Green s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall)

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1 Green s functions 3. G(t, τ) nd its derivtives G (k) t (t, τ), (k =,..., n 2) re continuous in the squre t, τ t with respect to both vribles, George Green (4 July My 84) In 828 Green privtely published "An Essy on the Appliction of Mthemticl Anlysis to the Theories of Electricity nd Mgnetism". The essy introduced severl importnt concepts, mong them theorem similr to modern Green's theorem, the ide of potentil functions s currently used in physics, nd the concept of wht re now clled Green's functions. It gined him dmittnce to Cmbridge s n undergrdute in 833. He grduted in 837 nd ws elected to fellowship in 839, two yers before his deth. The fellowship ws for bchelors; Green qulified becuse he hd never formlly mrried the mother of his six children. No portrit of Green ws ever mde For < τ < t G (n ) t (τ +, τ) G (n ) t (τ, τ) = p n (τ) [ jump of (n-)-th derivtive ] 4 Liner differentil eqution of integer order n: Consider the simplest exmple: n p k (t)y (k) (t) = f(t), t [, b] k= Structure of solution: y(t) = y H (t) + y f (t) b y f (t) = G(t, τ)f(τ)dτ { y (t) = f(t), t [, T ] y() = The solution is: y(t) = t G(t, τ) = f(τ)dτ = T G(t, τ)f(τ)dτ, {, τ t t < τ T t = H(t τ) τ Heviside function 2 5 Definition of G(t, τ) :. Prtil derivtives G (k) t (t, τ), (k =,..., n) exist nd re continuous with respect to both vribles in tringles t τ b nd τ t b. 2. As function of t, G(t, τ) stisfies eqution n p k (t)y (k) (t) = k= 3 The LT cn be used to obtin : { y (t) = f(t), t [, T ] y() = Consider the sme eqution with delt function in RHS: G (t) = δ(t) s g(s) = y(t) = g(s) = s T G(t τ)f(τ)dτ {, t G(t) = H(t) =, t < Heviside 6

2 Frctionl Frctionl Initil vlue problem for liner FODE Min property () Indeed, consider where 7 Frctionl Frctionl Definition Min property (2) Leopold Kronecker (823-89) nd similrly up to Kronecker believed tht mthemtics should del only with finite numbers nd with finite number of opertions. 8 Frctionl Frctionl Min property Min property (n) But: Solution of the problem is given by y(t) =!t G(t, τ )f (τ )dτ, Therefore, 9 2

3 Frctionl Frctionl Constructing solutions of equtions with zero RHS Constructing solutions of liner equtions with constnt coefficients Motivtion: consider n integer order exmple: G!! (t) + 2 G(t) = δ(t) g(s) = g(s)(s2 + 2 ) = G() (t) G(t) = sin t s2 + 2 G() (t) ψ () = sin t ψ2 () = ψ! () = ψ2! () = ψ (t) = cos t ψ2 (t) = 3 Frctionl 6 Frctionl Constructing solutions of equtions with zero RHS One-term eqution Motivtion: consider n integer order exmple: y!! (t) + 2 y(t) = f (t) y() = b g (s) = y () = b2! Solutions: y(t) = b ψ (t) + b2 ψ2 (t) +!t = b G! (t) + b2 G(t) + sα G(t τ )f (τ )dτ!t G(t τ )f (τ )dτ 4 Frctionl 7 Frctionl Constructing solutions of equtions with zero RHS Two-term eqution Consider the cse of constnt coefficients. Then Tke, g2 (s) = Then = sα + b sα + b due to zero initil conditions (definition of FGF) Also, 5 8

4 Frctionl Frctionl Three-term eqution g3 (s) = = Generl cse: n-term eqution cs α = α sβ + bsα + c c sβ α + b + scs β α +b " c #k+! s αk α ( )k $ %k+ c β α s + b k= multinomil coefficients 9 22 Frctionl Four-term eqution Tylor's expnsion: Brook Tylor (685-73) Methodus incrementorum direct et invers (75) Also: invented integrtion by prts clculus of finite differences chnge of vribles formul wy of relting the derivtive of function to the derivtive of the inverse function In 72 Tylor ws ppointed to the committee set up to djudicte on whether the clim of Newton or of Leibniz to hve invented the clculus ws correct. 2 Frctionl One term eqution: zero initil condition Four-term eqution Assume tht the RHS cn be expnded in Tylor series tht converges for t R : Knowing solution in the form: 2, we cn look for the

5 One term eqution: zero initil condition One term eqution: non-zero initil condition Zero initil conditions re stisfied; consider the eqution: The solution exists only if Comprison of the coefficients gives: Suppose (coeffs. re known) nd look for the solution in the form Therefore, 28 One term eqution: zero initil condition In the considered simple cse we hve: One term eqution: non-zero initil condition Consider the eqution: Comprison of the coefficients gives: 29 One term eqution: wek singulrity in the RHS One term eqution: initil condition in terms of R-L integrl Suppose nd α + β > Look for the solution in the form: Suppose tht the RHS cn be expnded in Tylor series: Then the coefficients in the solution re: We cn look for the solution in the form: 3

6 Eqution with non-constnt coefficients One term eqution: initil condition in terms of R-L integrl Consider the eqution: Consider the following problem: For some prticulr types of f (t) solution cn be obtined. Suppose Comprison of the coefficients gives: Then the solution cn hve the form: Still hve to determine 3 Eqution with non-constnt coefficients One term eqution: initil condition in terms of R-L integrl To determine 34 Initil condition is stisfied by the chosen form of solution., the initil condition must be used. The the comprison of the coefficients gives: Tking the limit s t we obtin: recurrence 32 Eqution with non-constnt coefficients: prticulr cse Exmine the initil conditions nd the RHS Initil conditions nd the RHS of the Tke, for exmple, eqution determine the clss of solutions nd dictte the form of the series 35 Then The key formul is 33 36

7 Eqution with non-constnt coefficients: even more prticulr cse If we tke then 37 Two-term nonliner eqution Consider the following problem: The solution cn hve the form: (becuse nd both give the series of the sme form) 38 Two-term nonliner eqution Initil condition is stisfied; use the eqution: nd we obtin the recurrence reltionships: 39

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