int_{\zeta}^{t} \,{e}^{||A||(t-\tau)}d\tau$
$\le||B||{e}^{||A||t}\int_{0}^{t} \,{e}^{||A||(t-\tau)}d\tau\le||B||{e}^{||A||t}t{e}^{||A||t}$
similarly $||{P}_{2}(t)||\le{e}^{||A||t}{(t||B||{e}^{||A||t})}^{2}$ etc...
Thus the series in (2) is absolutely convergent if $\epsilon t||B||{e}^{||A||t}<1$. According to Wieirstrass M-test, the series $\sum_{1}^{\infty}{\epsilon}^{n} {P}_{n}(t)$ converges uniformly on $[a,b]$

very grateful
Sarrah

NB: the first result is trivial, it's the 2nd result which i care for, i.e. the uniform convergence of the series.
thanks

 

[jvicens]

Dear Sarrah,

Thank you for sharing your Volterra integral equation with us. It is always exciting to see new equations and to explore their properties. I would like to offer some thoughts and comments on your results.

Firstly, I would like to clarify that the equation you have provided is a Volterra integral equation of the second kind, as it involves the unknown function $X(t)$ in the integral. This is in contrast to a Volterra integral equation of the first kind, where the unknown function appears in the limits of integration. This distinction is important as it affects the properties and solutions of the equation.

Now, onto your results. Your first result, which states that the solution can be written as a power series, is indeed trivial as you mentioned. It follows directly from substituting the power series into the equation and equating coefficients. However, it is still an important result as it shows that the solution can be approximated by a series, which can be useful for numerical methods.

Your second result, on the other hand, is quite interesting. The uniform convergence of the series is a crucial property and it is not always guaranteed for Volterra integral equations. In fact, there are many Volterra integral equations that do not have convergent solutions. Therefore, it is worth exploring the conditions under which the series in (2) converges, and your proof using the Weierstrass M-test is a good start.

However, I would like to point out that the condition you have derived, namely $\epsilon t||B||e^{||A||t}<1$, is not sufficient for the uniform convergence of the series. It is necessary but not sufficient. This can be seen by considering the simpler equation $X(t)=\int_{0}^{t} \,X(\tau) d\tau$, where $A=B=0$. In this case, the condition becomes $\epsilon t<1$, which is clearly not sufficient for the convergence of the series. Therefore, there must be additional conditions that need to be satisfied for the series to converge uniformly.

In conclusion, your results provide a good starting point for the analysis of this Volterra integral equation. I would suggest exploring different conditions and techniques to prove the convergence of the series. Perhaps, you can also investigate the properties of the solution and the behavior of the series for different values of $t$ and $\epsilon$. I wish you all the best in your research.[Your Name