本文關(guān)鍵詞：偏積分微分方程擬小波及緊致差分方法　出處：《湖南師范大學(xué)》2016年博士論文　論文類(lèi)型：學(xué)位論文

  更多相關(guān)文章： 積分微分方程 擬小波方法 向后歐拉方法 緊差分方法 弱奇異核 交替方向法

【摘要】：隨著科學(xué)技術(shù)的發(fā)展,人類(lèi)發(fā)現(xiàn)了一類(lèi)重要的方程：積分微分方程.這類(lèi)方程出現(xiàn)在很多領(lǐng)域,如：具有記憶材料中的熱傳導(dǎo)、粘彈性力學(xué)、人口動(dòng)力學(xué)等問(wèn)題.由于這類(lèi)方程的積分微分項(xiàng)通常帶有弱奇異性,因此這類(lèi)方程的求解是一件困難的事情.而解決這類(lèi)積分微分方程也是當(dāng)今研究的熱門(mén)課題之一本文主要采用擬小波方法、緊差分方法、有限差分方法來(lái)求解三類(lèi)不同的積分微分方程.第一類(lèi)方程是無(wú)界區(qū)域上的帶正定記憶項(xiàng)的積分微分方程.第二類(lèi)和第三類(lèi)方程均為帶弱奇異項(xiàng)的積分微分方程.本文由七章組成.第一章主要介紹積分微分方程的歷史背景、研究動(dòng)態(tài),以及本論文的研究?jī)?nèi)容的結(jié)構(gòu).第二章和第三章主要采用擬小波方法研究第一類(lèi)積分微分方程,第四章主要采用緊致差分方法研究第二類(lèi)方程,第五章主要采用交替方向隱式歐拉方法及有限差分方法研究第三類(lèi)方程,第六章主要采用交替方向隱式Crank-Nicolson方法及有限差分方法研究第三類(lèi)方程.第七章為總結(jié)與展望.在第二章對(duì)第一類(lèi)方程的研究中,我們首次采用了擬小波方法求解一類(lèi)無(wú)界區(qū)域上的積分微分方程.首先在無(wú)界區(qū)間上任意選取一定點(diǎn)x0,再取定任意正常數(shù)p,這樣我們就可以得到無(wú)界區(qū)域上的一有限區(qū)域[x0-p,x0+p]這樣我們就可以在這段區(qū)域上求出方程的數(shù)值解.一方面,由于該有限區(qū)域是任意確定的,而且擬小波是局部可解的,因此在端點(diǎn)附近的值與準(zhǔn)確值之間存在較大的誤差,為了得到更準(zhǔn)確的數(shù)值解,我們?cè)谇蠼庹`差時(shí)將舍去一些誤差較大的數(shù)值解,這在后文中將詳細(xì)描述.另一方面,由于數(shù)x0，p的任意性,我們可以模擬出無(wú)界區(qū)域上任意區(qū)間的數(shù)值解.在對(duì)方程進(jìn)行數(shù)值求解時(shí),我們?cè)跁r(shí)間上采用向前歐拉方法離散,空間方向采用擬小波方法離散,對(duì)于積分項(xiàng)采用三角形法則來(lái)逼近,即取右端點(diǎn)上的積分值作為逼近值.在這章中我們不僅求解了一維模型,還研究了二維模型,并給出了幾個(gè)數(shù)值例子加以證明.第三章我們采用Crank-Nicolson方法結(jié)合擬小波方法對(duì)第二章同類(lèi)無(wú)界區(qū)域上的積分微分方程進(jìn)行了研究探討.我們同樣采用上述方法來(lái)處理無(wú)界區(qū)間.時(shí)間方向采用Crank_Nicolson方法,空間方向同樣用擬小波方法,而積分項(xiàng)則采用連續(xù)分片線性插值逼近方法來(lái)逼近.在這一章里我們同樣研究了一維和二維模型,并采用了第二章的數(shù)值例子進(jìn)行比較.不管是在工程還是在數(shù)學(xué)領(lǐng)域,有限差分法被大部分工程師和研究學(xué)者采納用來(lái)解決各類(lèi)問(wèn)題.這是因?yàn)椴罘址ㄊ且环N簡(jiǎn)單有效的辦法,所以被大部分人采用.同樣過(guò)去許多學(xué)者采用差分法研究了積分微分方程,然而采用緊差分方法求解弱奇異積分微分方程卻工作較少.由于緊差分是一種四階格式,而有限差分方法是一種二階格式,因此本文第四章主要采用緊差分方法代替有限差分方法求解帶弱奇異項(xiàng)的積分微分方程.在得到離散格式后,我們還嚴(yán)格證明了全離散格式的穩(wěn)定性和收斂性分析.并用數(shù)值例子證明了分析的準(zhǔn)確性.在了解到弱奇異積分微分方程在時(shí)間上總是達(dá)不到豐滿階之后,我們對(duì)一種新的區(qū)間剖分方法產(chǎn)生興趣：等級(jí)網(wǎng)格剖分方法.這是因?yàn)椴捎玫燃?jí)網(wǎng)格方法剖分區(qū)間,使得區(qū)間在奇異點(diǎn)附近比較密集,在遠(yuǎn)離奇異點(diǎn)附近比較稀疏,這樣有效的彌補(bǔ)了方程的解的奇異性.另一方面,在了解到求解二維問(wèn)題需要大量的存儲(chǔ)空間來(lái)存儲(chǔ)過(guò)去的數(shù)據(jù)之后,我們想到了用交替方向隱式有限差分法來(lái)緩解存儲(chǔ)問(wèn)題,減少程序運(yùn)行時(shí)間.因此,在第五章采用基于非一致網(wǎng)格的交替方向有限差分法來(lái)研究弱奇異問(wèn)題,在時(shí)間方向采用隱式歐拉格式,空間方向采用二階差分格式離散,得到收斂階為O(k+h2x+h2y)還證明了該方法的穩(wěn)定性和收斂性.最后我們給出了兩個(gè)數(shù)值例子加以證明.第六章是在第五章的基礎(chǔ)上對(duì)等級(jí)網(wǎng)格法處理弱奇異核問(wèn)題的進(jìn)一步研究.第六章我們進(jìn)一步研究了二階Crank-Nicolson格式來(lái)處理弱奇異核問(wèn)題：時(shí)間方向采用交替方向隱式Crank-Nicolson格式替換隱式歐拉格式離散,空間方向同樣采用二階差分格式離散,這樣得到的離散格式具有收斂階O(k2+h2x+h2y)在這章同樣給出了該離散格式的穩(wěn)定性和收斂性分析,并采用第五章的兩個(gè)例子證明理論分析的可靠性.
[Abstract]:With the development of science and technology, human has found an important class of equations: Integro differential equations. These equations appear in many areas, such as: heat conduction in the material with memory, viscoelasticity, population dynamics and so on. Because of these integral differential equation usually with weak singularity, thus solving the equation is difficult. To solve this class of Integro differential equations is the current hot topic of research. This paper mainly adopts quasi wavelet method, compact difference method, finite difference method to solve the three kinds of integral differential equations. The first equation is positive definite integral differential equation of unbounded memory the band. Class second and third kinds of equations are Integro differential equations with weakly singular term. This paper consists of seven chapters. The first chapter mainly introduces the historical background, Integro differential equations and the dynamic research. The structure of the research content. The second chapter and the third chapter mainly adopts quasi wavelet method of first class of Integro differential equations, the fourth chapter mainly uses the compact difference method of second kinds of equations, the fifth chapter mainly adopts alternating direction implicit Euler method and the finite difference method of third kinds of equations, the sixth chapter mainly adopts the alternating direction implicit the Crank-Nicolson method and the finite difference method of third kinds of equations. The seventh chapter is the summary and outlook. In the second chapter of the first equation, we first use the quasi wavelet method for solving a class of Integro differential equations on unbounded domains. First in the unbounded interval arbitrary point x0, then any normal number P, so we can get a limited area of [x0-p on unbounded domain x0+p], so that we can in this area for the numerical solution of the equation. On the one hand, due to the The limited area is arbitrarily determined, and the quasi wavelet is locally solvable, so near the end of the value and the accurate value between the larger error, in order to get a more accurate numerical solution, we will give some numerical error in solving the solution error, which will be described in detail in this paper after another. Hand, as the number of x0, any p, we can simulate the numerical solution of arbitrary interval on unbounded domains. In the numerical solution of the equation, we use the forward Euler method in time discretization, spatial direction by using the quasi wavelet method for discrete integral using triangle rule approximation, namely takes the integral right the endpoint value as the approximate value. In this chapter, we not only solve the one-dimensional model, also studied the two-dimensional model, and several numerical examples are given to prove it. In the third chapter, we use the Crank-Nicolson method combined with quasi wavelet. The study of integral differential equations on unbounded domains in chapter second similar method. We also used the method to deal with unbounded intervals. The direction of time by using Crank_Nicolson method, spatial direction with the same quasi wavelet method, and integral using continuous piecewise linear interpolation approximation method to approximate. In this chapter we also study on a two-dimensional model, and using numerical examples. The second chapter analyzed. Whether in engineering or in the field of mathematics, the finite difference method is adopted to most engineers and researchers to solve all kinds of problems. This is because the difference method is a simple and effective way, so most people are using. The same past many scholars by difference method of integral differential equation to study, but the compact finite difference method for weakly singular integral differential equation has less work. Because of the compact difference is A four order scheme, and the finite difference method is a kind of two order format, so the fourth chapter of this paper mainly adopts the compact difference method instead of the finite difference method for solving Integro differential equations with weakly singular terms. In the discrete format, we strictly prove the stability and convergence of the fully discrete scheme. The accuracy of the analysis is proved by a numerical example. After understanding to weakly singular Integro differential equation in time always reach the fullness of our order, a new interval subdivision method: interest level mesh subdivision. This is because the level of region subdivision grid method, makes the interval in the singular point near the more intensive, relatively sparse in the vicinity of far away from the singular points, thus effectively compensate for equations singularity. On the other hand, to solve the two-dimensional problem in understanding the needs of large storage space to store the data in the past After that, we thought of using the alternating direction implicit finite difference method to solve the storage problem, reduce the running time of the program. Therefore, to study the problem of using weakly singular non uniform grid alternating direction based on finite difference method in the fifth chapter, using the implicit Euler scheme in time, space by two order difference scheme get the order of convergence is discrete, O (k+h2x+h2y) also proved the stability and convergence of the method. Finally, we give two numerical examples to prove it. The sixth chapter is on the basis of the fifth chapter with further study the problem of weakly singular kernel level grid method. In the sixth chapter, we further study the two order Crank-Nicolson format to deal with weakly singular kernel problem: the time direction by alternating direction implicit Crank-Nicolson scheme to replace the implicit Euler scheme, spatial direction using the same two order difference scheme, this kind of The discrete scheme has convergence order O (k2+h2x+h2y). In this chapter, the stability and convergence analysis of the discrete scheme are also given. Two examples of the fifth chapter are used to prove the reliability of the theoretical analysis.

【學(xué)位授予單位】：湖南師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O241.82

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