Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 當微分方程式具有可變的係數 (variable coefficients, 例如 x 的函數 ), 通常它必須以其他的方法求解. 雷建 德方程式 (Legendre’s equation), 超幾何方程式 (hypergeometric equation) 以及貝索方程式 (Bessel’s equation) 是此類型中非常重要的方程式. 一般有兩種標準的解法 : 產生級數型式之解的冪級 數法 (power series method) 以及以其延伸而來的弗賓 納斯法 (Frobenius method) 與其應用.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Power Series method ( 冪級數法 ) Infinite series Where the coefficients a m are constant, independent of x x 0 is the center of the series Maclaurin series Chapter 4 Series solutions of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Power Series method ( 冪級數法 ) 1. 先將 p(x) 與 q(x) 以 x[ 或 (x-x 0 )] 的冪級數來表示之. 2. 假設解為含待定係數的冪級數 3. 將此級數與逐項微分所得之級數帶入方程式中 4. 將 x 之冪次相同各項整合一起, 並令其係數為零來解 出各待定係數. Chapter 4 Series solutions of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Power Series method ( 冪級數法 ) Chapter 4 Series solutions of Differential Equations 前 n 項之總合 餘項 (remainder) 收斂

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Power Series method ( 冪級數法 ) Chapter 4 Series solutions of Differential Equations Convergent radius (a)(b)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre’s equation ( 雷建德方程式 ) 它通常出現在具有球形對稱 (spherical symmetry) 的問題. 對於雷建德方程 式的解均稱為雷建德函數 (Legendre function) 利用冪級數法 x 0 的係數 : x 1 的係數 : 當 s = 2,3,….

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre’s equation ( 雷建德方程式 ) s = 0,1,2,…. Recurrence relation ( 遞迴關係 ) Recursion formula ( 遞迴公式 ) 將上述係數代入整理後 其中 級數解

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre polynomials ( 雷建德多項式 ) 雷建德方程式中的參數 n 為非負值的整數時, 當 s = n, 上面公式等號右邊為零 這些多項式被稱為雷建德多項式 (Legendre polynomials) 若 n 為偶數, 則 y 1 (x) 縮減為 n 次的多項 式 若 n 為奇數, 則 y 2 (x) 縮減為 n 次的多項 式

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre polynomials ( 雷建德多項式 ) 此時, 我們可將其他非零的係數經由上式用係數 a n 來表 示 一般標準是選取 當 n = 0 此時係數 a n 的選取可使得這些多項式在 x = 1 時均會等於 1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre polynomials ( 雷建德多項式 ) In general

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Legendre polynomials ( 雷建德多項式 ) n 次的雷建德多項式 (Legendre polynomials) 以 P n (x) 來表示

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 弗賓納斯法 (Frobenius method) 對於具有以下形式的微分方程式 其中 b(x) 與 c(x) 在 x = 0 處為解析函數 則至少有一個解可以表示成 ( Frobenius method) 其中 r 可以是任何的實數或複數 (r 的選取需使 ) 另一個線性無關的解可以類似於上式 ( 有不同的 r 與係數 ) 或可含有對數項

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 弗賓納斯法 (Frobenius method) 適用的微分方程式例如 : 貝索微分方程式 (Bessel’s differential equation) 超幾何微分方程式 (hypergeometric differential equation) 其中 a,b 以及 c 為常數

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 正則與奇異點 (Regular and Singular Points) 對於微分方程式 使係數 p(x) 與 q(x) 可解析的點 x 0 稱為正則 點 使係數 p(x) 與 q(x) 不可解析的點 x 0 稱為奇異點 對於微分方程式 使係數 可解析的且 的點 x 0 稱為正則點 使係數 不可解析的點 x 0 稱為奇異點

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 指標方程式 (Indicial Equation) 先將 b(x) 與 c(x) 以冪級數展開 然後

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 指標方程式 (Indicial Equation) 整理後將 的每一冪次項的係數和為零 : 對應於 x r 的係數 和 上式稱為微分方程式 的指標方程式

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 指標方程式 (Indicial Equation) 的解 Case 1 : 相差不為整數的兩個相異根 r 1 與 r 2 : Case 2 : 重根 r 1 = r 2 : Case 3 : 相差為整數的兩個相異根 r 1 與 r 2 : 此處 r 1 - r 2 > 0 且 k 值可以為零

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 歐拉 - 科西方程式 (Euler-Cauchy equation) 其中 b 0, c 0 為常 數 將 代入得到的輔助方程式 (auxiliary equation) 即為指標方程式 Case 1 : 兩個相異根 r 1 與 r 2 : Case 2 : 重根 r 1 = r 2 : ( 特殊的形式 )

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 它通常出現在 electric field, vibrations 以及 heat conduction 等問題上, 特別是顯示出具有圓柱對稱 (cylindrical symmetry) 的問題. 此處假設 ν 為非負值實數. 將含待定係數的級數 及其導數帶入

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) ( 令 x s+r 之係數和等於零  第一, 二, 四項的 m = s, 第三項 m = s-2, 記得 m  0) (s = 0) : (s = 1) : (s = 2,3,…) : 由第一項可以得到 indicial equation 可以得到 兩根

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 當, 係數的遞迴性 (recurrence) 代入 由於 且

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) In general 當 ν 為整數 n 時

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 當 ν 為整數 n 時 我們可任意選取 所以當 r = ν = 整數 n 時, 一個特殊解以 J n (x) 來表示 n 階的第一種貝索函數, 此級數函數對於所有的 x 均收斂 ( 快速地 )

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 當 r = ν = 任一非負值 時 先介紹 Gamma 函數, 其定義為 利用分部積分, 我們可以計算出 而由定義可計算出 Gamma 函數將階乘函數 (factorial function) 廣義化

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 此時, 我們任意選取的 所以當 r = ν 時, 一個特殊解以 J ν (x) 來表示 ν 階的第一種貝索函數, 此級數函數對於所有的 x 均收斂 ( 快速地 )

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 所以當 r = ν = 非負值且非整數時, 一個特殊解以 J ν (x) 來表示 而另一個線性無關的特殊解可以直接將 ν 以 - ν 代入上式得之 所以, 如果 ν 不為整數, 則貝索方程式的通解可寫成

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 然而, 當 r = ν = 整數 n 時, 貝索函數 J n (x) 與 J -n (x) 為線性相依的 此時, 我們必須再尋求其他的 solution!( 參見 4-6 節 ) Bessel’s Theory

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) Proof :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) Proof :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) Proof :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) Proof :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 這兩個公式在包含貝索函數的積分計算中是常見的 這兩個公式常用於數值計算, 以及將高階貝索函數簡化為低 階貝索函數表示的運算.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 當 = ½ 時

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation) 此外, 在分母中

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 貝索微分方程式 (Bessel’s differential equation)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Elementary Bessel function Bessel function J of order =  1/2,  3/2,  5/2, …are elementary; they can be expressed by finitely many cosines and sines and powers of x.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) 記得, 當 r = ν = 整數 n 時, 貝索函數 J n (x) 與 J -n (x) 為線性相依的 此時, 我們必須再尋求其他的 solution! 當 接近一個正整數 n 時, 由於 Gamma 函數的特性, 當 m = n 時, 貝索函數的係數才不為零. 得證

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) 貝索方程式 當 = n = 0 時 double root r = 0 Case 2 : 重根 r 1 = r 2 :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) Consider even power x 2s

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) Consider odd power x 2s+1 For s = 0 For the other values of s

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) For s = 1 In general 令 A basis

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) 且 a,b 均為常數 Another basis 通常選取 Euler constant Particular solution 零階的第二種貝索函數, 或稱為 零階的 Neumann’s function

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations 第二種貝索函數 Y (x) 階的第二種貝索函數, 或稱為 階的 Neumann’s function 貝索方程式的通解 For all values of

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Sturm-Liouville Problems Legendre’s equation ( 雷建德方程式 ) 貝索微分方程式 (Bessel’s differential equation) Sturm-Liouville equation 令 r = 1-x 2, q = 0, p = 1, λ = n(n+1) 令 令 r = x, q = -n 2 /x, p = x, λ = k 2

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Sturm-Liouville Problems Sturm-Liouville equation 若此方程式在一個已知的區間 a  x  b 上, 滿足在兩端點 a 與 b 上的邊界條件 k 1,k 2 為已知的常數, 不全為零 l 1,l 2 為已知的常數, 不全為零 在此區間中, 假設 : p,q,r,r’ 的連續性, 以及 p(x) > 0 Sturm Liouville 方程式 + 邊界條件的邊界值問題 (boundary value problem) 稱之為 Sturm Liouville problem Sturm 與 Liouville 發展出此特殊解的級數解理論 其解稱之為本徵函數 (eigenfunction), 此時的 λ 稱之為本徴值 (eigenvalue)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Example 1 : Vibrating elastic string r = 1, q = 0, p = 1, a = 0, b = π, k 1 = l 1 = 1, k 2 = l 2 = 0 For negative λ = -ν 2 非本徵函數, 故 λ 非本徴值 or λ = 0 For positive λ = ν 2 Take B = 1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonality( 正交性 ) Sturm Liouville problem 中, 本徵值函數有所謂的正交性 正交的定義 : 定義在某一區間 : a  x  b 的函數 y 1,y 2,…. 與一個權函數 (weight function) p(x)>0 之間的關係為 則 y 1,y 2,…. 是稱為正交的 y m 的範數 (norm) : 定義為 如果函數在 a  x  b 的區間中是正交的, 且其範數為 1, 則可稱 這些函數在 a  x  b 的區間中為單範正交 (orthonormal)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonality of Eigenfunctions ( 本徵函數的正交性 ) 定理 : 假設在 Sturm-Liouville 方程式中的函數 p,q,r,r’ 在 a  x  b 的區間中是實 數值且連續的, 且 p(x)>0. 令 y m (x) 與 y n (x) 為 Sturm-Liouville 問題中分別對應 於不同本徵值 λ m 與 λ n 的本徵函數, 則在此區間中 y m (x) 與 y n (x) 關於權函數 p 為正交的. k 1,k 2 為已知的常數, 不全為零 l 1,l 2 為已知的常數, 不全為零 Sturm Liouville 方程式 + 兩個邊界條件的邊界值問題 (boundary value problem) 稱之為 Sturm Liouville regular problem 1 2 若 r(a) = 0, 則邊界條件 1 可省去 ; 若 r(b) = 0, 則邊界條件 2 可省去 ; 則此時可稱 為 Sturm Liouville singular problem 若 r(a) = r(b), 則邊界條件為 則此時可稱為 periodic Sturm Liouville problem

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonality of Legendre polynomials ( 雷建德多項式的正交性 ) Legendre’s equation ( 雷建德方程式 ) r = 1-x 2, q = 0, p = 1, λ = n(n+1) Sturm-Liouville equation 由於 r(-1) = r(1) = 0, 我們不需要邊界條件, 但在 -1  x  1 的區 間中這是一個奇異的 (singular)Sturm-Liouville 問題. 由於 n = 0,1,2,3,…. 所以 λ = 0,2,6,12,… 雷建德多項式 P n (x) 為此問題之解, 故由定理可知

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonality of Bessel function J n (x) ( 貝索函數的正交性 ) 貝索微分方程式 (Bessel’s differential equation) 對於固定整數 n  0 Sturm-Liouville equation r = x, q = -n 2 /x, p = x, λ = k 2 n 階的第一種貝索函數 由於 r(0) = 0, 對於那些在 x = R 處為零的解 J n (kx) 而言 ( 亦即 J n (kR) = 0), 定理說明了解 J n (kx) 在 0  x  R 的區間中的正交性.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonal Eigenfunction Expansions ( 正交本徵函數的展開 ) 定義在某一區間 : a  x  b 的函數 y 1,y 2,…. 與一個權函數 (weight function) p(x)>0 之間的關係為 則 y 1,y 2,…. 是稱為正交的 y m 的範數 (norm) : 定義為 對於單範正交 (orthonormal) 函數而言 Kronecker delta

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonal Eigenfunction Expansions ( 正交本徵函數的展開 ) 現在令 y 0,y 1,y 2,…. 為區間 a  x  b 上關於權函數 p(x) 的正 交集合, 而 f(x) 為一個可以由一收斂級數展開的函數 此可稱為正交展開 (orthogonal expansion) 或廣義傅立葉級數 (generalized Fourier series) 或是本徵函數展開 (eigenfunction expansion, 此時 y m 為 Sturm-Liouville 問題的本徵函數 ) 未知係數 a 0,a 1,a 2,… 為 f(x) 的傅立葉常數 (Fourier constant), 此可 利用正交性來求出.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Orthogonal Eigenfunction Expansions ( 正交本徵函數的展開 )

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Fourier series ( 傅立葉級數 ) 對於 periodic Sturm Liouville problem The general solution : 帶入邊界條件 eigenvalue eigenfunction Note that p(x) = 1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 4 Series solutions of Differential Equations Fourier series ( 傅立葉級數 ) 對於範數 : 其他的 此稱為 f(x) 的傅立葉級數, a m 與 b m 為傅立葉係數