本文選題：微分方程 + Laplace變換��； 參考：《華東師范大學(xué)》2015年碩士論文

【摘要】：非線性微分方程的解法研究是當(dāng)今非線性科學(xué)的一個(gè)重要研究?jī)?nèi)容.Adomian分解法是構(gòu)造非線性微分方程解析近似解的一種有效方法,該方法因?yàn)樗悸泛?jiǎn)單而獲得了廣泛應(yīng)用.但因?yàn)榉?hào)計(jì)算中間表達(dá)式急劇膨脹問題,致使單純使用Adomian分解法獲得的解析近似解的收斂區(qū)間往往很有限.近期有學(xué)者將Laplace變換法和Adomian分解法相結(jié)合,即所謂的Laplace分解法Laplace分解法較已有的Adomian分解法計(jì)算效率更高.本文將Laplace分解法推廣應(yīng)用到非線性偏微分方程情形,并針對(duì)已有算法的缺陷,提出了改進(jìn)的Laplace分解法.此外,本文還基于這兩種算法研發(fā)了自動(dòng)推導(dǎo)非線性微分系統(tǒng)解析近似解的軟件LDM P.本文的主要內(nèi)容如下：第一章主要介紹了和本文工作相關(guān)的研究背景,回顧了非線性微分系統(tǒng)解法研究的發(fā)展歷程,并簡(jiǎn)要總結(jié)了國內(nèi)外在該領(lǐng)域所取得的成果與發(fā)展現(xiàn)狀.第二章主要介紹了Laplace分解法及其改進(jìn)算法.首先闡述了直接推廣的Laplace分解法的思路與過程,然后通過具體實(shí)例對(duì)算法的缺陷進(jìn)行分析,進(jìn)而提出了改進(jìn)的Laplace分解法,并通過求解不同類型的方程對(duì)改進(jìn)前后兩種算法的適用范圍、優(yōu)缺點(diǎn)等作了比較,由此可知,Laplace分解法對(duì)擴(kuò)大級(jí)數(shù)解的收斂區(qū)間、提高級(jí)數(shù)解的精度均有很好的效果.第三章主要介紹了非線性微分系統(tǒng)解析近似解的自動(dòng)推導(dǎo)軟件LDMP.簡(jiǎn)要介紹了軟件的使用接口及其中主要模塊的功能和實(shí)現(xiàn)思路.通過應(yīng)用到不同類型的方程實(shí)例,進(jìn)一步驗(yàn)證了算法及軟件的有效性.軟件LDMP界面友好,使用方便,其編寫過程中局部也采用了并行化的思想和方法.用戶只要按照格式要求輸入待求解的方程及可能的初邊值條件,LDMP即可自動(dòng)輸出所獲得的解析近似解,還可輸出不同階解的比較曲線及誤差曲線,由此可進(jìn)一步佐證所獲結(jié)果的有效性.
[Abstract]:The study of solving nonlinear differential equations is an important research content of nonlinear science. Adomian decomposition method is an effective method to construct analytical approximate solutions of nonlinear differential equations. This method has been widely used because of its simple thinking. However, due to the problem of sharp expansion of intermediate expressions in symbolic computation, the convergence interval of analytical approximate solutions obtained by using Adomian decomposition method is often very limited. Recently, some scholars have combined the Laplace transform method with the Adomian decomposition method, that is, the so-called Laplace decomposition method is more efficient than the existing Adomian decomposition method. In this paper, the Laplace decomposition method is extended to nonlinear partial differential equations, and an improved Laplace decomposition method is proposed to overcome the defects of the existing algorithms. In addition, based on these two algorithms, a software called LDMP is developed for the automatic derivation of analytical approximate solutions of nonlinear differential systems. The main contents of this paper are as follows: the first chapter introduces the research background related to the work of this paper, reviews the development history of the nonlinear differential system solution, and briefly summarizes the achievements and development status in this field at home and abroad. In chapter 2, the Laplace decomposition method and its improved algorithm are introduced. In this paper, the idea and process of the Laplace decomposition method, which is extended directly, is introduced, and then the defects of the algorithm are analyzed through concrete examples, and the improved Laplace decomposition method is put forward. By solving different kinds of equations, the application range, advantages and disadvantages of the two algorithms before and after the improvement are compared. It can be seen that the Laplace decomposition method has a good effect on enlarging the convergence interval of the series solution and improving the accuracy of the series solution. In chapter 3, the analytical approximate solution of nonlinear differential system is introduced. This paper briefly introduces the interface of the software, the function and realization of the main modules. The validity of the algorithm and the software is further verified by the application of different kinds of equations. The software LDMP has friendly interface and easy to use. In the process of programming, it also adopts the idea and method of parallelization. The user can automatically output the analytical approximate solution obtained by the input of the equation to be solved and the possible initial boundary value condition according to the format requirement, and can also output the comparison curve and error curve of different order solutions. This can further verify the validity of the obtained results.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

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本文編號(hào)：2015247