【作者】 高博；

【导师】 曲小钢；

【作者基本信息】 西安建筑科技大学 ， 计算数学， 2011， 硕士

【摘要】 本论文研究了广义Burgers-Fisher方程初边值问题的Haar小波有限差分法。广义Burgers-Fisher方程作为描述反应机理、对流效应和扩散传播之间相互作用的典型模型,在现代物理学中具有重要的意义。本文利用周期小波定义了闭区间上的Haar小波系,对Haar小波系进行积分,从而建立了新的Haar小波n重积分矩阵,并将该矩阵应用到Haar小波有限差分法中。采用一阶精度的向后差分逼近时间微分,而空间导数则利用Haar小波系来展开,提出了一个求解广义Burgers-Fisher方程初边值问题的Haar小波有限差分法,进而给出了相应的算法和程序框图,编制了MATLAB程序代码。本文还分析了Haar小波有限差分法的稳定性,数值验证了文中所给出的算法是条件稳定的,并对算法进行了正定性和有界性测试,结果表明文中算法能够保持数值解的正定性和有界性。本文算法充分结合了Haar小波多分辨率分析计算灵活、计算效率高的特性和有限差分法易于实现的优点,同时,由于利用了Haar小波n重积分矩阵的稀疏性,因而有效地提高了计算速度和精度。通过计算机模拟所获得的数值结果,与有限差分法和Adomian分解法等方法进行比较,显示本文所给出的算法精度高于有限差分法,对于求解较小的时间问题时,精度高于Adomian分解法,这说明文中所给出的算法是可行的、有效的,并且精度较高,为研究广义Burgers-Fisher方程初边值问题的数值解法提供了新的途径和新的思路。更多还原

【Abstract】 In this paper, the numerical solution of the generalized Burgers-Fisher equation with the initial boundary value conditions based on the Haar wavelet-finite difference method is studied. The generalized Burgers-Fisher equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. It has the extremely vital significance in the modern physics. The Haar wavelet family on the closed interval is given. The Haar wavelet matrix of n-tuple integration is established. Haar wavelet-finite difference method adopted the matrix of n-tuple integration. The matrix is to convert the integral operations into the matrix operations. According to this method the spatial operators are approximated by the Haar wavelet family and the time derivation operators by the first-order precision backward difference quotient. The Haar wavelet-finite difference method is proposed to solve the generalized Burgers-Fisher equation with the general initial boundary value conditions. The paper gives the algorithm, program diagram and MATLAB code.This paper analyzes the stability of Haar wavelet-finite difference method, and numerical proves that the algorithm is conditional stability. The algorithm property of the positivity of the numerical solutions and their boundedness is testing. The results show that the algorithm can maintain property of the positivity of the numerical solutions and their boundedness.This paper fully combines the features of Haar wavelet multi-resolution analysis flexible, efficiency and the advantage of finite difference easy to realize. This method improves the computation speed and accuracy. It is due to the sparsity of the Haar wavelet matrix of n-tuple integration. Numerical results, obtained by computer simulation, are compared with the finite difference method and the Adomian decomposition method. The result showed that the method for solving problems of small time is more than Adomian decomposition method. The algorithm precision is higher than the finite difference method. The algorithm is feasible, effective and high accuracy. In order to study the numerical solution for the generalized Burgers-Fisher equation with initial boundary value problem providing new ideas and new methods.更多还原