本文的以一些分析和數(shù)值方法為基礎(chǔ),利用Sumudu變換,Adomian分解方法(ADM),和Pade近似技術(shù)等方法,求解一些重要的線性和非線性偏微分方程(PDEs)。Sumudu變換法只能求解線性PDE;Sumudu Adomian分解方法(SADM)是Sumudu變換和Adomian分解方法的結(jié)合,此方法適用于求解非線性PDE,但是該方法收斂半徑小,而且對(duì)于某些PDE,利用SADM方法所得的截?cái)嗉?jí)數(shù)在很多區(qū)域都是不準(zhǔn)確的.針對(duì)這種情況,本文將通過(guò)使用雙Pade近似的函數(shù)來(lái)執(zhí)行SADM解決方案.我們得到的PSADM解的收斂域大于SADM解.本文使用Adomian多項(xiàng)式計(jì)算非線性項(xiàng),并使用Pade近似來(lái)控制級(jí)數(shù)解的收斂性。我們從方法的基本定義,定理和性質(zhì)開始,介紹了所提數(shù)值方法的研究背景.以上方法可以用于研究一些科學(xué)和工程問(wèn)題的數(shù)值解,如線性和非線性薛定諤方程,線性克萊恩-戈登方程,非線性伯格斯方程等.接下來(lái)我們提出了該方法基本思想的基本數(shù)學(xué)公式,數(shù)值實(shí)驗(yàn)表明了該方法的有效性和高精度.此外,我們給出了該方法求解三維曲面的圖形化的數(shù)值模擬。所提出的方法為我們提供了一種利用不同階的帕德近似來(lái)...

【文章頁(yè)數(shù)】：102 頁(yè)

【學(xué)位級(jí)別】：碩士

【文章目錄】：
ABSTRACT
摘要
Chapter 1 Introduction
Chapter 2 Background Materials
    2.1 Sumudu transform
        2.1.1 Basic definitions and properties of the Sumudu transform
        2.1.2 The Laplace-Sumudu duality and the complex Sumudu in-version formula
    2.2 Adomian Decomposition Method
        2.2.1 Adomian Decomposition Method for Solving nonlinear prob-lems
    2.3 Application of the Sumudu decomposition method (SDM) for Lin-ear partrial differential equations
    2.4 Pade approximation
        2.4.1 Test experiments of Pade approximation
    2.5 SADM and PSADM for solving nonlinear PDEs
        2.5.1 Procedure for solving (2.54) by SADM
        2.5.2 Procedure for solving (2.54) by PSADM
        2.5.3 Theory on the nonlinear Schrodinger Equations
        2.5.4 Theory on the KdV Burger's equation
Chapter 3 SADM and PSADM for solving Schrodinger and KdVBurger's equations
    3.1 SADM and PSADM for solving General Nonlinear Schrodingerequation
        3.1.1 Application 1
        3.1.2 Application 2: One Dimensional Nonlinear Schrodinger E-quation with Harmonic Oscillator
    3.2 SADM and PSADM for solving compound KdV-Burger's equations
        3.2.1 Application: p＝-6, q=0, r=0,β=-1,u(x,0)=-2sech²(x)
        3.2.2 Application: p≠0,q=0,r≠0,β≠0, KdV-Burgersequation
        3.2.3 Application: p≠0,q=0,β=0, Burger's equation
        3.2.4 Application: q≠0,p＝0,β≠0, r=0, mKDV Burger'sequation
        3.2.5 Application:p=0,q≠0,r≠0,β≠0,mKdV-Burger'sequation
        3.2.6 Application: General KdV-Burger's equation
Chapter 4 Conclusions and Prospects
    4.1 Conclusions
    4.2 Prospects
References
Acknowledgment
學(xué)位論文評(píng)閱及答辯情況表



本文編號(hào)：3821920