爲什麼此係統解算器無法正常工作？

Question

syms a(t) b(t) c(t) d(t) e(t) f(t) 

S = dsolve(
     diff(a) == -kd*a*b^2, ... 
     diff(b) == -kf*b*c^2, ... 
     diff(c) == kd*a*b^2 - ke*a*d, ... 
     diff(d) == kd*a*b^2, ... 
     diff(e) == -ke*a*d, ... 
     diff(f) == kf*b*c^2) 

它返回此錯誤：

Warning: Explicit solution could not be found. 
> In dsolve at 194 
    In hw8 at 14 

S =  
    [ empty sym ] 

Answer 1

微分方程is not easily solved analytically的這種非線性系統。您可以嘗試Mathematica如果有的話（它往往是在符號數學比MATLAB的MuPad越好）：

DSolve[{a'(t) = -k*d(t)*a(t)*b(t)^2, b'(t) = -k*f(t)*b(t)*c(t)^2, c'(t) = k*d(t)*a(t)*b(t)^2 - k*e(t)*a(t)*d(t), d'(t) = k*d(t)*a(t)*b(t)^2, e'(t) = -k*e(t)*a(t)*d(t), f(t) = k*f(t)*b(t)*c(t)^2}, {a(t),b(t),c(t),d(t),e(t),f(t)}, t] 

（我不能對此進行測試，因爲輸入過長的Wolfram|Alpha免費版本:)

不管怎麼說，這是很容易做到數字它MATLAB：

function top 

    %// Initial values (random for this example) 
    y0 = 125*randn(6,1); 

    %// Time span to simulate 
    tspan = [0 +1]; 

    %// Solve system numerically 
    [t,y] = ode45(@deriv, tspan, y0); 

    %// Make a nice plot 
    plot(t,y) 
    xlabel('t'), ylabel('function values') 
    legend('a(t)', 'b(t)', 'c(t)', 'd(t)', 'e(t)', 'f(t)') 

end 

function dydt = deriv(~,y) 

    %// Set the value for your 'k' 
    k = 1e-4; 

    %// rename the variables for clarity 
    [a,b,c,d,e,f] = deal(y(1),y(2),y(3),y(4),y(5),y(6)); 

    %// Compute the derivative 
    dydt = k * [ 
     -d*a*b^2 
     -f*b*c^2 
     +a*(d*b^2 - e*d) 
     +d*a*b^2 
     -e*a*d 
     +f*b*c^2 
    ]; 

end 

其中funkier解決方案，我用這個得了的：

出於好奇：做這些方程描述？我想說的經歷化學反應物質的濃度，但它會是非常奇怪的（對於一些初始值，你會發現負濃度，奇異等，東西你只是不會在這些系統中期望，所以...我的好奇心被觸發:)