Numerical Methods for Ordinary Differential Equations

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Abstract

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.

1. Introduction

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.

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

1.1 Linear Multistep Methods

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

α_ky_n+k + α_k-1y_n+k-1 + ... + α₁y_n+1 + α₀y_n = h(β_kf_n+k + β_k-1f_n+k-1 + ... + β₁f_n+1 + β₀f_n)

Alternatively, it can be expressed as:

Σ_j=0 α_jy_n+j = hΣ_j=0 β_jf_n+j

Here, α_k is always non-zero, and at least one of α₀ and β₀ will be non-zero. The specific values of k, α₀, α₁, ..., α_k, β₀, β₁, ..., 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.

2. Adams-Bashforth Method (Explicit Method)

2.1 Derivation

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

y'(t) = f(t, y), a ≤ t ≤ b, y(a) = y₀

The solution of this IVP satisfies the integral equation:

y(t_n+1) = y(t_n) + ∫_{t_n}^t_n+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 = t₀ + 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!h²) ∇²f_n + ... + (t - t_n)(t - t_n-1)...(t - t_n-k+1) / (k-1)!h^k-1 ∇^k-1f_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 ∇^mf_n + (-1)^k (-u)^k h^k f^k(ξ)

Where -u^m = (-1)^mu(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 γ₀^m ∇^mf_n

2.2 Numerical Results

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	f⁰⁰(ξ)h²/3
3	y_n + h(23/12)f_n - 16/12f_n-1 + 5/12f_n-2	f⁰⁰⁰(ξ)h³/8
4	y_n + h(55/24)f_n - 59/24f_n-1 + 37/24f_n-2 - 9/24f_n-3	f⁰⁰⁰⁰(ξ)h⁴/19

3. Adams-Moulton Method (Implicit Method)

3.1 Derivation

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

y'(t) = f(t, y), a ≤ t ≤ b, y(a) = y₀

The solution of the IVP satisfies the integral equation:

y(t_n+1) = y(t_n) + ∫_{t_n}^t_n+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 = t₀ + 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!h²) ∇²f_n+1 + ... + (t - t_n+1)(t - t_n)...(t - t_n-k+1) / k!h^k ∇^kf_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) ∇^mf_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 γ₀^m ∇^mf_n+1 + T_k+1

3.2 Numerical Results

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	f⁰⁰(ξ)h²/12
2	y_n + h(5/12)f_n+1 + 8/12f_n - 1/12f_n-1	f⁰⁰⁰(ξ)h³/24
3	y_n + h(9/24)f_n+1 + 19/24f_n - 5/24f_n-1 + 1/24f_n-2	f⁰⁰⁰⁰(ξ)h⁴/72

4. Nystrom Method (Explicit Method)

4.1 Derivation

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) + ∫_{t_n}^t_n+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 ∇^mf_n

4.2 Numerical Results

The equation provides us with the following table:

Number of Steps	y_n+1	T_k(h)
1	y_n + h f_n	f⁰⁰(ξ)h²/2
2	y_n + h(3/2)f_n - 1/2f_n-1	f⁰⁰⁰(ξ)h³/4
3	y_n + h(11/6)f_n - 7/6f_n-1 + 1/3f_n-2	f⁰⁰⁰⁰(ξ)h⁴/6

5. Milne Method (Implicit Method)

5.1 Derivation

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) + ∫_{t_n}^t_n+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 ∇^mf_n+1

5.2 Numerical Results

The equation provides us with the following table:

Number of Steps	y_n+1	T_k(h)
1	y_n-1 + h f_n	f⁰⁰(ξ)h²/2
2	y_n-1 + h(3/2)f_n - 1/2f_n-1	f⁰⁰⁰(ξ)h³/4
3	y_n-1 + h(11/6)f_n - 7/6f_n-1 + 1/3f_n-2	f⁰⁰⁰⁰(ξ)h⁴/6

6. Predictor-Corrector Method

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.

7. Conclusion

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.