Perturbation methods, Physics 2400 - Mathematical …

1 Physics 2400 Spring 2017perturbation methodslecture notes, spring semester 2017 rozman/Courses/P2400_17S/Last modified: April 25, 20171 IntroductionPerturbation theory is a collection of methods for obtaining approximate solutions toproblems involving a small parameter . These methods are very powerful, thus sometimesit is actually advisable to introduce a parameter temporarily into a difficult problemhaving no small parameter, and then finally to set = 1 to recover the original approach of Perturbation theory is to decompose a tough problem into an (infinite)number of relatively easy ones.

2 The Perturbation theory is most useful when the firstfew steps reveal the important features of the solution and the remaining ones give classify Perturbation solutions into two types. A basic feature ofregular perturbationproblems is that the exact solution for small but nonzero smoothly approaches theunperturbed solution as define asingular perturbationproblem as one whose solution for = 0 is fundamentallydifferent in character from the neighboring solutions obtained in the limit 1 of 18 Physics 2400 Perturbation methodsSpring 20172 Regular Perturbation An example of perturbative analysis: roots of a polynomialWe consider first an elementary example to introduce the ideas of regular perturbationtheory.

3 Let us find approximations to the roots of the following 16x+ 1 = 0.(1)For the reference, Eq. (1) has three real roots and two complex conjugate ones. Thenumerical values of the roots1arex1= ,x2= ,x3= , andx4,5= it stands, this problem is not a Perturbation problem because there is no small parameterin Eq. (1). In general it may not be easy to convert a particular problem into a tractableperturbation problem, but in the present case the trick is to replace the constant term inEq. (1) with a parameter:x5 16x+ = 0.(2)When = 1, the original Eq. (1) is consider the values of roots to be functions of.

4 We further assume a perturbationseries in powers of :x( ) = n=0an n.(3)To obtain the first term in this series, we set = 0 in Eq. (2) and factor it as followingx5 16x= 0 x(x2 4)(x2+ 4) = 0 x(x 2)(x+ 2)(x 2i)(x+ 2i) = 0.(4)Thus, in the zeroth-order Perturbation theory the equation s roots are:xm= 2,0, 2i, m= 1,..,5.(5)A second-order Perturbation approximation to the first of these roots consists of writingx1= 2 +a1 +a1 2,(6)1 The roots can be determined using a computer algebra system, with the followingMathematicacommand:NSolve[x 5-16x+1==0,x].Page 2 of 18 Physics 2400 Perturbation methodsSpring 2017substituting this expression into Eq.

5 (2), and neglecting powers of beyond is(1 + 64a1) + ( 80a1+ 64a2) 2= 0.(7)Since is a variable, we conclude that the coefficient of each power of in Eq. (7) areseparately equal to gives a sequence of equations for the expansion coefficientsai:1 + 64a1= 0, 80a21+ 64a2= 0,(8)with the solutionsa1= 164, a2=54a21=516384.(9)Therefore, the Perturbation expansion for the rootx1isx1= 2 164 +516384 2.(10)If we now set = 1, we obtainx1= accurate to better than 10 5. The sameprocedure givesx2= 2 164 516384 2= (11)x3=116 = (12)x4= 2i 164 5i16384 2= .015625 (13)x5= 2i 164 +5i16384 2= .015625 + (14)This example illustrates the three steps of perturbative the original problem into a Perturbation problem by introducing the smallparameter.

6 An expression for the answer in the form of a Perturbation series and computethe coefficients of that the answer to the original problem by summing the Perturbation series forthe appropriate value of .2 Computer algebra systems are perfectly suited for the tasks like this. the commandord=3; li= CoefficientList[Series[x 5-16*x+ /.x -> Sum[a[n]* n,{n,0,ord}],{ ,0,ord}], ]; sol =Solve[li==0]; r = Sum[a[n] n,{n,0,ord}]/.sol; r/. ->1does the 3 of 18 Physics 2400 Perturbation methodsSpring 2017 Step 1 is sometimes ambiguous because there may be many ways to introduce a smallparameter in the equation.

7 It is preferable to introduce in such a way that the zeroth-order solution (the leading term in the Perturbation series) is obtainable as a closed-formanalytic expression. Step 1 may be omitted when the original problem already has a smallparameter if a Perturbation series can be developed in powers of that Perturbative solution of a boundary-value problemLet s apply the Perturbation theory to a boundary value problem for an ordinary differen-tial the following nonlinear two-point boundary-value problem:y +y=cos(x)2 + 2y2, y(0) = 2, y( 2)= 1.(15)As a first step, we convert Eq. (15) into a Perturbation problem by introducing in theright side of the equation.

8 Then we obtain a first-order approximation to the , we return to the original equation by assigning = +y= cos(x)2 + 2y2.(16)We assume a Perturbation expansion fory(x) in the formy(x) = n=0 nyn(x),(17)wherey0(0) = 2,y0( 2)= 1,yn(0) =yn( 2)= 0 forn zeroth-order problemy +y= 0 is obtained by setting = 0. The solution that satisfiesthe boundary conditions isy0(x) = 2 cos(x) + sin(x).(18)The first order problem is obtained by substituting Eqs. (17), (18) into Eq. (16) and equatingthe coefficient of in the left and right hand sides of the equation:y 1+y1=cos(x)2 + 2y20=cos(x)7 + 3 cos(2x) + 4 sin(2x).

9 (19)Page 4 of 18 Physics 2400 Perturbation methodsSpring 2017Eq. (19) is linear inhomogeneous differential equation. It can be solved by variation ofparameters. The solution of Eq. (19) is as following:y1(x) =a(x) cos(x) +b(x) sin(x),(20)wherea(0) = 0 andb( 2)= 0,a(x) = x 0cos(y) sin(y)7 + 3 cos(2y) + 4 sin(2y)dy=1300 24x 14 6 arctan 23 + 14 6 arctan 23(1 + tan(x)) 9 ln(10) + 9 ln(7 + 3 cos(2x) + 4 sin(2x)) ,(21)b(x) =x 2cos2(y)7 + 3 cos(2y) + 4 sin(2y)dy=1300 18x 9 6 + 2 6 arctan 23(1 + tan(x)) 24 ln(2) + 12 ln(7 + 3 cos(2x) + 4 sin(2x)) .(22)The graph of the first order solutiony(x) =y0(x) +y1(x)(23)is presented in Fig.

10 1 together with the numerical solution of Eq. (15).Page 5 of 18 Physics 2400 Perturbation methodsSpring 2017 Figure 1: The graphs of the nu-merical solution of the bound-ary value problem Eq. (15)(solid line), Perturbation solu-tion Eq. (23) (dashed line), andperturbation solution Eq. (18)(dotted line). Perturbation ap-proximation Eq. (23) is practi-cally indistinguishable on thegraph from the numerical (x)x3 Singular Perturbation The method of dominant balanceLet s introduce an approach for attacking unsolvable problems that is called themethodof dominant you are given an equation in the formA+B+C= 0,(24)whereA,B,Care different terms in the equation.