本文選題：微分方程 + 積分方程��； 參考：《蘭州大學(xué)》2016年博士論文

【摘要】：小波數(shù)值方法是近二十多年來發(fā)展起來的一類新興數(shù)值方法。隨著其自身的發(fā)展,小波數(shù)值方法的應(yīng)用范圍越來越廣泛。而發(fā)展統(tǒng)一求解弱非線性和強(qiáng)非線性問題的小波方法這一重要課題也越來越受到重視。立足于小波封閉解法的基礎(chǔ)之上,本文拓展了小波方法在具有非線性、奇異性及微分積分算子共存的復(fù)雜力學(xué)問題中的應(yīng)用。另外,通過改進(jìn)小波逼近方式和提出新的求解思路,本文針對(duì)一般非線性初值問題和邊值問題分別提出了新的高精度小波算法。本文首先介紹了緊支正交的Coiflet小波函數(shù)基及其具有擬插值特性的小波逼近公式,它們是小波封閉解法的理論基礎(chǔ)。接著介紹了構(gòu)造有限區(qū)間上平方可積函數(shù)Coiflet小波逼近公式的邊界延拓技術(shù),它是小波數(shù)值方法的應(yīng)用基礎(chǔ)。數(shù)值研究表明消失矩?cái)?shù)目為6的Coiflet是現(xiàn)有小波方法較好的基函數(shù)選擇。在這些基礎(chǔ)之上,本文通過將非線性項(xiàng)中的導(dǎo)數(shù)定義為新函數(shù),拓展了現(xiàn)有小波方法在一維和二維擬線性微分方程中的應(yīng)用,以及結(jié)合分部積分和函數(shù)變換等技術(shù)和小波伽遼金法,還提出了非線性奇異積分方程的幾類高精度小波方法。而通過十余個(gè)具體數(shù)值算例和與其他方法的對(duì)比均顯示了這些小波方法在計(jì)算精度和收斂性方面的優(yōu)勢(shì)。非線彈性梁桿的大撓度彎曲屈曲問題和矩形薄板的大變形問題均是現(xiàn)代工程中的典型結(jié)構(gòu)非線性問題,細(xì)胞特異性粘附問題是具有彈性-隨機(jī)耦合特性的非線性生物力學(xué)問題。本文發(fā)展的小波方法提供了定量求解這些問題的技術(shù)。在分析屈曲問題時(shí),小波方法得到的離散代數(shù)方程組形式簡(jiǎn)單,便于結(jié)合擴(kuò)展系統(tǒng)法來直接求解屈曲問題中的臨界荷載。在分析大變形問題時(shí),小波方法相對(duì)于傳統(tǒng)的有限元方法具有更高的計(jì)算效率且不出現(xiàn)剪力鎖死現(xiàn)象。在分析粘附問題時(shí),小波方法提供了穩(wěn)定狀態(tài)下細(xì)胞間歸一化的力與界面位移非線性關(guān)系的定量描述。同時(shí)可以注意到在具體的求解過程中,本文的小波方法均能處理任意形式的非線性項(xiàng)以及具有對(duì)問題非線性強(qiáng)弱特征不敏感的特性。最后通過推導(dǎo)基于Coiflet的數(shù)值微分公式,提高了有限區(qū)間上平方可積函數(shù)小波逼近公式的逼近精度。在此基礎(chǔ)之上,本文構(gòu)造了一般非線性初值問題的小波時(shí)間積分法,并結(jié)合空間離散的小波伽遼金法提出了非線性初邊值問題的小波時(shí)空統(tǒng)一求解法。理論分析表明,該小波時(shí)間積分法具有N階精度和良好的穩(wěn)定性。數(shù)值算例則表明,該小波方法適用于追蹤激波或者孤立波等劇烈變化的時(shí)空演化問題。另外,本文還提出了求解一般邊值問題的新的高精度小波積分配點(diǎn)法。理論分析和數(shù)值算例均表明,該小波積分配點(diǎn)法的收斂速度大約為O(2~(-nN)),n為小波分解尺度,N為Coiflet小波消失矩階數(shù)。與之前的小波伽遼金法相比,小波積分配點(diǎn)法不僅提高了方程的求解精度而且其收斂階數(shù)與方程的階數(shù)無關(guān)。
[Abstract]:The wavelet numerical method is a new kind of new numerical method developed in the last more than 20 years. With its own development, the application range of the wavelet numerical method is becoming more and more extensive. And the important topic of the wavelet method to develop the unified solution to the weak nonlinear and strong nonlinear problems is becoming more and more important. Based on the wavelet closed method On the basis of this, this paper extends the application of wavelet method to the complex mechanics problem with nonlinear, singular and differential integral operators. In addition, a new high precision wavelet algorithm is proposed for the general nonlinear initial value problem and the boundary value problem by improving the method of wavelet approximation and the new solution. This paper first introduces the Coiflet wavelet function base of tight branch orthogonal and the wavelet approximation formula with quasi interpolation properties. They are the theoretical basis of the wavelet closed method. Then, the boundary extension technique for constructing the Coiflet wavelet approximation formula of the square integrable function on the finite interval is introduced. It is the application basis of the small wave numerical method. Coiflet with the number of vanishing moments is 6 is a better basis function choice for the existing wavelet method. On these basis, by defining the derivative in the nonlinear term as a new function, the application of the existing wavelet method in the one and two dimensional quasilinear differential equations is extended, as well as the combination of partial integral and function transformation and small wave gamma. Several high precision wavelet methods for nonlinear singular integral equations are also proposed by the Liao and Jin method. Through more than ten specific numerical examples and the comparison with other methods, the advantages of these methods in calculating precision and convergence are shown. The large deflection flexural buckling of non linear elastic beams and the large deformation of rectangular thin plates It is a typical structural nonlinear problem in modern engineering. The problem of cell specific adhesion is a nonlinear biomechanical problem with elastic random coupling characteristics. The wavelet method developed in this paper provides a quantitative solution to these problems. In the analysis of the problem of buckling, the discrete algebraic equations obtained by the wavelet method are simple. In the analysis of the large deformation problem, the wavelet method has higher computational efficiency and no shear locking phenomenon when analyzing the large deformation problem. When analyzing the adhesion problem, the wavelet method provides the force and the interface position of the cell normalization under the stable state. At the same time, it can be noted that in the specific solving process, the wavelet method in this paper can both deal with any form of nonlinear term and is insensitive to the nonlinear strong and weak characteristic of the problem. Finally, the square integrable function on the finite interval is improved by deriving the Coiflet based numerical differential common formula. On this basis, the wavelet time integration method for the general nonlinear initial value problem is constructed, and the wavelet space-time unified solution method for nonlinear initial boundary value problem is proposed with the space discrete wavelet Galerkin method. The theoretical analysis shows that the wavelet time integration method has the N order accuracy and good quality. The numerical example shows that the wavelet method is suitable for the spatio-temporal evolution of intense changes in the shock wave or the solitary wave. In addition, a new high precision wavelet integral point method for solving general boundary value problems is also proposed. Both theoretical analysis and numerical examples show that the convergence rate of the wavelet integral point method is about O (2~). -nN)), n is a wavelet decomposition scale, and N is the order of vanishing moment of Coiflet wavelets. Compared with the previous small Galerkin method, the wavelet integral point method not only improves the solution accuracy of the equation, but also has nothing to do with the order of the equation.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O344.1;O241
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本文編號(hào)：2063856