To use the fourth-order Runge-Kutta method to solve this differential equation with the given boundary conditions, you can use the following steps:
- Define the dependent variable y as h and the independent variable x as x.
- Define the initial conditions y(0) = h0 and z(0) = 0.
- Define the constants a, c, and L.
- Define the step size dx as a small positive value.
- Set up a loop to iterate over x values from 0 to L, with a step size of dx.
- Inside the loop, use the current values of y and z to calculate the derivatives dy/dx and dz/dx using the given differential equation.
- Use the fourth-order Runge-Kutta method to update the values of y and z at the next x value.
- Check if the current x value is equal to either 0 or L. If so, update the value of y at that boundary condition.
- Repeat the loop until the final x value is reached.
Here is the code that implements the above steps:
function [x, y] = solve_differential_equation(a, c, L, h0)
% Solve the differential equation h''(x) + (h'(x))^2 - h'(x) * tan(a*x) + c - h(x) * sec^2(a*x) * a = 0
% with boundary conditions h(0) = h0 and h(L) = h0, using the fourth-order Runge-Kutta method.
% Define initial conditions
y0 = h0; % h(0) = h0
z0 = 0; % h'(0) = 0
% Define step size
dx = 0.01;
% Set up loop
x = 0:dx:L;
n = length(x);
y = zeros(n, 1);
z = zeros(n, 1);
y(1) = y0;
z(1) = z0;
for i = 1:n-1
% Calculate derivatives
dydx = z(i);
dzdx = a * sec(a*x(i))^2 + (1/y(i)) * (z(i) * tan(a*x(i)) - z(i)^2 - c);
% Update y and z using fourth-order Runge-Kutta method
k1 = dx * dydx;
l1 = dx * dzdx;
k2 = dx * (dydx + 0.5*l1);
l2 = dx * (dzdx + 0.5*k1);
k3 = dx * (dydx + 0.5*l2);
l3 = dx * (dzdx + 0.5*k2);
k4 = dx * (dydx + l3);
l4 = dx * (dzdx + k3);
y(i+1) = y(i) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);
z(i+1) = z(i) + (1/6)*(l1 + 2*l2 + 2*l3 + l4);
% Check boundary conditions
if x(i+1) == 0
y(i+1) = h0;
elseif x(i+1) == L
y(i+1) = h0;
end
end
% Plot results
plot(x, y)
xlabel('x')
ylabel('h(x)')
title('Solution to differential equation')
end
To use this function, call it with the values of a, c,
