Consider the initial value problemy′(t) = cos(y), y(0) = 0.(i) Apply 4 steps of forward Euler method with the step size h = 12 to approximately findy(2). Show at least 4 decimal places in your calculations. [10 marks](ii) Apply 2 steps of Heun’s method with the step size h = 1 to approximately find y(2).Show at least 4 decimal places in your calculations.

(i) Forward Euler Method:

The Forward Euler method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is an explicit method for solving initial value problems (IVPs), as a single step method, and is the simplest Runge–Kutta method. The Forward Euler method is based on the tangent line and the slope of the tangent line.

Given the initial value problem y′(t) = cos(y), y(0) = 0, we can apply 4 steps of the forward Euler method with the step size h = 1/2 to approximately find y(2).

Step 1: y1 = y0 + h * f(t0, y0) = 0 + 1/2 * cos(0) = 0.5

Step 2: y2 = y1 + h * f(t1, y1) = 0.5 + 1/2 * cos(0.5) = 0.5 + 0.5 * 0.8776 = 0.9388

Step 3: y3 = y2 + h * f(t2, y2) = 0.9388 + 1/2 * cos(0.9388) = 0.9388 + 0.5 * 0.5918 = 1.2347

Step 4: y4 = y3 + h * f(t3, y3) = 1.2347 + 1/2 * cos(1.2347) = 1.2347 + 0.5 * 0.3302 = 1.3998

So, y(2) ≈ 1.3998

(ii) Heun’s Method:

Heun's method is a numerical method to solve ordinary differential equations. It is an improved version of the Euler method which is both a single-step method and a two-stage method. It has the property of being a predictor-corrector type of method.

Given the same initial value problem, we can apply 2 steps of Heun’s method with the step size h = 1 to approximately find y(2).

Step 1: Predictor: y1' = y0 + h * f(t0, y0) = 0 + 1 * cos(0) = 1

Corrector: y1 = y0 + h/2 * (f(t0, y0) + f(t1, y1')) = 0 + 1/2 * (cos(0) + cos(1)) = 0.5403

Step 2: Predictor: y2' = y1 + h * f(t1, y1) = 0.5403 + 1 * cos(0.5403) = 1.4125

Corrector: y2 = y1 + h/2 * (f(t1, y1) + f(t2, y2')) = 0.5403 + 1/2 * (cos(0.5403) + cos(1.4125)) = 1.0652

So, y(2) ≈ 1.0652