Q5.3-18E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Combat Model. A simplified mathematical model for conventional versus guerrilla combat is given by the system $\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{;} {\mathbf{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10}\mathbf{;} \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}\mathbf{;}{\mathbf{x}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{15}$ where ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the combat effectiveness coefficients. Who will win the conflict: the conventional troops or the guerrillas? [Hint: Use the vectorized Runge–Kutta algorithm for systems with h=0.1 to approximate the solutions.]

Step-by-Step Solution

Answer

The solution is convention loop will win.

Transform the equation

Write the equation as $x{\text{'}}_1=-\left(0.1\right)x_1{x}_2$ and $x{\text{'}}_2=-x_1$

The initial conditions are:

$$\begin{gathered} {\mathbf{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10} \\ {\mathbf{x}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{15} \end{gathered}$$

Apply Runge –Kutta method

For the solution, apply the Runge-Kutta method in Matlab for h=0.1.

Now the algorithm is:

 

Function n[t,x] = Runge_Kutta(f,t0,t_end,init_cond,h)
 %we begin at time t0 and end when we reach t_end
 %init_cond(i) contains the initial value of x_i
 %f contains functions such that x_i'=f_i(t,x1,x2,...)
 
 t(:,1)=t0; % t0 is the initial value of t
 x(:,1)=init_cond; % initial conditions are set
 
 i=1;
while t(:,i) < t_end 
 
     k1=f(t(i),x(:,i));
     k2=f(t(i)+0.5*h,x(:,i)+0.5*h*k1);
     k3=f(t(i)+0.5*h,x(:,i)+0.5*h*k2);
     k4=f(t(i)+h,x(:,i)+h*k3);
 
      x(:,i+1)=x(:,i)+(h/6)*(k1+2*k2+2*k3+k4);

  t(:,i+1)=t(:,i)+ h;
i=i+1;
 
 end

 

Now for the solution 

clear all
 
 init_cond=[10;15];
 f=@(t,X) [-0.1*X(1)*X(2);-X(1)];
 [t,x] = Runge_Kutta(f,0,10,init_cond,0.1);
 
 plot(t,x(1,:),t,x(2,:));
 legend('guerilla','conventional')

 

Therefore, the convention loop will win.