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Given the wave equation 10²u 9 მე2 subject to the boundary conditions and initial conditions J²u Ət² where r = u(x,0) = k 3h = 0 < x < 2, t> 0, u(0,t) = 0, u(2, t) = 0, t>0, i. Use finite difference method to show that the numerical solution for the equation above can be written as U₁j+1 = r²u₁-1₁j + (2 − 2r²)u₁j+r²U₂+1,j — Uj₂j-1 = sin 2x, ut(x,0)=3, 0≤x≤ 2. (3 marks) ii. If h = 0.5 and k = 0.1, sketch a suitable diagram to show the mesh point where the numerical solutions are calculated up to t = 0.3. (3 marks) iii. Using results in part (i) and (ii), find the numerical solutions up to t = 0.2. (12 marks) iv. Using results obtained in part (iii), estimate u(1.5,0.3). (2 marks)