Given the wave equation J²u 10²u Ət² 90x² subject to the boundary conditions and initial conditions SSCE 2393 NUMERICAL

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Given the wave equation J²u 10²u Ət² 90x² subject to the boundary conditions and initial conditions SSCE 2393 NUMERICAL METHODS ASSIGNMENT 2 where r = k 3h = * 0<x<2, t> 0, u(x,0) = sin 2x, ut(x,0)=3, 0≤x≤2. i. Use finite difference method to show that the numerical solution for the equation above can be written as U₁j+1 = r²ui-1j+(2-2r²)u₁j+r²u₁+1j - Uį, j-1 u(0,t) = 0, u(2, t) = 0, t>0, (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. iv. Using results obtained in part (iii), estimate u(1.5,0.3). (3 marks) iii. Using results in part (i) and (ii), find the numerical solutions up to t = 0.2. (12 marks) (2 marks)