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This lab report explores various numerical methods for solving ordinary differential equations (ODEs) using linear multistep methods. We discuss the Adams-Bashforth, Adams-Moulton, Nyström, and Milne methods, presenting their derivations, equations, and co-efficients. Additionally, we investigate the use of predictor-corrector methods in solving ODEs.

Ordinary Differential Equations (ODEs) play a crucial role in modeling various physical phenomena. Solving ODEs analytically can be challenging or even impossible in some cases. Therefore, numerical methods are essential for approximating the solutions of ODEs.

In this lab report, we focus on linear multistep methods for solving ODEs, including explicit and implicit methods.

The general k-step linear multistep method is given by the following equation:

α_{k}y_{n+k} + α_{k-1}y_{n+k-1} + ... + α_{1}y_{n+1} + α_{0}y_{n} = h(β_{k}f_{n+k} + β_{k-1}f_{n+k-1} + ... + β_{1}f_{n+1} + β_{0}f_{n})

Alternatively, it can be expressed as:

Σ_{j=0} α_{j}y_{n+j} = hΣ_{j=0} β_{j}f_{n+j}

Here, α_{k} is always non-zero, and at least one of α_{0} and β_{0} will be non-zero. The specific values of k, α_{0}, α_{1}, ..., α_{k}, β_{0}, β_{1}, ..., β_{k} determine the characteristics of the method.

If β_{k} = 0, the method is called **explicit** since there is no appearance of the unknown y_{n+k} on the right-hand side of the equation at each step.

On the other hand, if β_{k} ≠ 0, the method is called **implicit** because y_{n+k} appears on both sides of the equation, making it impossible to rearrange for an explicit formula for y_{n+k}.

Let's consider the initial value problem for an ordinary differential equation:

y'(t) = f(t, y), a ≤ t ≤ b, y(a) = y_{0}

The solution of this IVP satisfies the integral equation:

y(t_{n+1}) = y(t_{n}) + ∫_{tn}^{tn+1} f(t, y(t)) dt

Assuming we have already found f_{i} = f(t_{i}, y_{i}) for i = n - k + 1, ..., n with t_{i} = t_{0} + ih, we can approximate f(t, y) by a polynomial that interpolates it at k points (t_{n}, y_{n}), (t_{n-1}, y_{n-1}), ..., (t_{n-k+1}, y_{n-k+1}). These interpolation points are equidistributed, allowing us to use Newton's Backward Difference Formula of degree k - 1.

The interpolated polynomial P_{k-1}(t) can be expressed as:

P_{k-1}(t) = f_{n} + (t - t_{n}) / h ∇f_{n} + (t - t_{n})(t - t_{n-1}) / (2!h^{2}) ∇^{2}f_{n} + ... + (t - t_{n})(t - t_{n-1})...(t - t_{n-k+1}) / (k-1)!h^{k-1} ∇^{k-1}f_{n} + (t - t_{n})(t - t_{n-1})...(t - t_{n-k+1}) / k! f^{k}(ξ)

Substituting t = t_{n} + uh, we get:

P_{k-1}(t) = P_{k-1}(t_{n} + uh) ≈ Σ_{m=0} (-1)^{m} (-u)^{m} ∇^{m}f_{n} + (-1)^{k} (-u)^{k} h^{k} f^{k}(ξ)

Where -u^{m} = (-1)^{m}u(u+1)...(u+m-1)/m! and ξ ∈ [t_{n-k+1}, t_{n}].

Now, we can approximate y_{n+1} as:

y_{n+1} ≈ y_{n} + h Σ_{m=0} γ_{0}^{m} ∇^{m}f_{n}

The equation provides us with the following table:

Number of Steps | y_{n+1} |
T_{k+1}(h) |
---|---|---|

2 | y_{n} + h(3/2)f_{n} - 1/2f_{n-1} |
f^{00}(ξ)h^{2}/3 |

3 | y_{n} + h(23/12)f_{n} - 16/12f_{n-1} + 5/12f_{n-2} |
f^{000}(ξ)h^{3}/8 |

4 | y_{n} + h(55/24)f_{n} - 59/24f_{n-1} + 37/24f_{n-2} - 9/24f_{n-3} |
f^{0000}(ξ)h^{4}/19 |

Consider the same initial value problem for an ordinary differential equation:

y'(t) = f(t, y), a ≤ t ≤ b, y(a) = y_{0}

The solution of the IVP satisfies the integral equation:

y(t_{n+1}) = y(t_{n}) + ∫_{tn}^{tn+1} f(t, y(t)) dt

Assuming we have already found f_{i} = f(t_{i}, y_{i}) for i = n - k + 1, ..., n + 1 with t_{i} = t_{0} + ih, we can approximate f(t, y) by a polynomial that interpolates it at k + 1 points (t_{n+1}, y_{n+1}), (t_{n}, y_{n}), ..., (t_{n-k+1}, y_{n-k+1}). These interpolation points are equidistributed, allowing us to use Newton's Backward Difference Formula of degree k.

The interpolated polynomial P_{k}(t) can be expressed as:

P_{k}(t) = f_{n+1} + (t - t_{n+1}) / h ∇f_{n+1} + (t - t_{n+1})(t - t_{n}) / (2!h^{2}) ∇^{2}f_{n+1} + ... + (t - t_{n+1})(t - t_{n})...(t - t_{n-k+1}) / k!h^{k} ∇^{k}f_{n+1} + (t - t_{n+1})(t - t_{n})...(t - t_{n-k+1}) / (k + 1)!h^{k+1} f^{k+1}(ξ)

Substituting t = t_{n} + uh, we get:

P_{k}(t) = P_{k}(t_{n} + uh) ≈ Σ_{m=0} (-1)^{m} (1 - u^{m}) ∇^{m}f_{n+1} + (-1)^{k+1} (1 - u^{k+1}) h^{k+1} f^{k+1}(ξ)

Where (1 - u^{m}) = (-1)^{m}(u-1)(u)(u+1)...(u+m-1)/m! and ξ ∈ [t_{n-k+1}, t_{n+1}].

Now, we can approximate y_{n+1} as:

y_{n+1} ≈ y_{n} - j + h Σ_{m=0} γ_{0}^{m} ∇^{m}f_{n+1} + T_{k+1}

The equation provides us with the following table:

Number of Steps | y_{n+1} |
T_{k+1}(h) |
---|---|---|

1 | y_{n} + h(1/2)f_{n+1} + 1/2f_{n} |
f^{00}(ξ)h^{2}/12 |

2 | y_{n} + h(5/12)f_{n+1} + 8/12f_{n} - 1/12f_{n-1} |
f^{000}(ξ)h^{3}/24 |

3 | y_{n} + h(9/24)f_{n+1} + 19/24f_{n} - 5/24f_{n-1} + 1/24f_{n-2} |
f^{0000}(ξ)h^{4}/72 |

The Nystrom method is an explicit method that works similarly to the Adams-Bashforth method, but with an increment of y_{n} calculated over two steps instead of one. The solution of the IVP satisfies the integral equation:

y(t_{n+1}) = y(t_{n}) + ∫_{tn}^{tn+1} f(t, y(t)) dt

Thus, y_{n+1} can be approximated as:

y_{n+1} ≈ y_{n} + h Σ_{m=0} γ_{m} ∇^{m}f_{n}

The equation provides us with the following table:

Number of Steps | y_{n+1} |
T_{k}(h) |
---|---|---|

1 | y_{n} + h f_{n} |
f^{00}(ξ)h^{2}/2 |

2 | y_{n} + h(3/2)f_{n} - 1/2f_{n-1} |
f^{000}(ξ)h^{3}/4 |

3 | y_{n} + h(11/6)f_{n} - 7/6f_{n-1} + 1/3f_{n-2} |
f^{0000}(ξ)h^{4}/6 |

The Milne method is an implicit method that works similarly to the Adams-Moulton method but with an increment of y_{n} calculated over two steps instead of one. The solution of the IVP satisfies the integral equation:

y(t_{n+1}) = y(t_{n-1}) + ∫_{tn}^{tn+1} f(t, y(t)) dt

Thus, y_{n+1} can be approximated as:

y_{n+1} ≈ y_{n-1} + h Σ_{m=0} γ_{m} ∇^{m}f_{n+1}

The equation provides us with the following table:

Number of Steps | y_{n+1} |
T_{k}(h) |
---|---|---|

1 | y_{n-1} + h f_{n} |
f^{00}(ξ)h^{2}/2 |

2 | y_{n-1} + h(3/2)f_{n} - 1/2f_{n-1} |
f^{000}(ξ)h^{3}/4 |

3 | y_{n-1} + h(11/6)f_{n} - 7/6f_{n-1} + 1/3f_{n-2} |
f^{0000}(ξ)h^{4}/6 |

Linear multistep methods are often implemented in a predictor-corrector form. A preliminary calculation is done using the explicit form of the multistep method, and then the solutions are corrected using the implicit form.

In this lab report, we discussed several linear multistep methods for solving ordinary differential equations, including the Adams-Bashforth, Adams-Moulton, Nyström, and Milne methods. We presented their derivations, equations, and numerical results for different numbers of steps. Additionally, we explored the use of predictor-corrector methods in solving ODEs. These methods provide valuable tools for approximating solutions to ODEs, allowing us to tackle a wide range of practical problems.

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