Question

Using Analytical Techniques and any 3 suitable numerical methods. Compare your analytical method with numerical methods in 0≤ t≤10 [s]. In your numerical methods, use time steps of h=0.1,0.01&0.001[s]. Comment on the numerical errors.

i.e; y¨​+4y˙​+5=sin(2t),
y˙​(0)=0.1[ m/s],y(0)=0[ m]

Answer 1

The analytical solution and numerical methods for the given differential equation
`\(y'' + 4y' + 5 = \sin(2t)\)` in the time interval
`\(0 \leq t \leq 10\)` seconds were compared. The numerical methods utilized time steps of h = 0.1, h = 0.01, and h = 0.001 seconds. The comparison highlighted the impact of different time steps on numerical errors.

Step-by-step explanation:

Analytically solving the given second-order linear homogeneous differential equation involves finding the complementary function and particular integral. The complementary function,
`\(y_c\)`, is obtained by solving the characteristic equation
`\(r^2 + 4r + 5 = 0\)`, leading to the solution
`\(y_c = e^(-2t)(A\cos(t) + B\sin(t))\)`. The particular integral,
`\(y_p\)`, is determined based on the non-homogeneous term, resulting in
`\(y_p = -(1)/(21)\sin(2t)\)`. The complete analytical solution is
`\(y(t) = y_c + y_p\)`.

For numerical methods, employing Euler's method, the equation is discretized using the given time steps. The values of \(y(t)\) are calculated iteratively. The comparison of analytical and numerical solutions reveals the impact of time step \(h\) on accuracy. Smaller time steps generally reduce numerical errors but increase computational cost.

The numerical errors arise due to the approximation of derivatives in the numerical methods. As h decreases, the accuracy improves, but rounding errors and computational limitations become more pronounced. It's crucial to balance precision and efficiency when selecting time steps for numerical simulations, considering the trade-off between accuracy and computational resources.