Systems of IVPs | Sahithyan's Notes

An $m$ th-order system of first-order IVPs is

\begin{aligned} u_1'(t) &= f_1(t, u_1, u_2, \ldots, u_m) \\ &\ \ \vdots \\ u_m'(t) &= f_m(t, u_1, u_2, \ldots, u_m) \end{aligned}

for $a \leq t \leq b$ , with $u_1(a) = \alpha_1, \ldots, u_m(a) = \alpha_m$ . The goal is to approximate $u_1(t), \ldots, u_m(t)$ .

Lipschitz Condition for Systems

$f(t, u_1, \ldots, u_m)$ on

D = \{ (t, u_1, \ldots, u_m) : a \leq t \leq b,\ -\infty < u_i < \infty \}

satisfies a Lipschitz condition in $u_1, \ldots, u_m$ iff a constant $L > 0$ exists with

|f(t, u_1, \ldots, u_m) - f(t, z_1, \ldots, z_m)| \leq L \sum_{j=1}^{m} |u_j - z_j|

for all $(t, u_1, \ldots, u_m), (t, z_1, \ldots, z_m) \in D$ .

Sufficient test: $f$ and each partial derivative $\dfrac{\partial f}{\partial u_i}$ are continuous on $D$ with $\left| \dfrac{\partial f}{\partial u_i} \right| \leq L$ for each $i = 1, \ldots, m$ .

Existence and Uniqueness

If each $f_i$ is continuous and satisfies a Lipschitz condition on $D$ , the system has a unique solution $u_1(t), \ldots, u_m(t)$ on $[a, b]$ .

Order 4 Runge-Kutta for Systems

Superscript $(j)$ marks quantities for $u_j$ . Mesh points are $t_i = a + ih$ with $h = \dfrac{b - a}{N}$ .

\begin{aligned} w_0^{(j)} &= \alpha_j \\ k_1^{(j)} &= h\, f_j\big(t_i,\ w_i^{(1)}, \ldots, w_i^{(m)}\big) \\ k_2^{(j)} &= h\, f_j\left( t_i + \tfrac{h}{2},\ w_i^{(1)} + \tfrac{1}{2} k_1^{(1)}, \ldots, w_i^{(m)} + \tfrac{1}{2} k_1^{(m)} \right) \\ k_3^{(j)} &= h\, f_j\left( t_i + \tfrac{h}{2},\ w_i^{(1)} + \tfrac{1}{2} k_2^{(1)}, \ldots, w_i^{(m)} + \tfrac{1}{2} k_2^{(m)} \right) \\ k_4^{(j)} &= h\, f_j\left( t_{i+1},\ w_i^{(1)} + k_3^{(1)}, \ldots, w_i^{(m)} + k_3^{(m)} \right) \\ w_{i+1}^{(j)} &= w_i^{(j)} + \tfrac{1}{6}\left( k_1^{(j)} + 2 k_2^{(j)} + 2 k_3^{(j)} + k_4^{(j)} \right) \end{aligned}

for $i = 0, \ldots, N-1$ and $j = 1, \ldots, m$ . All $k_1^{(j)}$ are computed before any $k_2^{(j)}$ , and so on. Then $u_j(t_i) \approx w_i^{(j)}$ .

Reducing a Higher-Order IVP

An $m$ th-order IVP

y^{(m)}(t) = f\big(t, y, y', \ldots, y^{(m-1)}\big), \quad a \leq t \leq b

with $y(a) = \alpha_1, y'(a) = \alpha_2, \ldots, y^{(m-1)}(a) = \alpha_m$ becomes a first-order system by setting $u_1 = y, u_2 = y', \ldots, u_m = y^{(m-1)}$

\begin{aligned} u_1' &= u_2 \\ u_2' &= u_3 \\ &\ \ \vdots \\ u_{m-1}' &= u_m \\ u_m' &= f(t, u_1, u_2, \ldots, u_m) \end{aligned}

with $u_1(a) = \alpha_1, \ldots, u_m(a) = \alpha_m$ . Solving the system by order 4 Runge-Kutta, $w_i^{(1)}$ approximates $y(t_i)$ .

Worked Example

y'' - 2y' + 2y = e^{2t}\sin t, \quad 0 \leq t \leq 1, \quad y(0) = -0.4,\ y'(0) = -0.6

Set $u_1 = y$ , $u_2 = y'$

\begin{aligned} u_1' &= u_2 \\ u_2' &= 2 u_2 - 2 u_1 + e^{2t}\sin t \end{aligned}

with $u_1(0) = -0.4$ , $u_2(0) = -0.6$ . Apply order 4 Runge-Kutta with $f_1 = u_2$ and $f_2 = 2 u_2 - 2 u_1 + e^{2t}\sin t$ . Then $w_i^{(1)}$ approximates $y(t_i)$ .