本文選題：非線性演化方程 + 行波解��； 參考：《華東師范大學(xué)》2017年碩士論文

【摘要】：非線性演化方程在非線性科學(xué)領(lǐng)域的研究中起著非常重要的作用,這些方程的解析解,特別是行波解,可以準(zhǔn)確地描述許多物理現(xiàn)象的內(nèi)在規(guī)律,例如振動(dòng)、傳播波以及孤立子等。隨著計(jì)算機(jī)科學(xué)和符號(hào)計(jì)算方法的快速發(fā)展,涌現(xiàn)了很多構(gòu)造非線性演化方程解析解的方法,如雙曲正切方法,橢圓函數(shù)方法等。這些方法被稱為直接代數(shù)方法。然而,由于符號(hào)計(jì)算的精確性及表達(dá)式快速膨脹等原因,直接代數(shù)方法普遍存在計(jì)算效率較低的問題。為了有效提高求解效率,本文將并行計(jì)算的思想應(yīng)用到了直接代數(shù)方法求解非線性演化方程的過程中�；赗iccati方程方法和并行計(jì)算的理念,本文提出了一種并行化構(gòu)造非線性演化方程精確行波解的新算法。特別是在并行求解的過程中,將多項(xiàng)式因式分解與負(fù)載均衡技術(shù)有機(jī)結(jié)合,有效地提高了多個(gè)CPU的利用率。而且,相比于其他已有的方法,通過分解算法和運(yùn)行時(shí)間限制,我們可以獲得更多的解。本文提出的并行算法已經(jīng)在Maple 18上進(jìn)行了實(shí)現(xiàn),并封裝成一個(gè)具有靈活接口和輸入輸出形式的軟件包PREM。通過將該軟件應(yīng)用于一些具體實(shí)例及對(duì)結(jié)果進(jìn)行比較分析,驗(yàn)證了本文提出的并行算法不僅在效率上比串行求解算法有了顯著的提高,而且求解能力也強(qiáng)于原有算法。從自然科學(xué)到社會(huì)科學(xué),幾乎所有的學(xué)科領(lǐng)域都越來越傾向于采用微分方程方法求解問題。為了便于不同領(lǐng)域的學(xué)者及工程技術(shù)人員學(xué)習(xí)微分方程知識(shí)、進(jìn)行學(xué)術(shù)交流和合作,本文研發(fā)了一個(gè)在線的開放的方程知識(shí)庫系統(tǒng)。該知識(shí)庫的研發(fā)綜合使用了多種較新的技術(shù)來實(shí)現(xiàn)方程的存儲(chǔ)、編輯和展示。特別地,系統(tǒng)中數(shù)學(xué)公式的可視化顯示均采用了二維形式。另外,對(duì)每一個(gè)方程,除了用一條記錄保存其重要信息,如方程類型、方程名稱、方程表達(dá)式等,還為每個(gè)方程建立了一個(gè)方程頁面,在其中展示該方程的基本信息、方程背景知識(shí)、方程的相關(guān)研究成果等。在方程頁面中也內(nèi)置了評(píng)論區(qū)域以便學(xué)術(shù)討論。該系統(tǒng)不僅功能完備,而且能快速響應(yīng)各種操作,具有良好的用戶體驗(yàn)。
[Abstract]:Nonlinear evolution equations play an important role in the research of nonlinear science. The analytical solutions of these equations, especially the traveling wave solutions, can accurately describe the inherent laws of many physical phenomena, such as vibration.Propagation waves and solitons, etc.With the rapid development of computer science and symbolic computing methods, there are many methods to construct analytical solutions of nonlinear evolution equations, such as hyperbolic tangent method, elliptic function method and so on.These methods are called direct algebraic methods.However, due to the accuracy of symbolic computation and the rapid expansion of expressions, direct algebraic methods generally have the problem of low computational efficiency.In order to improve the efficiency of solution, the idea of parallel computing is applied to the direct algebraic method for solving nonlinear evolution equations.Based on the Riccati equation method and the idea of parallel computing, a new algorithm for constructing exact traveling wave solutions of nonlinear evolution equations is proposed in this paper.Especially in the process of parallel solution, combining polynomial factorization with load balancing technology can effectively improve the utilization rate of multiple CPU.Moreover, compared with other existing methods, we can obtain more solutions by decomposing algorithms and running time constraints.The parallel algorithm proposed in this paper has been implemented on Maple 18 and encapsulated into a software package PREMwith flexible interface and input and output forms.By applying the software to some concrete examples and comparing and analyzing the results, it is verified that the proposed parallel algorithm is not only more efficient than the serial algorithm, but also better than the original algorithm.From natural science to social science, almost all disciplines are more and more inclined to use differential equations to solve problems.In order to facilitate scholars and engineers in different fields to learn the knowledge of differential equations and to carry out academic exchanges and cooperation, an online and open knowledge base system of equations is developed in this paper.The research and development of the knowledge base uses a variety of new technologies to store, edit and display equations.In particular, the visualization of mathematical formulas in the system is in two-dimensional form.In addition, for each equation, in addition to keeping its important information with a record, such as the type of equation, the name of the equation, the expression of the equation, and so on, an equation page is established for each equation, in which the basic information of the equation is displayed.Equation background knowledge, equation related research results and so on.Comment areas are also built into the equation page for academic discussion.The system not only has complete functions, but also can quickly respond to various operations and has a good user experience.
【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】：2017
【分類號(hào)】：O175

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