Question

我在编写这个matlab差分问题的精确等式时遇到了麻烦。我正在使用Forward Euler，Backward Euler和Crank-Nicolson方法。一位朋友暗示如何编写确切的解决方案，但我仍然遇到麻烦。目前我的错误是“Matrix Dimensions必须同意”。我已经尝试过一些试验和错误，但到目前为止还没有任何工作。错误在倒数第二行。有人可以引导我朝着正确的解决方案方向迈进吗？

clear;                                 % clears all memory
clc;                                   % clears Command Window screen
close all;                             % closes all open figure windows 

for input_dt = 1:1:3
    prompt = 'enter dt: ';


% 1. Initialize
dt          =  input(prompt);          % choose delta t
t_start     = 0;                       % starting time
t_end       = 50;                       % end time
nstep       = ( t_end - t_start ) / dt;    
x           = zeros(2,1);
y           = zeros(2,1);
z           = zeros(2,1);
b           = zeros(2,1);
I           = eye(2);                       
all_t       = ( 1 : nstep) * dt;
all_Euler_x = zeros( nstep, 2);
all_exact = zeros( nstep, 2);
all_Euler_y = zeros( nstep, 2);
all_Euler_z = zeros( nstep, 2);
initial_displacement = 0.5;
initial_velocity = 1.0;

% 1.2 System Characteristics
m        = 1.0;
c        = 0.05;
k        = 4.0;
A(1,1) = 1.0 ; A(1,2) = 0.5;
A(2,1) = -k/m; A(2,2) = -c/m;

x(1) = initial_displacement;
x(2) = initial_velocity;


% 2. Forward Euler time integration
for n = 1 : nstep     
    x = x + dt * (A * x + b);     
    all_Euler_x (n, :) = x;                  
end

y(1) = initial_displacement;
y(2) = initial_velocity;

for n = 1 : nstep
    y = (I - dt * A) \ (y + dt * b);
    all_Euler_y (n, :) = y;
end

z(1) = initial_displacement;
z(2) = initial_velocity;

for n = 1 : nstep
    z = 0.5*(y + x); %Crank-Nicolson
    all_Euler_z (n, :) = z;
end


%3. Compute the exact solution

%3.1 Parameters needed to compute the exact solution
c_c      = 2.0 * sqrt (k*m);
zeta     = c / c_c;
omega_n  = c_c / (2.0 * m );
c1       = (initial_velocity + zeta * omega_n * initial_displacement) / (omega_n * sqrt (1.0 - zeta^2));     
c2       = initial_displacement;

%3.2 exact solution
for n = 1 : nstep
    t = n * dt;
    all_exact (n,1) =  all_exact - all_Euler_x / all_Euler_y / all_Euler_z; %Error on this line
end

编写Matlab微分方程的精确方程

0 个答案: