a) Consider the following wave equation Utu = 4uzz, 0 < x < 1, t>0, with...

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a) Consider the following wave equation Utu = 4uzz, 0 < x < 1, t>0, with x(1 – 1) u(x,0) 4(1,0) 8x. %3D Use d'Alembert's solution to determine the solution of the above equation. (5 marks) b) Consider the following heat equation Pu t>0 'I> * >0 with boundary conditions uz(0, t) = 0, Uz(1, t) = 0, t> 0 %3D and initial condition u(x, 0) = f(x), 0 < x < 1. i. Use the method of separation of variables to show that the heat equation can be reduced to X" – kX = 0, X'(0) = 0, X'(1) = 0 0, %3D T' - kT where k is a constant. (3 marks) ii. Show that k > 0 leads to trivial solution (you may let k = ²). (2 marks) iii. Show that if k = 0 and k < 0, then the solution is u(x, t) = Po + P, cos(A,r) e t n=1 where A = nn, Po and P are constants for n = 1,2, 3, .. (6 marks) iv. Hence, find the solution if 0 < r <1/2 0 , 100(x - 1/2) , 1/2 <x <1. (4 marks) a) Consider the following wave equation Utu = 4uzz, 0 < x < 1, t>0, with x(1 – 1) u(x,0) 4(1,0) 8x. %3D Use d'Alembert's solution to determine the solution of the above equation. (5 marks) b) Consider the following heat equation Pu t>0 'I> * >0 with boundary conditions uz(0, t) = 0, Uz(1, t) = 0, t> 0 %3D and initial condition u(x, 0) = f(x), 0 < x < 1. i. Use the method of separation of variables to show that the heat equation can be reduced to X" – kX = 0, X'(0) = 0, X'(1) = 0 0, %3D T' - kT where k is a constant. (3 marks) ii. Show that k > 0 leads to trivial solution (you may let k = ²). (2 marks) iii. Show that if k = 0 and k < 0, then the solution is u(x, t) = Po + P, cos(A,r) e t n=1 where A = nn, Po and P are constants for n = 1,2, 3, .. (6 marks) iv. Hence, find the solution if 0 < r <1/2 0 , 100(x - 1/2) , 1/2 <x <1. (4 marks)