.539 Lecture 2 Introduction to Perturbation Methods Prof. Dean Wang. Perturbation Theory Perturbation theory is a large collection of iterative methods to obtaining approximate solutions to problems involving a small parameter ε. The approach of perturbation theory is to decompose a tough problem into an infinite number of relatively easy ones. Hence, perturbation theory is most useful when the first few steps reveal the important features of the solution and the remaining ones give small corrections. Example. Find approximations to the roots of x! 4.x +.2 = () Solution: Step : Introduce ε as x! 4 + ε x + 2ε = (2) Note: When ε =., we recover (). Step 2: Assume a perturbation series in the powers of ε as x ε =!!!! a! ε! (3) Step 3: To obtain the first term in this series, we set ε = in (2) and solve x! 4x = (4) and get three roots x = a! = 2,, 2 Step 4: Let us first look at a = 2. A second-order perturbation approximation consists of writing (3) as x = 2 + a! ε + a! ε! + O ε!, ε (5) Step 5: Substituting (5) into (2) gives 8 + 8 + 2a! 4a! + 2 + 2 ε + 2a! a! 6a!! 4a! ε! = O ε! (6) Step 6: The above Eq (6) gives!:!!!!!!!!! :!!!!!!!!!!!!!!!!!!!! Step 7: Substituting a! and a! into (5) gives x = 2! ε +!!! ε! + O ε!, ε (8) Step 8: Verification ε =.: x = 2.499875 Exact solution: x = 2.4998756 format longeng -2-.5*.+/8*.^2 syms x!!!!! (7)
solve(x^3-4.*x +.2==,x) Step 9: a = and 2 are left as exercises. This example illustrates the three steps of perturbative analysis: () Convert the original problem into a perturbation problem by introducing the small parameter ε. (2) Assume an expression for the answer in the form of a perturbation series and compute the coefficients of that series. (3) Recover the answer to the original problem by summing the perturbation series for approximate value of ε. 2. Regular and Singular Perturbation Theory The formal techniques of perturbation theory are a natural generalization of the ideas of local analysis of differential equations as we learned in the previous lectures. Local analysis involves approximating the solution to a differential equation near the point x = x! by developing a series solution about x! in powers of a small parameter, x x!. Once the leading behavior of the solution near x = x! is known, the remaining coefficients in the series can be computed recursively. For local analysis, we recall that there are two different types of series. A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point generally does not have this form. Definition: Regular perturbation problem: its perturbation series is a power series in ε having a nonvanishing radius of convergence. A basic feature of all regular perturbation problems is that the exact solution for small but nonzero ε smoothly approaches the unperturbed solution as ε. Singular perturbation problem: its perturbation series either does not take the form of a power series or, if it does, the power series has a vanishing radius of convergence. In singular perturbation theory there is sometimes no solution to the unperturbed problem (the exact solution as function of ε may cease to exist when ε =. Example is a regular perturbation problem. The following are three examples of singular perturbation problems. Example 2: Appearance of a boundary layer. The boundary-value problem ε! y y = y = This problem is a singular perturbation problem because the associated unperturbed problem (9) 2
y! = () y = has no solution. The solution to (9) cannot have a regular perturbation expansion of the form y =!!!! a! x ε! because a! x does not exist. The exact solution to (9) is easy to find: y x =!!/!!! ()!!/!!!.9 =.2 =.5 =.25.8.7.6 y.5.4.3.2...2.3.4.5.6.7.8.9 x In this example we saw that the exact solution varies rapidly in the neighborhood of x = for small ε and develops a discontinuity there in the limit ε +. A solution to a boundary-value problem may also develop discontinuities through a large region as well as in the neighborhood of a point as shown in the next example. Example 3: Appearance of rapid variation on a global scale. ε! y!! + y = (2) y = This problem is a singular perturbation problem because when ε =, the solution to the perturbation problem, y =, does not satisfy the boundary condition y =. The exact solution to (2) is y x =!"#!!!"#!! for ε nπ!!, n =,, 2, (3) 3
4 3 =.5 =.4 2 - -2-3 -4..2.3.4.5.6.7.8.9 x=linspace(,,2) y=sin(x/.5^.5)/sin(/.5^.5) y2=sin(x/.4^.5)/sin(/.4^.5) plot(x,y,x,y2) legend('\epsilon =.5', '\epsilon =.4') Example 4: Perturbation theory on an infinite interval. The initial-value problem y!! + εx y = y = (4) y = is a regular perturbation problem in ε over the finite interval x L. In fact, the perturbation solution is just y x = cosx + ε!! x! sinx +! xcosx! sinx!! +ε!!!" x! cosx +!!" x! sinx +!!" x! cosx! xsinx + (5)!" which converges for all x and ε, with increasing rapidity as ε + for fixed x. 4
3 = = /4 2 - -2-3 5 5 2 25 3 35 4 45 5 x=linspace(,5,5) y=cos(x) y2=cos(x)+/4*(/4*x.^2.*sin(x)+/4*x.*cos(x)-/4*sin(x))+(/4)^2*(- /32*x.^4.*cos(x)+5/48*x.^3.*sin(x)+7/6*x.^2.*cos(x)-7/6*x.*sin(x)) 5