以波動方程理論建立單車道及號誌路口車流模式之基礎研究

Abstract

本研究主要乃利用一階雙曲型偏微分方程式(或稱一階單向性波動方 程式)的數學理論與方法 ,在巨觀的角度下來探討動態車流密度的現象 ,我們根據車輛數守恆原理建構動態車流模式 ,並且引進偏微分方程式的 觀點 ,以交通車流的理論對所建構的模式提出完整的分類 ,同時提出模式 解的存在性與唯一性和一般解的理論與方法 o 本研究的主要內容 ,可 分為以下五點 :(一)單車道動態車流模式的推導 :在本研究中我們以車輛 數守恆原理配合交通車流理論 , 來推導模式 o(二)單車道動態車流模 式的分類 :引進偏微分方程式中線性 ,半線性 ,擬似線性與非線性 方 程式的觀點 ,對於本研究所推導的模式加以分類 o(三)單車道動態車流模 式解的存在性與唯一性及求解公式的探討 :利用特徵曲線的數學方 法 將所求的偏微分方程式轉成常微分方程式 ,藉助常微分方程式中解存在性 與唯一性 定理的理論來探討解行為 o(四)單車道號誌路口中號誌變化 時動態車流模式的推導 :我們首先引進初始值 -邊界值( Neumann型態 的邊界值)的模式提出一種新的號誌路口車流模式 o(五)單車道號誌路口 中號誌變化時動態車流模式解的存在性及求解公式的探討 :藉助在二 階波動方程式上解理論的分析方法 ,我們提出號誌路口模式解存在與解計 算理論與方 法 o 正確的巨觀動態車流模式與解理論之建立是一切 電腦計算與模擬的基礎 ,本研究嘗試以嚴謹的態度在交通車流的理論上建 立模式與分類並提出相關的解行為理論 ,希望能提供初步的理論給予以動 態車流方法探討交通現象的學者 o Our research mainly used the first-order hyperbolic partial differentialequations ( or fisrt-order one-way wave equations ) to study the behavior of the dynamic traffic flow density from the macroscopic point of view.Based on the principle of vehicle conservation law,this study formulated some dynamic trafffic flow models.Furthermore,partial differential equations and traffic flow theory were used to raise the complete classification of the formulated models.In additions, this study also raised existence,uniqueness of solutions and the methods to get general solutions. The points of our research can be concluded as follows:(1) One-lane dynamic traffic flow model formulation : our research formulated models based on the principle of vehicle conservation law andtraffic flow theory.(2) One-lane dynamic traffic flow model classification : linear, semilinear,quasilinear and nonlinear partial differential equationswere used to classify the models.(3) Existence and uniqueness of solution andformula of one-lane dynamic traffic flow models : our research used character-istic curve method to transform partial differential equations into ordinarydifferential equations,and studied solutions by means of ordinary differentialequation based on existence and uniqueness theorem.(4) One-lane intersectiondynamic traffic flow model formulation : our research first introduced initial -boundary value ( Neumann boundary condition ) problems to raise a new inter-section traffic flow model. (5) Existence of solutions and solution formula ofone-lane intersection dynamic traffic flow models : our research raised exist-tence and calculation method of solutions by means of second-order wave equation analytical method.Correct macroscopic dynamic traffic flow models andsolution theory are the basis of computerized calculation and simulation.Ourresearch strictly triedto formulate models, classify models and develop corre-lated solutions.Our researchcould be a great contribution to the researchs whostudy traffic phenomenon bydynamic traffic flow method.