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Picard's Method
Hello 
I am having some problems with this question:
When u satisfies the IVP u'' = -(x^2)(u(x)), u(0) =1, u'(0) = 0, show that it is a solution of the integral equation u(x) = 1 - INT(0 to x) [(x-s) s^2 u(s)] ds. I have done this.
Assuming that a solution exists, show that it is unique. This is where I get stuck. I start by saying there are 2 solutions, u(x) and v(x), and try to show e(x) = u(x) - v(x) = 0.
e(x) = INT(0 to x) [(x-s) s^2 (v(s) - u(s))] ds
I don't know how this would be/ tend to 0.
The question goes on to say: show that if {un} is the sequence of Picard approximations for the integral equation then the functions un are the partial sums of the power series sum(n=0 to infinity) an x^4n where a0=1, an+1 = -an/((4n+3)(4n+4)) n>=0.
I don't know what to attempt for this part.
Any insights would be really helpful, thanks.
I am having some problems with this question:
When u satisfies the IVP u'' = -(x^2)(u(x)), u(0) =1, u'(0) = 0, show that it is a solution of the integral equation u(x) = 1 - INT(0 to x) [(x-s) s^2 u(s)] ds. I have done this.
Assuming that a solution exists, show that it is unique. This is where I get stuck. I start by saying there are 2 solutions, u(x) and v(x), and try to show e(x) = u(x) - v(x) = 0.
e(x) = INT(0 to x) [(x-s) s^2 (v(s) - u(s))] ds
I don't know how this would be/ tend to 0.
The question goes on to say: show that if {un} is the sequence of Picard approximations for the integral equation then the functions un are the partial sums of the power series sum(n=0 to infinity) an x^4n where a0=1, an+1 = -an/((4n+3)(4n+4)) n>=0.
I don't know what to attempt for this part.
Any insights would be really helpful, thanks.
I think he used the warp drive to make the ship appear to be in two places at once then fired all phasers and photon torpedoes into the Klingon Warbird, thereby crippling it.
http://memory-alpha.org/wiki/Picard_Maneuver
http://memory-alpha.org/wiki/Picard_Maneuver
(edited 13 years ago)
Without knowing much (anything, really) about Picard's theorem, note that if you limit yourself to, say x <= 2, and define (i.e. the sup norm on [0,2]), then we have
.
As this is true for all x in [0,2] which we deduce and so . Therefore, e = 0 on the interval [0,2].
If you need it to be true for all x, I think you can push this argument to work: let S = {y: e(x) = 0 \for 0 \leq x \leq y}. We know S is nonempty (since 2 is in S); if S is unbounded we're done. So suppose S bounded, let m = sup S. Then for epsilon sufficiently small, we can apply the argument above on the interval (instead of [0,2]) to prove e(x) = 0 for 0 <= x <= m+epsilon and deduce that m cannot infact bound S after all, contradiction.
.
As this is true for all x in [0,2] which we deduce and so . Therefore, e = 0 on the interval [0,2].
If you need it to be true for all x, I think you can push this argument to work: let S = {y: e(x) = 0 \for 0 \leq x \leq y}. We know S is nonempty (since 2 is in S); if S is unbounded we're done. So suppose S bounded, let m = sup S. Then for epsilon sufficiently small, we can apply the argument above on the interval (instead of [0,2]) to prove e(x) = 0 for 0 <= x <= m+epsilon and deduce that m cannot infact bound S after all, contradiction.
(edited 13 years ago)
k05
Hello
I am having some problems with this question:
When u satisfies the IVP u'' = -(x^2)(u(x)), u(0) =1, u'(0) = 0, show that it is a solution of the integral equation u(x) = 1 - INT(0 to x) [(x-s) s^2 u(s)] ds. I have done this.
Assuming that a solution exists, show that it is unique. This is where I get stuck. I start by saying there are 2 solutions, u(x) and v(x), and try to show e(x) = u(x) - v(x) = 0.
e(x) = INT(0 to x) [(x-s) s^2 (v(s) - u(s))] ds
I don't know how this would be/ tend to 0.
The question goes on to say: show that if {un} is the sequence of Picard approximations for the integral equation then the functions un are the partial sums of the power series sum(n=0 to infinity) an x^4n where a0=1, an+1 = -an/((4n+3)(4n+4)) n>=0.
I don't know what to attempt for this part.
Any insights would be really helpful, thanks.
I am having some problems with this question:
When u satisfies the IVP u'' = -(x^2)(u(x)), u(0) =1, u'(0) = 0, show that it is a solution of the integral equation u(x) = 1 - INT(0 to x) [(x-s) s^2 u(s)] ds. I have done this.
Assuming that a solution exists, show that it is unique. This is where I get stuck. I start by saying there are 2 solutions, u(x) and v(x), and try to show e(x) = u(x) - v(x) = 0.
e(x) = INT(0 to x) [(x-s) s^2 (v(s) - u(s))] ds
I don't know how this would be/ tend to 0.
The question goes on to say: show that if {un} is the sequence of Picard approximations for the integral equation then the functions un are the partial sums of the power series sum(n=0 to infinity) an x^4n where a0=1, an+1 = -an/((4n+3)(4n+4)) n>=0.
I don't know what to attempt for this part.
Any insights would be really helpful, thanks.
The picard method? doesn't he do a short warp jump to make the ship appear in two places and fires everything on the target before they got chance to find the real target?
Make it so number one!
Thanks, that's really helpful. I have one question. Working through the same argument for m + epsilon, I get to ||eZ|| <= ((m+epsilon)^3 / 12) ||e||. If ((m+epsilon)^3 / 12) >= 1 then would ||e|| necessarily be 0?
Do you know how I would start working out the Picard approximations to get to the power series? Would it be un+1(x) = 1 - INT(0 to x)(x-s) s^2 un(s) ds, u0(x) =1
Do you know how I would start working out the Picard approximations to get to the power series? Would it be un+1(x) = 1 - INT(0 to x)(x-s) s^2 un(s) ds, u0(x) =1
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