【摘要】：非線性波動方程和二維粘彈性方程是兩大重要的微分方程,被廣泛應用在物理學,經(jīng)濟學,自然學等許多領域。因而有很多學者對這兩類方程做了大量的研究,但是他們只是得到了方程的解的近似值。在本文中,我借助塊中心差分方法對二維粘彈性方程和非線性波動方程在剖分的網(wǎng)格上進行討論,不僅可以得到它們的近似解和解的一階導數(shù)的近似值,而且還得到了近似解的L2模誤差估計,解得一階導數(shù)的近似值具有超收斂性。首先,簡要介紹這篇論文的所討論問題的有關研究背景和基本的理論知識。其次,對有界區(qū)域上的二維粘彈性方程,在非等距剖分的網(wǎng)格上用塊中心差分方法求得了其近似解和解的一階導數(shù)的近似值。從理論上給出了近似解的L2模誤差估計,并且得到了解的一階導數(shù)的近似值具有超收斂性,時間上的精度也有了提高。同理,用塊中心差分方法對有界區(qū)域上的非線性波動方程進行研究,在等距剖分的網(wǎng)格上,也得到了近似解和解的一階導數(shù)的近似值,在理論上給出了近似解的L2模誤差估計,并且得到了解的一階導數(shù)的近似值具有超收斂性。最后,對全文進行小結,并提出以后要進一步研究的問題。
[Abstract]:Nonlinear wave equation and two-dimensional viscoelastic equation are two important differential equations, which are widely used in many fields such as physics, economics, nature and so on. Therefore, many scholars have done a lot of research on these two kinds of equations, but they only get the approximate value of the solutions of the equations. In this paper, I discuss two-dimensional viscoelastic equations and nonlinear wave equations on meshes by means of block central difference method. Moreover, the L2-norm error estimates of the approximate solution are obtained, and the approximate value of the first order derivative is superconvergent. Firstly, the research background and basic theoretical knowledge of the problems discussed in this paper are briefly introduced. Secondly, for the two-dimensional viscoelastic equations in the bounded region, the approximate solutions and the first order derivatives are obtained by using the block central difference method on the non-equidistant meshes. The L2-norm error estimates of the approximate solution are given theoretically, and the approximate value of the first derivative of the solution is superconvergent, and the accuracy of the time is improved. Similarly, the block central difference method is used to study the nonlinear wave equation in the bounded region. The approximation of the approximate solution and the first order derivative of the approximate solution is also obtained on the meshes of equidistant subdivision. In theory, the L2 norm error estimates of the approximate solution are given. The approximate value of the first order derivative of the solution is superconvergent. Finally, the paper summarizes the whole paper and puts forward some problems to be further studied.
【學位授予單位】：河南師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8

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