Fourier series of odd functions with period 2 l
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1) The document provides review material for Exam 3 of Math 285 including definitions of piecewise continuous and piecewise smooth functions and the convergence theorem for Fourier series. 2) It also defines Fourier series for periodic piecewise continuous functions and discusses Fourier sine and cosine series for odd and even functions. 3) Applications of Fourier series include using them to find formal solutions to boundary value problems involving differential equations.
Fourier series of odd functions with period 2 l
- 1. Math 285 - Spring 2012 - Review Material - Exam 3 Section 9.2 - Fourier Series and Convergence • State the definition of a Piecewise Continuous function. Answer: f is Piecewise Continuous if the following to conditions are satisfied: 1) It is continuous except possibly at some isolated points. 2) The left and right limits f(x+) and f(x−) exist (finite) at the points of discontinuity. For example, Square-Wave functions are piecewise continuous. But f(x) = sin 1 x for 0 < x < 1, extended to R periodically, is not piecewise continuous, since the right limit at zero DNE. Also f(x) = tanx for 0 < x < π 2 and for π 2 < x < π, extended to R periodically, is not piecewise continuous, since the right and left limits at π 2 DNE (infinite). • Define Fourier Series for functions for a Piecewise Continuous periodic function with period 2L. Answer: f(x) ∼ a0 2 + ∞ ∑ n=1 an cosn π L x+bn sinn π L x where for n ≥ 0 an = 1 L L −L f(x)cosn π L x dx and for n ≥ 1 bn = 1 L L −L f(x)sinn π L x dx • State the definition of a Piecewise Smooth function. Answer: f(x) is Piecewise Smooth if both f(x) and f (x) are piecewise continuous. For example, Square-Wave functions are piecewise smooth. • State the Convergence Theorem for Fourier Series. Answer: If f(x) is periodic and piecewise smooth, then its Fourier Series converges to 1) f(x) at each point x where f is continuous. 2) 1 2(f(x+)+ f(x−)) at each point where f is NOT continuous. • Compute the Fourier Series of Square-Wave Function with period 2L: f(x) = +1 0 < x < L −1 L < x < 2L For which values of f(x) at the points of discontinuity the Fourier series is convergent for all x? 1
- 2. 2 Answer: an = 0 for all n ≥ 0, and bn = 2 nπ(1−cosnπ) for all n ≥ 1. Thus f(x) = ∑ n=odd 4 nπ sinn π L x Note that f(x) is discontinuous at x = integer multiples of L at which the average of left and right limit of f(x) is zero. Thus by Convergence Theorem f(x) must be 0 at those points. • Letting x = L 2 in the Fourier Series representation of the Square-Wave Function, obtain the following relation: ∞ ∑ k=0 (−1)k 2k +1 = 1− 1 3 + 1 5 − 1 7 +··· = π 4 Answer: Note that if we let x = L 2 , then the Fourier series is convergent to f(x) = 1, thus we have 1 = f L 2 = 4 π ∞ ∑ k=0 sin(2k +1)π 2 2k +1 = 4 π ∞ ∑ k=0 (−1)k 2k +1 Hence ∞ ∑ k=0 (−1)k 2k +1 = 1− 1 3 + 1 5 − 1 7 +··· = π 4
- 3. 3 Section 9.3 - Fourier Sine and Cosine Series • Recall f(x) is odd if f(−x) = −f(x), i.e. its graph is symmetric w.r.t y-axis. Example: x2n+1,sinnx for all integers n = 0, the square-wave functions f(x) = −1 −L < x < 0 1 0 < x < L • Recall f(x) is even if f(−x) = f(x), i.e. its graph is symmetric w.r.t origin. Example: x2n,cosnx for all integers n. • Remarks: 1) If f(x) is odd, then L −L f(x)dx = 0 for any L 2) If f(x) is even, then L −L f(x)dx = 2 L 0 f(x)dx for any L 3) If f and g are odd, then fg is even. 4) If f and g are even, then fg is even. 3) If f is odd and g is even, then fg is odd. • Fourier series of odd functions with period 2L: a0 = 1 L L −L f(x)dx = 0, an = 1 L L −L f(x)cosnπ Lxdx = 0 since f(x)cosnπ Lx is odd. bn = 1 L L −L f(x)sinnπ Lxdx = 2 L L 0 f(x)sinnπ Lxdx since f(x)sinnπ Lx is even. In this case, if f is piecewise smooth, f(x) = ∑bn sinnπ Lx only involves sine. Example: f(x) = x 2 on (−π,π), then f(x) = ∑∞ n=1 (−1)n+1 n sinnx • Fourier series of even functions with period 2L: an = 1 L L −L f(x)cosnπ Lxdx = 2 L L 0 f(x)cosnπ Lxdx since f(x)cosnπ Lx is even. bn = 1 L L −L f(x)sinnπ Lxdx = 0 since f(x)sinnπ Lx is odd. In this case, if f is piecewise smooth, f(x) = a0 2 +∑an cosnπ Lx only involves cosine. Example: Try f(x) = x2 on (−π,π). • Even and odd extensions of a function: Suppose f is a piecewise continuous function defined on interval (0,L). Even extension of f to the interval (−L,0) is fE(x) = f(x) 0 < x < L f(−x) −L < x < 0 Example: f(x) = x2 +x+1 on (0,L) Then fE(x) = x2 +x+1 0 < x < L x2 −x+1 −L < x < 0 Odd extension of f to the interval (−L,0) is fO(x) = f(x) 0 < x < L −f(−x) −L < x < 0
- 4. 4 Example: f(x) = x2 +x+1 on (0,L) Then fO(x) = x2 +x+1 0 < x < L −x2 +x−1 −L < x < 0 Remark: 1) fE(x) is an even function with Fourier Series of the form a0 2 +∑an cosnπ Lx Hence f(x) = a0 2 +∑an cosnπ Lx for x in (0,L). This is called the Fourier cosine series of f 2) fO(x) is an odd function with Fourier Series of the form ∑bn sinnπ Lx Hence f(x) = ∑bn sinnπ Lx for x in (0,L). This is called the Fourier sine series of f • Remark: Note that for x in (0,L), f(x) = fE(x) = fO(x). In many cases we are not concerned about f(x) on (−L,0), so the choice between (1) and (2) depends on our need for representing f by sine or cosine. • Example: f(t) = 1 on (0,π). Compute the Fourier sine and cosine series and graph the two extensions. 1) The Even extension is fE(t) = 1 on (−π,π), period 2π. Then a0 = 2,an = 0,bn = 0, so the cosine series is just f(t) = a0 2 = 1 2) The Odd extension is fO(t) = 1 on (0,π) and −1 on (−π,0), period 2π. Then a0 = an = 0,bn = 2 nπ(1−(−1)n), so the cosine series is fO(t) = ∑ n=odd 4 nπ sinnx • Example: f(t) = 1−t on (0,1). Compute the Fourier sine and cosine series and graph to the to extensions. 1) The Even extension is fE(t) = 1−t on (0,1) and 1+t on (−1,0), period 2L, L = 1. Then a0 = 1,an = 21−cosnπ n2π2 ,bn = 0, so the cosine series is f(t) = 1 2 + ∑ n=odd 4 n2π2 cosnπx 2) The Odd extension is fO(t) = 1−t on (0,1) and −1−t on (−1,0), period 2L, L = 1. Then an = 0 for all n ≥ 0 and bn = 2 nπ, so the cosine series is f(t) = ∞ ∑ n=1 2 nπ sinnπx • Termwise differentiation of a Fourier series Theorem: Suppose f(x) is Continuous for all x, Periodic with period 2L, and f is
- 5. 5 Piecewise Smooth for all t. If f(x) = a0 2 + ∞ ∑ n=1 an cosn π L x+bn sinn π L x then f (x) = ∞ ∑ n=1 (−an nπ L )sinn π L x+(bn nπ L )cosn π L x Remarks: 1) RHS it the Fourier series of f (x). 2) It is obtained by termwise differentiation of the RHS for f(x). 3) Note that the constant term in the FS of f (x) is zero as L −L f (x)dx = f(L)− f(−L) = 0 4) The Theorem fails if f is not continuous! For example: consider f(x) = x on (−L,L) Then f(x) = x = ∑ 2L nπ (−1)n+1 sinn π L x if we differentiate 1 = ∑ 2 π (−1)n+1 cosn π L x For example equality fails at t = 0 or L! • Example: Verify the above Theorem for f(x) = x on (0,L) and f(x) = −x on (−L,0). Answer: bn = 0 as f is even, a0 = L, an = 2L n2π2 ((−1)n −1). Thus f(x) = 1 2 L+ ∑ n=odd − 4L n2π2 cosn π L x Then term wise derivative gives f (x) = ∑ n=odd 4 nπ sinn π L x On the other hand, directly computing the derivative of f(x) we have f (x) = 1 on (0,L) and f(x) = −1 on (−L,0). Thus an = 0 for all n and bn = 2 nπ(1 − cosnπ) which gives the same Fourier Series as in above. • Applications to BVP’s Consider the BVP of the form ax +bx +cx = f(t), x(0) = x(L) = 0 or x (0) = x (L) = 0 • Example: x +2x = 1, x(0) = x(π) = 0 Here f(t) = 1, restrict to the interval (0,π) as in Boundary Values. The idea is to find a formal Fourier Series solution of the equation. Since we want x(0) = x(π) = 0, we prefer to consider the Fourier sine series of x(t) on the interval (0,π), x(t) = ∑bn sinnt
- 6. 6 substitute in the equation and compare the coefficients with that of the sine Fourier Series of f(t) = 1 which is f(t) = ∑ n=odd 4 nπ sinnt Note that by term wise differentiation, x (t)+2x(t) = (2−n2 )bn sinnt Thus bn = 0 for n even and bn = 4 πn(2−n2) for n odd. Hence a formal power series of x(t) is x(t) = ∑ n=odd 4 πn(2−n2) sinnt • Example: x +2x = t, x (0) = x (π) = 0 Here f(t) = t, restrict to the interval (0,π) as in Boundary Values. Since we want x (0) = x (π) = 0, we prefer to consider the Fourier Cosine Series of x(t) on (0,π), x(t) = a0 2 +∑an cosnt substitute in the equation and compare the coefficients with that of the Cosine FS of f(t) = t which is f(t) = π 2 − ∑ n=odd 4 πn2 cosnt Note that by termwise differentiation, x (t)+2x(t) = a0 +(2−n2 )an cosnt Thus a0 = π 2 , an = 0 for n even and an = − 4 πn2(2−n2) for n odd. Hence a formal power series of x(t) is x(t) = π 4 − ∑ n=odd 4 πn2(2−n2) cosnt • Termwise Integration of the Fourier Series. Theorem: Suppose f(t) is Piecewise Continuous (not necessarily piecewise smooth) periodic with period 2L with FS representation f(t) ∼ a0 2 + ∞ ∑ n=1 an cosn π L t +bn sinn π L t Then we can integrate term by term as t 0 f(x)dx = t 0 a0 2 +∑∞ n=1 an t 0 cosnπ Lxdx+bn t 0 sinnπ Lxdx = a0 2 t +∑∞ n=1 an L nπ sinnπ Lx−bn L nπ(cosnπ Lx−1) Note that the RHS is not the Fourier series of LHS unless a0 = 0. • Example: Consider f(x) = 1 on (0,π) and f(x) = −1 on (−π,0), then f(t) = ∑ n=odd 4 πn sinnt
- 7. 7 Then F(t) = t 0 f(x)dx = t for t ∈ (0,1) and −t for t ∈ (−π,0), whose Fourier Series is F(t) = 1 2 π− 4 π ∑ n=odd 1 n2 cosnt Term by term integration gives the same thing! 4 π ∑ n=odd 1 n2 (1−cosnt) = 4 π ∑ n=odd 1 n2 = π2 8 − 4 π ∑ n=odd 1 n2 cosnt
- 8. 8 Section 9.4 - Applications of Fourier Series • Finding general solutions of 2nd order linear DE’s with constant coefficients: Example: x + 5x = F(t), where F(t) = 3 on (0,π) and −3 on (−π,0), is odd with period 2π. We obtain a particular solution in the following way: Since L = π and the FS of F(t) is ∑n odd 12 nπ sinnt, we may assume x(t) is odd and we consider its Fourier sine series F(t) = ∑∞ n=1 bn sinnt. Substitute in the equation: x +5x = (5−n2 )bn sinnt = F(t) = ∑ n odd 12 nπ sinnt Hence, comparing the coefficients of sinnt on both sides we get bn = 12 n(5−n2)π if n is odd and zero otherwise. Thus a particular solution is x(t) = ∑n odd 12 n(5−n2)π sinnt Definition: We call this a steady periodic solution, denoted by xsp(t). Thus, if x1(t),x2(t) are the solutions of the associated homogeneous equation, then the general solution is x(t) = c1x1(t)+c2x2(t)+xsp(t) = c1 cos( √ 5t)+c2 sin( √ 5t)+∑n odd 12 n(5−n2)π sinnt • Remark: Consider the equation x +9x = F(t). Then when trying to find a particular solution we get x +9x = (9−n2 )bn sinnt = F(t) = ∑ n odd 12 nπ sinnt We can not find b3 as 9−n2 = 0 for n = 3. In this case we need to use the method of undetermined coefficients to find a function y(t) such that y +9y = 12 3π sin3t Take y = At sin3t +Bt cos3t, and substitute to find A = 0 and B = − 2 3π. Therefore, the general solution is x(t) = c1 cos(3t)+c2 sin(3t)+ ∑ n odd,n=3 12 n(9−n2)π sinnt − 2 3π t cos3t Definition: We say in this case a pure resonance occurs. Remark: To determine the occurrence of pure resonance, just check if for some n, sinnπ Lt is a solution of the associated homogeneous equation.
- 9. 9 • Application: Forced Mass-Spring Systems Let m be the mass, c be the damping constant, and k the constant of spring. Then mx +cx +kx = F(t) Consider the case that the external force F(t) is odd or even periodic function. Remark: If F(t) is periodic for t ≥ 0, it can be arranged to be odd or even by passing to odd or even extension for values of time t. Case 1) Undamped Forced Mass-Spring Systems: c = 0 mx +kx = F(t) Let ω0 = k m be the natural frequency of the system, then we can write x +ω2 0x = 1 m F(t) Assume F(t) is periodic odd function with period 2L. Then F(t) = ∞ ∑ n=1 Fn sinn π L t Consider the odd extension of x(t), so that x(t) = ∞ ∑ n=1 bn sinn π L t substituting in the equation we get x +ω2 0x = ω2 0 −( nπ L )2 bn sinn π L t = 1 m F(t) = ∞ ∑ n=1 1 m Fn sinn π L t Thus ω2 0 −( nπ L )2 bn = 1 m Fn If for all n ≥ 1, ω2 0 −(nπ L )2 = 0, or equivalently k m · π L is not a positive integer, then we can solve for bn as bn = 1 mFn ω2 0 −(nπ L )2 Hence we have a steady periodic solution xsp(t) = ∞ ∑ n=1 1 mFn ω2 0 −(nπ L )2 sinn π L t Example: If m = 1,k = 5,L = π, x +5x = F(t) then ω2 0 −(nπ L )2 = 5−n2 = 0 for all positive integers n ≥ 1. On the other hand, if n0 = k m · π L is a positive integer, then a particular solution is x(t) = ∞ ∑ n=1,n=n0 1 mFn ω2 0 −(nπ L )2 sinn π L t − Fn0 2mω0 t cosω0t
- 10. 10 Example: m = 1,k = 9,L = π, x +9x = F(t) then ω2 0 −(nπ L )2 = 9−n2 = 0 for n = 3. There will be a pure resonance. Example: m = 1,k = 9,L = 1, x +9x = F(t) then ω2 0 − (nπ L )2 = 9 − (nπ)2 = 0 for all positive integers n. We have a steady periodic solution. Case 2) Damped Forced Mass-Spring Systems: c = 0 mx +cx +kx = F(t) Assume F(t) is periodic odd function with period 2L. Then F(t) = ∞ ∑ n=1 Fn sinn π L t For each n ≥ 1, we seek a function xn(t) such that mxn +cxn +kxn = Fn sinn π L t = Fn sinωnt where ωn = nπ L. Note that there will never be a duplicate solution as c = 0. Hence using the method of undetermined coefficients, we can show xn(t) = Fn (k −mω2 n)2 +(cωn)2 sin(ωnt −αn) where αn = tan−1 cωn k −mω2 n 0 ≤ α ≤ π Therefore, we have a steady periodic solution xsp(t) = ∞ ∑ n=1 xn(t) = ∞ ∑ n=1 Fn (k −mω2 n)2 +(cωn)2 sin(ωnt −αn) Example: m = 3,c = 1,k = 30, F(t) = t −t2 for 0 ≤ t ≤ 1 is odd and periodic with L = 1. Compute the first few terms of the steady periodic solution. 3x +x +30x = F(t) = ∞ ∑ n=1 Fn sinn π L t = ∑ n=odd 8 n3π3 sinnπt So, we have xsp(t) = ∑∞ n=1 Fn√ (k−mω2 n)2+(cωn)2 sin(ωnt −αn) = ∑n= odd 8 n3π3 √ (30−3n2π2)2+n2π2 sin(nπt −αn) αn = tan−1 nπ 30−n2π2 0 ≤ α ≤ π The fist two terms are 0..0815sin(πt −1.44692)+0.00004sin(3π3−3.10176)+···
- 11. 11 Section 9.5 - Heat Conduction and Separation of Variables • Until now, we studied ODE’s - which involved single variable functions. In this section will consider some special PDE’s - Differential Equations of several vari- able functions involving their Partial Derivatives - and we will apply Fourier Series method to solve them. • Heat Equations: Let u(x,t) denote the temperature at pint x and time t in an ideal heated rod that extends along x-axis. then u satisfies the following equation: ut = kuxx where k is a constant - thermal diffiusivity of the material - that depends on the material of the rod. Boundary Conditions: Suppose the rod has a finite length L, then 0 ≤ x ≤ L. 1) Assume the temperature of the rod at time t = 0 at every point x is given. Then we are given a function f(x) such that u(x,0) = f(x) for all 0 ≤ x ≤ L. 2) Assume the temperature at the two ends of the rod is fixed (zero) all the time - say by putting two ice cubes! Then u(0,t) = u(L,t) = 0 for all t ≥ 0. Thus we obtain a BVP, ut = kuxx u(x,0) = f(x) for all 0 ≤ x ≤ L u(0,t) = u(L,t) = 0 for all t ≥ 0 Remark: Other possible boundary conditions are, insulating the endpoints of the rod, so that there is no heat flow. This means ux(0,t) = ux(L,t) = 0 for all t ≥ 0 Remark: Geometric Interpretation of the BVP. We would like to find a function u(x,t) such that on the boundary of the infinite strip t ≥ 0 and 0 ≤ x ≤ L satisfies the conditions u = 0 and u = f(x). Remark: If f(x) is ”piecewise smooth”, then the solution of the BVP is unique. • Important observations: 1) Superposition of solutions: If u1,u2,... satisfy the equation ut = kuxx, then so does any linear combination of the ui’s. In other words, the equation ut = kuxx is linear! 2) Same is true about the boundary condition u(0,t) = u(L,t) = 0 for all t ≥ 0. We say this is a linear or homogeneous condition. 3) The condition u(x,0) = f(x) for all 0 ≤ x ≤ L is not homogenous, or not linear!
- 12. 12 • General Strategy: Find solutions that satisfy the linear conditions and then take a suitable linear condition that satisfies the non-linear conditions. • Example: Verify that un(x,t) = e−n2t sinnx is a solution of ut = uxx (here k = 1) for any positive integer n. For example, u1(x,t) = e−t sinx and u2(x,t) = e−4t sin2x. • Example: Use the above example to construct a solution of the following BVP. ut = uxx u(0,t) = u(π,t) = 0 for all t ≥ 0 u(x,0) = 2sinx+3sin2x for all 0 ≤ x ≤ π Answer: Here L = π. Note that un(x,t) = e−n2t sinnx also satisfy the linear condition u(0,t) = u(π,t) = 0. Thus it is enough to take u(x,t) to be a linear combination of un’s that satisfies the non homogenous condition u(x,0) = 2sinx + 3sin2x. Since un(x,0) = sinnx, we take u(x,t) = 2u1 +3u2 = 2e−t sinx+3e−4t sin2x so that u(x,0) = 2e0 sinx+3e0 sin2x = 2sinx+3sin2x Remark: The above method in the example for ut = uxx works whenever f(x) is a finite linear combination of sinx,sin2x,... • Example: Use the above example to construct a solution of the following BVP. ut = uxx u(0,t) = u(π,t) = 0 for all t ≥ 0 u(x,0) = sin4xcosx for all 0 ≤ x ≤ L Solution: Again we have L = π. Note that f(x) = sin4xcosx = 1 2 sin(4x+x)+ 1 2 sin(4x−x) = 1 2 sin5x+ 1 2 sin3x Thus we take u(x,t) = 1 2 u3 + 1 2 u5 = 1 2 e−9t sin3x+ 1 2 e−25t sin5x • Remark: When f(x) is a not a finite linear combination of the sine functions, then rep- resent it as an infinite sum using Fourier sine series. • Example: Construct a solution of the following BVP. ut = uxx u(0,t) = u(π,t) = 0 for all t ≥ 0 u(x,0) = 1 for all 0 ≤ x ≤ π Answer: Note that f(x) = 1 and L = π, so represent f(x) as a Fourier sine series with period 2L = 2π f(x) = ∑ n= odd 4 nπ sinnx
- 13. 13 Thus we take u(x,t) = ∑ n= odd 4 nπ un(x,t) = ∑ n= odd 4 nπ en2t sinnx This is a formal series solution of the BVP, one needs to check the convergence, ... We only take finitely many terms for many applications. • In general, to solve the following BVP (k is anything, not necessarily 1, and L is any- thing, not just π) ut = kuxx u(0,t) = u(π,t) = 0 for all t ≥ 0 u(x,0) = f(x) for all 0 ≤ x ≤ L we observe that un(x,t) = e−k(nπ L )2t sinn π L x satisfies the equation ut = kuxx and the homogenous boundary condition u(0,t) = u(L,t) = 0 for all t ≥ 0. Thus, represent f(x) as a Fourier sine series with period 2L, f(x) = ∞ ∑ n=1 bn sinn π L x Then u(x,t) := ∞ ∑ n=1 bnun(x,t) = ∞ ∑ n=1 bne−k(nπ L )2t sinn π L x satisfies the non homogenous condition u(x,0) = f(x) for all 0 ≤ x ≤ L. • Case of a rod with insulated endpoints: Consider the BVP corresponding to a heated rod with insulated endpoint, ut = kuxx ux(0,t) = ux(π,t) = 0 for all t ≥ 0 u(x,0) = f(x) for all 0 ≤ x ≤ L we observe that for n ≥ 0, un(x,t) = e−k(nπ L )2t cosn π L x satisfies the equation ut = kuxx and the homogenous boundary condition ux(0,t) = ux(L,t) = 0 for all t ≥ 0. Remark: for n = 0, we get u0 = 1 which satisfies the linear conditions! Thus, represent f(x) as a Fourier cosine series with period 2L, f(x) = a0 2 + ∞ ∑ n=1 an cosn π L x Then u(x,t) := a0 2 + ∞ ∑ n=1 anun(x,t) = a0 2 + ∞ ∑ n=1 ane−k(nπ L )2t cosn π L x satisfies the non homogenous condition u(x,0) = f(x) for all 0 ≤ x ≤ L.
- 14. 14 • Remarks: 1) In the BVP for heated rod with zero temperature in the endpoints, we have lim t→∞ u(x,t) = lim t→∞ ∞ ∑ n=1 bne−k(nπ L )2t sinn π L x = 0 in other words, heat goes away with no insulation, thus temperature is zero at the end! 2) In the BVP for heated rod with insulated endpoints, we have lim t→∞ u(x,t) = lim t→∞ a0 2 + ∞ ∑ n=1 bne−k(nπ L )2t cosn π L x = a0 2 which means, with insulation heat distributes evenly throughout the rod, which is the average of the initial temperature as a0 2 = 1 L L 0 f(x)dx
- 15. 15 Section 9.6 - Vibrating Strings and the One-Dimensional Wave Equation • Consider a uniform flexible string of length L with fixed endpoints, stretched along x-axis in the xy-plane from x = 0 to x = L. Let y(x,t) denote the displacement of the points x on the string at time t (we assume the points move parallel to y-axis). Then y satisfies the One-dimensional wave equation: ytt = a2 yxx where a is a constant that depend on the material of the string and the tension! Boundary Conditions: 1) Since the endpoints are fixed, y(0,t) = y(L,t) = 0. 2) The initial position of the string y(x,0) at each x is given as a function y(x,0) = f(x). 3) The solution also depends on the initial velocity yt(x,0) of the string at each x, given as a function yt(x,0) = g(x). Thus we obtain the following BVP ytt = a2yxx y(0,t) = y(L,t) = 0 for all t ≥ 0 y(x,0) = f(x) for all 0 ≤ x ≤ L yt(x,0) = g(x) for all 0 ≤ x ≤ L • Important Observations: 1) ytt = a2yxx is a linear equation. Thus the superposition of solutions applies. 2) Condition y(0,t) = y(L,t) = 0 is linear. 3) Conditions y(x,0) = f(x) and yt(x,0) = g(x) are not linear. • General Strategy: Split the BVP into two problems, (A) ytt = a2yxx y(0,t) = y(L,t) = 0 for all t ≥ 0 y(x,0) = f(x) for all 0 ≤ x ≤ L yt(x,0) = 0 for all 0 ≤ x ≤ L (B) ytt = a2yxx y(0,t) = y(L,t) = 0 for all t ≥ 0 y(x,0) = 0 for all 0 ≤ x ≤ L yt(x,0) = g(x) for all 0 ≤ x ≤ L If yA(x,t) and yB(x,t) are the respective solutions, then y(x,t) = yA(x,t)+yB(x,t) satisfies the original BVP as y(x,0) = yA(x,0)+yB(x,0) = f(x)+0 = f(x) and yt(x,0) = (yA)t(x,0)+(yB)t(x,0) = 0+g(x) = g(x) • Solving a BVP of type (A) (A) ytt = a2yxx y(0,t) = y(L,t) = 0 for all t ≥ 0 y(x,0) = f(x) for all 0 ≤ x ≤ L yt(x,0) = 0 for all 0 ≤ x ≤ L
- 16. 16 Verify directly that for all positive integers n, the function yn(x,t) = cosn π L at ·sinn π L x satisfies the equation ytt = a2yxx and the linear conditions y(0,t) = y(L,t) = 0 and yt(x,0) = 0. Thus, by superposition law, we need to find coefficients bn such that y(x,t) = ∞ ∑ n=1 bnyn(x,t) = ∞ ∑ n=1 bn cosn π L at ·sinn π L x satisfies y(x,0) = f(x). Note that f(x) = y(x,0) = ∞ ∑ n=1 bnyn(x,0) = ∞ ∑ n=1 bn sinn π L x Thus bn’s are the Fourier sine coefficients of f(x). • Example: Triangle initial position (pulled from the midpoint) with zero initial velocity Assume a = 1, L = π and f(x) = x if 0 < x < π 2 π−x if π 2 < x < π . Then the BVP is type (A) (A) ytt = yxx y(0,t) = y(π,t) = 0 for all t ≥ 0 y(x,0) = f(x) = x if 0 < x < π 2 π−x if π 2 < x < π yt(x,0) = 0 for all 0 ≤ x ≤ π Fourier sine series of f(x) is ∞ ∑ n=1 4sin nπ 2 πn2 sinnx = ∑ n=odd 4(−1) n−1 2 πn2 sinnx Thus, since a = 1 and L = π, y(x,t) = ∑∞ n=1 bn cosnπ Lat ·sinnπ Lx = ∑∞ n=1 4sin nπ 2 πn2 ·cosnt ·sinnx = ∑n=odd 4(−1) n−1 2 πn2 ·cosnt ·sinnx = ∑n=odd 4(−1) n−1 2 πn2 ·cosnt ·sinnx = 4 π cost sinx− 4 9π cos3t sin3x+ 4 25π cos5t sin5x+··· • Remark: Using the identity 2sinAcosB = sin(A+B)+sin(A−B) we can write the solution as
- 17. 17 y(x,t) = ∑n=odd 4(−1) n−1 2 πn2 ·cosnt ·sinnx = 1 2 ∑n=odd 4(−1) n−1 2 πn2 ·2cosnt ·sinnx = 1 2 ∑n=odd 4(−1) n−1 2 πn2 ·(sin(nx+nt)+sin(nx−nt)) = 1 2 ∑n=odd 4(−1) n−1 2 πn2 ·sinn(x+t)+ 1 2 ∑n=odd 4(−1) n−1 2 πn2 ·sinn(x−t) = 1 2 fO(x+t)+ 1 2 fO(x−t) • Solving a BVP of type (B) (A) ytt = a2yxx y(0,t) = y(L,t) = 0 for all t ≥ 0 y(x,0) = 0 for all 0 ≤ x ≤ L yt(x,0) = g(x) for all 0 ≤ x ≤ L Verify directly that for all positive integers n, the function yn(x,t) = sinn π L at ·sinn π L x satisfies the equation ytt = a2yxx and the linear conditions y(0,t) = y(L,t) = 0 and y(x,0) = 0. Thus, by superposition law, we need to find coefficients cn such that y(x,t) = ∞ ∑ n=1 cnyn(x,t) = ∞ ∑ n=1 cn sinn π L at ·sinn π L x satisfies yt(x,0) = g(x). Note that by termwise differentiation with respect to variable t we have yt(x,t) = ∞ ∑ n=1 cn(n π L a)cosn π L at ·sinn π L x Thus g(x) = yt(x,0) = ∞ ∑ n=1 cn(n π L a)sinn π L x Then cn(nπ La)’s are the Fourier sine coefficients of g(x). Hence cn = π nLa ·n-th Fourier sine coefficients of g(x) • Example: Assume a = 1, L = π and g(x) = 1. Then the following BVP is type (B) (B) ytt = yxx y(0,t) = y(π,t) = 0 for all t ≥ 0 y(x,0) = 0 yt(x,0) = g(x) = 1 for all 0 ≤ x ≤ π Fourier sine series of g(x) = 1 with L = π is ∑ n=odd 4 πn sinnx
- 18. 18 Thus, since a = 1 and L = π y(x,t) = ∑∞ n=1 cn sinnπ Lat ·sinnπ Lx = ∑n=odd π nπ(1) · 4 πn ·sinnt ·sinnx = ∑n=odd 4 πn2 ·sinnt ·sinnx
- 19. 19 Section 9.7 - Steady-State Temperature and Laplace’s Equation • Consider the temperature in a 2-dimensional uniform thin plate in xy-plane bounded by a piecewise smooth curve C. Let u(x,y,t) denote the temperature of the point (x,y) at time t. Then the 2-dimensional eat equation states that ut = k(uxx +uyy) where k is a constant that depends on the material of the plate. If we let ∇2u = uxx +uyy, which is called the laplacian of u, then we can write ut = k∇2 u Remark: The 2-dimensional wave equation is ztt = a2 (zxx +zyy) = a2 ∇2 z where z(x,y,t) is the position of the point (x,y) in a vibration elastic surface at time t. • Case of the steady state temperature: i.e. we consider a 2-dimensional heat equation in which the temperature does not change in time (The assumption is after a while temperature becomes steady). Thus ut = 0 Therefore, ∇2u = uxx +uyy = 0. This equation is called the 2-dimensional Laplace equation. • Boundary problems and Laplace equation: If we know the temperature on the boundary C of a plate, as a function f(x,y), can we determine the temp. at every point inside the plate? In other word, can we solve the BVP uxx +uyy = 0 u(x,y) = f(x,y) on the boundary of the plate This BVP is called a Dirichlet Problem. • Remark: If the Boundary C is Piecewise Smooth and the function f(x,y) is Nice!, then Dirichlet Problem has a unique solution. We will consider the cases that C is rectangular or circular! • Case of Rectangular Plates: Suppose the plate is a rectangle positioned in xy-plane with vertices (0,0),(0,b),(a,b),(a,0). Assume we are given the temp. at each side of the rectangle. Then we have the following type BVP.
- 20. 20 uxx +uyy = 0 u(x,0) = f1(x,y),0 < x < a u(x,b) = f2(x,y),0 < x < a u(0,y) = g1(x,y),0 < y < b u(a,y) = g1(x,y),0 < y < b • General Strategy: The main equation uxx + uyy = 0 is linear, but all of the boundary conditions are non- linear, the idea is to split this BVP into 4 simpler BVP denotes by A, B, C, D, in which only one of the boundary conditions in non-linear and apply Fourier series method there, and and the end, u(x,y) = uA(x,y)+uB(x,y)+uC(x,y)+uD(x,y) • Solving a type BVP of type (A) For 0 < x < a and 0 < y < b, uxx +uyy = 0 u(x,0) = f1(x,y) u(x,b) = 0 u(0,y) = 0 u(a,y) = 0 In this case, one can show un(x,y) = sin nπ a x·sinh nπ a (b−y) satisfies the linear conditions of the BVP. Recall that sinhx = 1 2(ex −e−x) and coshx = 1 2(ex +e−x), so (sinhx) = coshx and (coshx) = −sinhx • Method of separation of variables: To actually find un’s Assume u(x,y) = X(x)Y(y). Then uxx +uyy = 0 implies X Y +XY = 0. Thus X X = −Y Y . Since RHS only depends on y and LHS only depends on x, these fractions must be con- stant, say −λ. Then X X = −λ and Y Y = λ. Thus we obtain X and Y are nonzero solutions of X +λX = 0 and Y −λY = 0, such that Y(b) = 0, X(0) = X(a) = 0 This is an endpoint problem on X, we seek those values of λ for which there are nonzero solutions X, such that X(0) = X(a) = 0. The eigen values are λn = (nπ a )2 and the corresponding eigenfunctions are scalar multiples of Xn(x) = sin nπ a x. Now substitute λn = (nπ a )2 in Y −λY = 0 with Y(b) = 0, and solve for Y(y). We obtain the solutions are a scalar multiple of Yn(y) = sinh nπ a (b−y). Hence un(x,y) = Xn(x)Yn(y) = sin nπ a x·sinh nπ a (b−y)
- 21. 21 • Back to solving a type BVP of type (A) We want u(x,y) = ∑cnun(x,y) such that u(x,0) = f1(x). Hence f1(x) = u(x,0) = ∑cnun(x,0) = ∑cn ·sin nπ a x·sinh nπ a (b) Therefore, cn · sinh nπ a (b) is the n-th Fourier sine coefficient bn of f1(x) over interval (0,a). Hence cn = bn/sinh nπb a • Example: Solve the BVP of type (A) if a = b = π and f1(x) = 1. Compute u(π/2,π/2), the temp at the center of the rectangle. Solution: Note that f(x) = ∑ n=odd 4 nπ sin nπ a x = ∑ n=odd 4 nπ sinnx Thus u(x,y) = ∑ n=odd 4 nπ /sinh nπb a ·sin nπ a x·sinh nπ a (b−y) Then u(x,y) = ∑ n=odd 4 nπsinhnπ ·sinnx·sinhn(π−y) Also note that after computing the first few terms and using sinh2t = 2sinht cosht, u(π/2,π/2) = ∑n=odd 4 nπsinhnπ ·sinnπ/2·sinhnπ/2 = ∑n=odd 2 nπsinhnπ/2 ·sinnπ/2 ∼ .25 In fact, one can argue by symmetry that is is exactly .25. • Case of a semi-infinite strip plate! Assume the plate is an infinite plate in the first quadrant whose vertices are (0,0) and (0,b), and u = 0 along the horizontal sides, u(0,y) = g(y) and u(x,y) < ∞ as x → ∞. Apply the separation of variable method to solve the corresponding Dirichlet problem. Answer: Let u(x,y) = X(x)Y(y). Then uxx +uyy = 0 implies X Y +XY = 0. Thus X X = −Y Y . Since RHS only depends on y and LHS only depends on x, these fractions must be con- stant, say λ. Then X X = λ and Y Y = −λ. Thus we obtain X and Y are nonzero solutions of X −λX = 0 and Y +λY = 0, such that Y(0) = Y(b) = 0, and u(x,y) = X(x)Y(y) in bounded as x → ∞ This is an endpoint problem on Y. We want those values of λ for which there are nonzero solutions Y, such that Y(0) = Y(b) = 0. The eigen values are λn = (nπ b )2
- 22. 22 and the corresponding eigenfunctions are scalar multiples of Yn(y) = sin nπ b y. Now substitute λn = (nπ b )2 in X −λX = 0 and solve for X(x). We obtain Xn(x) = Ane nπ b x +Bne−nπ b x Since un = XnYn and u and Yn = sin nπ b y are bounded, then Xn must be bounded too. Hence An = 0. Suppress bn. Thus un(x,y) = Xn(x)Yn(y) = e−nπ b x ·sin nπ b y Note that un(x,0) = un(x,b) = 0 and un(x,y) < ∞ as x → ∞. Now, to solve the BVP: We want u(x,y) = ∑cnun(x,y) such that u(0,y) = g(y). Hence g(y) = u(0,y) = ∑cnun(0,y) = ∑cn ·(1)·sin nπ b y Therefore, cn is the n-th Fourier sine coefficient bn of g(y) over interval (0,b). • Example: If b = 1 and g(y) = 1, then compute u(x,y). Solution: g(y) = ∑ n=odd 4 nπ sin nπ b x = ∑ n=odd 4 nπ sinnx Thus u(x,y) = ∑ n=odd 4 nπ ·e−nπx ·sinnπy • Case of a circular disk: Using polar coordinates (r,θ) to represent the points in a disk, on can transform the Laplace equation into urr + 1 r ur + 1 r2 uθθ = 0 Note that 0 < r < a = the radius of the disk, and 0 < θ < 2π. It turns out that the solution is of the form u(r,θ) = a0 2 + ∞ ∑ n=1 (an cosnθ+bn sinnθ)rn Boundary conditions: If we know the temp. on the boundary of the disk, want to determine the temp. inside. To give the temp. on the boundary means to give a function f(θ) such that u(a,θ) = f(θ) for 0 < θ < 2π Thus f(θ) = u(a,θ) = a0 2 +∑∞ n=1(an cosnθ+bn sinnθ)an = a0 2 +∑∞ n=1 anan cosnθ+anbn sinnθ Therefore anan and anbn are the n-th Fourier series coefficient of f(θ) over the interval (0,2π).
- 23. 23 • Example: If f(θ) = 1 on (0,π) and −1 on (π,2π), and radius r = 1, then compute u(r,θ). Answer: We compute the Fourier series of f(θ) = ∑n=odd 4 nπ sinnθ. Thus an = 0 for all n. Hence u(r,θ) = a0 2 + ∞ ∑ n=1 (an cosnθ+bn sinnθ)rn = ∑ n=odd 4 nπ rn sinnθ • Remark: Intuitively, in the above example u(r,0) = 0, which is consistent with the solution: If θ = 0 or π, u(r,θ) = ∑ n=odd 4 nπ rn sinnθ = 0 • Remark: In the above example, if we change the boundary condition to f(θ) = 1 on (0,2π), then intuitively, u(r,θ) = 1 everywhere by symmetry. This is consistent with the actual solution to BVP: In this case, the Fourier coefficients of f are a0 = 2,an = bn = 0 for all n ≥ 1. Thus u(r,θ) = 1 for all 0 < r < 1 and 0 < θ < 2π.