MatrixPolynomial_Solving_withMATLAB

在数值线性代数中，雅可比方法是一种迭代算法，用于确定严格对角占优线性方程组的解。该方法通过求解每个对角线元素并插入一个近似值，随后迭代该过程直到收敛。此算法是矩阵对角化雅可比变换方法的精简版。该方法以卡尔·古斯塔夫·雅各比（Carl Gustav Jacobi）的名字命名。

Matlab 开发 - Newton-Raphson 方法。牛顿-拉斐逊法用于求解多项式的所有实根。该方法通过迭代不断逼近函数的零点，适用于求解非线性方程的根。具体步骤如下： 定义多项式和它的导数。 选择一个初始猜测值（x0）。 使用 Newton-Raphson 迭代公式： x_{n+1} = x_n - f(x_n)/f'(x_n) 重复步骤3直到满足精度要求。 代码示例： function roots = newtonRaphson(f, df, x0, tol, maxIter) x = x0; for i = 1:maxIter x = x - f(x) / df(x); if abs(f(x)) < tol xss=removed> 该方法常用于数值分析中，能够快速地找到多项式的实根。

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.

The HowarthsTransformation.m file provides a framework for solving boundary layer problems using the Howarth's Transformation. The function takes the following parameters: y3y5_0 = HowarthsTransformation(rhofun, miufun, hw, M, Pr, Gamma, y3y5_0guess), with default values:- rhofun = @(h) h^(-1)- miufun = @(h) h^(2/3)- hw = 2- M = 0- Pr = 0.7- Gamma = 1.4- y3y5_0guess = [0.1; 2] Example 1: y3y5_0 = HowarthsTransformation() (using only the default values);Example 2: y3y5_0 = HowarthsTransformation with custom input parameters. For more details, please refer to the Wikipedia page on Howarth's transformation: https://en.wikipedia.org/wiki/Blasius_boundary_layer#Howarth_transformation

This code uses the Newton-Raphson method to calculate the roots of transcendental equations. The method includes enhanced features, such as handling cases where the function's derivative disappears, or when the initial approximation is poor, leading to infinite loops due to the non-existence of the derivative or roots. It is recommended to use the Symbolic Toolbox for better accuracy and handling of symbolic differentiation in MATLAB.

给定组装质量、刚度矩阵和阻尼矩阵，n个自由度的外部载荷矢量，系统对外部载荷的响应使用Wilson Theta逐步积分法计算。在时间步长上，可以根据问题更改Theta值，以调整数值精度和稳定性。 步骤：1. 输入质量、刚度和阻尼矩阵。2. 设定外部载荷矢量。3. 选择合适的Theta值（通常为0.5、1.0等）。4. 逐步计算系统响应。5. 输出计算结果并进行分析。

The following differential schemes are used to solve the given problem: First Scheme: Combining equations (18), (21), and (22), we derive a differential scheme for solving problem (7). The general form is given as follows:[u^{n+1} = u^n + \Delta t \left( -\left( \frac{d^2 u}{dx^2} \right)^n \right)]In this case, the value of ( u_0 ) at layer ( j=0 ) is known. Using the above scheme, we can compute approximate values for nodes in the first layer ( j=1 ), then continue calculating successively for each layer. Classical Implicit Scheme: Rearranging equation (19) and combining with equations (21) and (22), we arrive at the following implicit differential scheme:[u^{n+1} = u^n + \Delta t \left( -\left( \frac{d^2 u}{dx^2} \right)^n + f(x, t) \right)]In this implicit scheme, though ( u_0 ) at the 0th layer is known, the calculation of values for subsequent layers ( j \geq 1 ) cannot be done directly. Hence, this scheme is termed the classical implicit format. Dufort-Frankel Scheme: The Dufort-Frankel scheme is a three-layer explicit scheme derived by combining equations (24), (25), and (26). Its specific form is:[u^{n+1} = u^n + \Delta t \left( \frac{1}{2} \left( \frac{d^2 u}{dx^2} \right)^n \right)]In this scheme, the value ( u_0 ) at the 0th layer is determined by the initial condition, and then values for subsequent layers are computed iteratively, starting from the 1st layer using a two-layer format. Hyperbolic Equation and Differential Solutions: For the second-order wave equation:[\frac{\partial^2 u}{\partial x^2} = a^2 \frac{\partial^2 u}{\partial t^2}]we define:[v = \frac{\partial u}{\partial x}, \quad \frac{\partial v}{\partial t} = \frac{\partial u}{\partial t}]This transforms the equation into a first-order linear system of hyperbolic equations."

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.