【摘要】：隨著計(jì)算機(jī)技術(shù)的不斷應(yīng)用與發(fā)展,以及非線(xiàn)性科學(xué)理論的進(jìn)一步完善,求解非線(xiàn)性偏微分方程的精確解已經(jīng)越來(lái)越成為一項(xiàng)富有重要意義的科研工作。近幾年,分?jǐn)?shù)階非線(xiàn)性偏微分方程相比于整數(shù)階更具有一般性,有許多學(xué)者都在致力于對(duì)求解分?jǐn)?shù)階非線(xiàn)性偏微分方程,故而該研究方向已成為研究熱點(diǎn)。目前已經(jīng)有許多方法可以求解出分?jǐn)?shù)階非線(xiàn)性偏微分方程的精確解,如:齊次平衡法,B?cklund變換法等等。文章主要在總結(jié)前人研究的基礎(chǔ)上,利用輔助方程進(jìn)一步研究B?cklund變換以及非線(xiàn)性疊加公式在非線(xiàn)性偏微分方程中的應(yīng)用,以及求出分?jǐn)?shù)階非線(xiàn)性偏微分方程的精確解和各種形式的無(wú)窮序列解。文章對(duì)分?jǐn)?shù)階非線(xiàn)性偏微分方程進(jìn)行了大量研究和敘述,內(nèi)容安排如下:第一章敘述了非線(xiàn)性科學(xué)、孤立子理論研究與發(fā)展、分?jǐn)?shù)階偏微分方程研究與發(fā)展以及研究中所用到的方法簡(jiǎn)介。第二章用Riccati作輔助方程研究了時(shí)間-空間分?jǐn)?shù)階非線(xiàn)性偏微分方程PKP方程和Gardner方程。第三章研究了分別用Riccati方程和第一種橢圓方程作輔助方程時(shí)間-空間分?jǐn)?shù)階非線(xiàn)性偏微分方程mBBM方程并分析兩種輔助方程的優(yōu)缺點(diǎn)。第四章對(duì)全文進(jìn)行了總結(jié)性概括,并對(duì)文章研究課題以后的發(fā)展做出了展望。
[Abstract]:With the continuous application and development of computer technology and the further improvement of nonlinear scientific theory, the exact solution of nonlinear partial differential equations has become an important research work. In recent years, fractional nonlinear partial differential equations are more general than integer order. Many scholars are devoted to solving fractional nonlinear partial differential equations. At present, there are many methods to solve the exact solutions of fractional nonlinear partial differential equations, such as the homogeneous equilibrium method and Bcklund transform method and so on. In this paper, the application of B?cklund transform and nonlinear superposition formula in nonlinear partial differential equations is studied by using auxiliary equations on the basis of summarizing previous studies. The exact solutions of fractional nonlinear partial differential equations and various forms of infinite sequence solutions are obtained. In this paper, the fractional order nonlinear partial differential equations are studied and described. The contents are arranged as follows: in chapter one, the nonlinear science, the research and development of soliton theory are described. The research and development of fractional partial differential equation and the methods used in the study. In chapter 2, the PKP equation and Gardner equation of fractional partial differential equation in time space are studied by using Riccati as auxiliary equation. In chapter 3, the mBBM equations of fractional partial differential equations with Riccati equation and the first elliptic equation are studied, and the advantages and disadvantages of the two auxiliary equations are analyzed. The fourth chapter summarizes the whole paper and prospects the future development of the research topic.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175.29

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