Equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\varphi(x)=x+1-\int^{x}_0 \varphi(y)dy[/tex]

If I start from ##\varphi_0(x)=1## or ##\varphi_0(x)=x+1## I will get solution of this equation using Picard method in following way

[tex]\varphi_1(x)=x+1-\int^{x}_0 \varphi_0(y)dy[/tex]

[tex]\varphi_2(x)=x+1-\int^{x}_0 \varphi_1(y)dy[/tex]

[tex]\varphi_3(x)=x+1-\int^{x}_0 \varphi_2(y)dy[/tex]

...

Then solution is given by

[tex]\varphi(x)=\lim_{n \to \infty}\varphi_n(x)[/tex].

When I could say that this sequence will converge to solution of integral equation. How to see if there is some fixed point? I know how to use this method, but I am not sure from the form of equation, when I can use this method. Thanks for the answer.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Integral equations -- Picard method of succesive approximation

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**