MATLAB Mathematics - SERC - Index of

5 Differential Equations
The MATLAB kneeode Demo. The MATLAB kneeode demo solves the “knee
problem” by imposing a nonnegativity constraint on the numerical solution.
The initial value problem is
ε *y' = (1-x)*y - y^2, y(0) = 1
For 0 < ε < 1, the solution of this problem approaches null isoclines y = 1 -
x and y = 0 for x < 1 and x > 1 respectively. The numerical solution, when
computed with default tolerances, follows the y = 1 - x isocline for the whole
interval of integration. Imposing nonnegativity constraints results in the
correct solution.
Here is the code that comprises the kneeode demo:
function kneeode
%KNEEODE The "knee problem" with Nonnegativity constraints.
% Problem parameter
epsilon = 1e-6;
y0 = 1;
xspan = [0, 2];
% Solve without imposing constraints
options = [];
[x1,y1] = ode15s(@odefcn,xspan,y0,options);
% Impose nonnegativity constraint
options = odeset('NonNegative',1);
[x2,y2] = ode15s(@odefcn,xspan,y0,options);
figure
plot(x1,y1,'b.-`,x2,y2,'g-')
axis([0,2,-1,1]);
title('The "knee problem"');
legend('No constraints','nonnegativity')
xlabel('x');
ylabel('solution y')
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