Numerical Solutions of PDEs in MATLAB

3.4 Numerical Solutions of Linear Systems The solution of linear systems of equations is a topic that is widely applicable not only in engineering and technology but also in many other fields. There are two main categories of numerical methods for solving linear systems: Direct methods, where an accurate solution to the system is found through a finite number of arithmetic operations, assuming no rounding errors. Direct methods include matrix division methods and elimination methods. Iterative methods, where an initial guess for the solution is provided, and successive approximations are made to refine the solution over iterations.

本书提供了用Matlab进行数值计算的丰富资料，内容可读性、知识性和实用性都非常强。

很不错的Matlab代码，可以很好的解决奇异值分解问题。

数值方法(MATLAB版)(第四版)中文版.pdf

3.2 Genetic Operators (1) Crossover Operator The crossover operator randomly pairs individuals from the parent population for crossover operations, generating ( m ) offspring individuals to form the next generation. Two types of crossover are employed: single-point crossover and two-point crossover. Given two individuals for crossover ( P = {p_1, p_2, p_3, \dots, p_n} ) and ( Q = {q_1, q_2, q_3, \dots, q_n} ), a random crossover point ( b_1 ) is chosen from the range [1, n] for single-point crossover. The elements before ( b_1 ) in ( P ) are copied to offspring individual ( \text{new Individual1} ), while the remaining elements are copied from ( Q ). Similarly, a second offspring ( \text{new Individual2} ) is generated by swapping the roles of ( P ) and ( Q ). In two-point crossover, two random crossover points ( b_1 ) and ( b_2 ) are chosen, and the elements between ( b_1 ) and ( b_2 ) in ( P ) are copied to the offspring, with the remaining elements taken from ( Q ). (2) Mutation Operator After the crossover operation, two mutation operators are applied to the offspring individuals. The first is rotation mutation, where a random position ( \text{bit} ) is chosen, and with probability ( p_m1 ), the portion of the individual after ( \text{bit} ) is rotated. The second is position mutation, with a smaller probability ( p_m2 ), two integers ( \text{bit1} ) and ( \text{bit2} ) are randomly chosen from the range [1, n], and the corresponding parts of the individual are swapped. (3) Selection Operator The fitness of the mutated offspring individuals is evaluated using the lowest level line method. The parent and offspring individuals are ranked by their fitness in descending order, and the top ( m ) individuals are selected as the next generation's parents. 3.3 Termination Criteria The steps in sections 3.2(1), 3.2(2), and 3.2(3) are repeated until the fitness of the best solution meets the required threshold or the pre-defined number of generations is reached. At this point, the optimal solution is output. 4. Case Study To test the performance of the algorithm, two cases from literature [3] are solved. In Case 1, a large rectangle of size ( 15 \times 40 ) is divided into 25 smaller rectangles. Based on the lowest level line method, the corresponding coding sequence is ( \text{Opt} = {1, -9, 11, -15, 17, -24, -25, -10, -14, -22, -23, -2, -3, -5, 18, 7, -8, -12, 19, -20, 21, 6, 13, 4} ). The width is set at 40, and height considerations follow suit for the genetic algorithm implementation.

目前的脚本近似于原始和相关联的对偶（伴随）布拉修斯方程，如Kuehl等人[~11/2020]在关于“连续伴随补充到布拉修斯方程”中的研究所述。数值边值问题使用射击方法近似，其中要解决的初值问题采用4阶Runge-Kutta方法(RK4)。

在Volterra的封闭系统中，人口增长模型的无量纲形式为 k(du/dt) = u - u^2 - u ∫_0^t u(x) dx。该GUI允许用户输入初始总体 u0、无量纲常数 k、最终时间 Tmax 和网格点数 M。通过单击适当的按钮，用户可以使用各种数值方法生成图。 \"毒性项\" 是积分 ∫_0^t u(x) dx。面板“毒性术语的梯形规则”和“毒性术语的辛普森规则”首先对毒性术语应用正交规则，然后使用指定的数值方法求解所得系统。有关更多信息，请参阅 (1)。有关问题的全面分析，请参阅： 1. Kevin G. TeBeest，Volterra*人口模型的数值和解析解，SIAM Rev. 39 (1997)，第1期。3, 484-493。 2. RD Small，《封闭系统中的人口增长》，SIAM评论25（1983），第1期。1, 93-95。

This method can solve various partial differential equations and represents the latest numerical solution techniques. It is based on MATLAB programming, making it easier to understand and implement. By utilizing MATLAB, complex mathematical models become more accessible and the process of solving PDEs is streamlined for better clarity and efficiency.

This code solves the Fokker-Planck equation for the dynamic mass-spring-damper system depicted in the ForceBalance.png, considering only a single 'substrate' interaction. It can be used to validate the numerical simulations of the modified multi-bond model under the condition of a single substrate interaction. A publication with full explanations is soon to be submitted. Instructions for use: The following MATLAB files should be located in the same folder:- CallFokkerPlanckPDEMovingBC(Periodic or Harmonic)Potential.m- PlotAllProbs.m- CalcResults(Periodic or Harmonic)Potential.m In lines 13-21, adjust the desired mechanical and dynamic parameters. Modify the spatial and temporal resolution in lines 24 and 25, respectively. Run CallFokkerPlanckPDEMovingBC to begin the simulation.