本文選題：第三類非線性VIE + 非緊算子　； 參考：《哈爾濱工業(yè)大學》2017年碩士論文

【摘要】：積分方程是數(shù)學的一個重要分支,而Volterra積分方程(VIE)在積分方程中占有重要地位。VIE的研究遍及物理、生物、化學等多個領域。常見的熱傳導模型、Lighthill模型、等時擺問題等都涉及到VIE。但是,對于一般的VIE,其解析解很難得到,所以求解VIE的數(shù)值解受到人們的廣泛關注。近年來,配置法被許多學者應用到求解VIE中,并取得了一些成果。本文研究了配置法求解非線性第三類VIE,數(shù)值解的存在唯一性和收斂性得到系統(tǒng)的研究。首先,我們回顧了一些積分方程和積分算子的背景知識:cordial Volterra積分算子及其緊性、cordial Volterra積分方程(CVIE)解析解的存在唯一性。特別地,我們討論了非線性第三類VIE相關算子的緊性。此外,我們給出了非線性第三類VIE解析解的存在唯一性。其次,配置法運用到非線性第三類VIE,我們采用與第二類VIE相似的方法討論了帶有緊算子的第三類VIE對應配置方程的可解性。但是對于非緊算子,為了保證可解性,我們在第一個區(qū)間運用了隱函數(shù)定理,在后面的區(qū)間則采用了改進的幾何網(wǎng)格方法。最后,我們討論了配置法的收斂階。我們定義了誤差函數(shù),它滿足一個線性離散VIE.收斂性是由這個線性離散Volterra積分算子的逆算子的一致有界性得出的,而其逆算子可以用一種適當?shù)姆稊?shù)進行估計。
[Abstract]:Integral equation is an important branch of mathematics, and Volterra integral equation has played an important role in the integral equation. Vie has been studied in many fields, such as physics, biology, chemistry and so on.Some common heat conduction models, such as Lighthill model and isochronous pendulum, are all related to VIE.However, for the general VIEs, the analytical solutions are difficult to obtain, so the numerical solutions to the VIE are paid more and more attention.In recent years, collocation method has been applied to solve VIE by many scholars, and some achievements have been made.In this paper, we study the solution of nonlinear VIEs by collocation method. The existence, uniqueness and convergence of numerical solutions are studied systematically.First of all, we review the background knowledge of some integral equations and integral operators. The existence and uniqueness of analytic solutions of the compact cordial Volterra integral operators and their compact cordial Volterra integral equations are reviewed.In particular, we discuss the compactness of the nonlinear third class of VIE correlation operators.In addition, we give the existence and uniqueness of the nonlinear third class VIE analytic solution.Secondly, the collocation method is applied to the nonlinear third class of VIEs. The solvability of the third type of VIE corresponding collocation equations with compact operators is discussed by using the method similar to that of the second kind of VIE.But for noncompact operators, in order to ensure solvability, we use implicit function theorem in the first interval and the improved geometric mesh method in the following interval.Finally, we discuss the convergence order of collocation method.We define the error function, which satisfies a linear discrete VIE.The convergence is derived from the uniform boundedness of the inverse operator of the linear discrete Volterra integral operator, which can be estimated with an appropriate norm.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.83
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本文編號：1741621