Consider the initial value problem dy/dt=a*t^(a-1), y(0)=0, where a0. The true solution is y(t)=t^a. Use the Euler method to solve the initial value problem for a=2.5,1.5,1.1 with stepsize h=0.2,0.1,0.05. Compute the solution errors at the nodes, and determine numerically the convergence orders of the Euler method for these problems. Does anyone have any ideas?

Question

Consider the initial value problem dy/dt=a*t^(a-1), y(0)=0, where a>0. The true solution is y(t)=t^a. Use the Euler method to solve the initial value problem for a=2.5,1.5,1.1 with stepsize h=0.2,0.1,0.05. Compute the solution errors at the nodes, and determine numerically the convergence orders of the Euler method for these problems.  Does anyone have any ideas?

unklerhaukus · Answer

Forward Euler method for  $y'=f(t,y)$$$\hat y_{j+1}=\hat y_j+h\cdot f(t_j,\hat y_j)$$


Your equation is
$$y'=a\times t^{a-1}$$

unklerhaukus · Answer

$$\hat y_0=y(0)$$

unklerhaukus · Answer

$$\hat y_1=\hat y_0+h(a(0)^{a-1})$$
$$\hat y_2=\hat y_1+h(a(1)^{a-1})$$ ...