project 1 math lab

You need to include an introduction, primary discussion, and summary. Include graphs, tables, and images, as necessary, to improve the clarity of your discussion. Your project needs to be both correct and well written. Communication remains a critical component of our modern, technological society. A few notes about format: you MUST use MS Word for your project and use Equation Editor for all mathematical symbols, e.g. 𝑧(𝑡) = sin(𝑡) + 1 ln(𝑡) . Problem 1: Consider the following Initial Value Problem (IVP) where 𝑦 is the dependent variable and 𝑡 is the independent variable: 𝑦 ′ = sin(𝑡) ∗ (1 − 𝑦) with 𝑦(0) = 𝑦0 and 𝑡 ≥ 0 Note: the analytic solution for this IVP is: 𝑦(𝑡) = 1 + (𝑦0 − 1)𝑒 cos(𝑡)−1

Part 1A:

Approximate the solution to the IVP using Euler’s method with the following conditions: Initial condition 𝑦0 = − 1 2 ; time step ℎ = 1 16 ; and time interval 𝑡 ∈ [0,20] + Derive the recursive formula for Euler’s method applied to this IVP + Plot the Euler’s method approximation + Plot the absolute error between the approximation and the exact solution using a semi-log plot

Part 1B:

Approximate the solution to the IVP using the Improved Euler’s method with the following conditions: Initial condition 𝑦0 = − 1 2 ; time step ℎ = 1 16 ; and time interval 𝑡 ∈ [0,20] + Derive the recursive formula for the Improved Euler’s method applied to this IVP + Plot the Improved Euler’s method approximation + Plot the absolute error between the approximation and the exact solution using a semilog plot

Part 1C:

Approximate the solution to the IVP using the RK4 method with the following conditions: Initial condition 𝑦0 = − 1 2 ; time step ℎ = 1 16 ; and time interval 𝑡 ∈ [0,20] + Plot the RK4 method approximation + Plot the absolute error between the approximation and the exact solution using a semilog plot

Problem 2: Consider the following Initial Value Problem (IVP) where 𝑦(𝑡) is the dependent function:

𝑦 ′ = 𝑦 − 𝑦 2 + 1.14 cos(𝑒 𝑡/2 ) with 𝑦(0) = 𝑦0 and 𝑡 ≥ 0

Part 2A:

Approximate the solution to the IVP using the Improved Euler’s method with the following conditions: Initial condition 𝑦0 = 1; time steps ℎ = 1 8 , 1 16 , 1 32 , 1 64 ; and time interval 𝑡 ∈ [0,20]

Plot the Improved Euler’s method approximation for all 4 time steps

Discuss the results of these approximations

Part 2B:

Approximate the solution to the IVP using the RK4 method with the following conditions:

Initial condition 𝑦0 = 1;

time steps ℎ = 1 8 , 1 16 , 1 32 , 1 64 ; and time interval 𝑡 ∈ [0,20]

Plot the RK4 approximation for all 4 time steps

Discuss the results of these approximations

 

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