第五章 結論與建議
5.2 建議
本研究受限於時間之限制，對於非線性效應所產生之諸多影響，
僅能反應出波速之改變，對於其它部份並無深入之探討。建議未來可 以考量高階解之推導，對非線性效應之影響將會有更精確的描述，同 時將結果與實際實驗相比較，以期能夠更加精確地描述造波過程之水 面變化。
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附錄 Hypergeometric Function
differential equation），具有三個正則奇點（regular singular point）於(
0,1,∞)
。若利用級數解析，可假設解的形式為：為了得到非顯性解（nontrivial solution），a0之值必不為零。若考 慮s=0的情況，則（A-6）式可改寫為：
0
（A-11）式一般稱為超幾何級數（hypergeometric series）。若利 用 Pochammer symbol，可將（A-11）式表示為：
( ) ( ) ( )
式中 Pochammer symbol 之定義為：
( )
a n =a(
a+1)(
a+2) (
L a+n−1)
(A-13) 根據微分方程的理論，只有方程式的奇點才可能是解的奇點，因 此級數（A-11）式在單位圓|z|<1內所表示的解析函數可以解析開拓 到全 z 平面，故開拓後以（A-12）式定義之函數2F1(
a,b;c;z)
即為超幾 何函數（hypergeometric function），亦意指 hypergeometric 函數為超 幾何方程的一個級數解析解（另一解為z1−c⋅2F1(
a+1−c,b+1−c;2−c;z)
）。Hypergeometric 函數是重要的一類特殊函數，凡具有三個正則奇點的 微分方程之解均可以 hypergeometric 函數表達，許多基本函數亦可 以 hypergeometric 函 數 表 示 ， 可 參 考 Erdélyi （ 1953 ）。 關 於 hypergeometric 函數之推廣及應用，則可參考 Seaborn（1991）與 Exton（1978）等人的著作。
故應用參數變異法（method of variation of parameters）求取將
（3-10）式代回（2-23）式之特解，設方程式之齊次解為Anp1
( )
t 與Anp 2( )
t ， 其特解可假設為：( )
t u( ) ( )
t A t u( ) ( )
t A tAnp = 1 np1 + 2 np2 (A-14) 將（A-14）式代回（2-23）式並整理之，可得到：