ODE problems

Solve ordianary diffential equations with finit differential method.

Boundary value problem

The general govening equation can be written as

\[ u' = f(u,t), \]

with two point boundary value

\[ u(0) = a,\quad u(1) = b.\]

The equations can be reduced to first-order vector form, which take the matrix form

\[ A\boldsymbol{u(t)}' + D\boldsymbol{u(t)} = N(t,\boldsymbol{u(t)}) + F(t). \]

For the linear problem, the nonlinear term $N(t,u)\equiv 0$.

The shooting method is used for the nonlinear boundary value problem.

Besides, the FDM with Newton-Raphson method can be use to solve the problem as well.

4-order center differential is used

\[ \frac{\mathrm{d}y_i}{\mathrm{d}t} = \frac{y_{i-2}-8y_{i-1}+8y_{i+1}-y_{i+2}}{12\Delta t}, \]

Airy function

\[u'' - t u = 0,\]

with the boundary condition

\[u(0) = \operatorname{Ai}(0),\\ u(10) = \operatorname{Ai}(10),\]

where $\operatorname{Ai}(x)$ is the Airy function of the first kind.

The equation can be written as the first order form

\[ y_1' = y_2,\\ y_2' = ty_1.\]

Furthmore, write it in the matrix form

\[ \varGamma\frac{\mathrm{d} y_{i}}{\mathrm{d} t} + D y_{i} = 0,\]

where

\[ \varGamma=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad D=\begin{pmatrix} 0 & -1 \\ -t & 0 \end{pmatrix}.\]

Blasius equation

\[f''' + \frac{1}{2} f f'' = 0,\]

with the boundary condition

\[f(0) = f'(0) = 0,\quad f' → 1 \quad\mathrm{as}\quad y → ∞.\]