5. (a) A unifom rod of length 10 meters is held at both ends. The initial temperature at any point on the rod is f(x). T

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5 A A Unifom Rod Of Length 10 Meters Is Held At Both Ends The Initial Temperature At Any Point On The Rod Is F X T 1 (109.78 KiB) Viewed 61 times

5. (a) A unifom rod of length 10 meters is held at both ends. The initial temperature at any point on the rod is f(x). The temperature at both ends is held fixed at a constant temperature of 0°C. The temperature u(x, t) at any time t of a point P which is a distance away from any of the two endpoints is du (0 < x < 10, t > 0), Ət i. Use the method of separation of variables to show that the general solution is given by u(x, t) = (A cos Ar + B sin Ar)e-4X²t J²u Əx²¹ = such that A, B and X are constant. ii. State the boundary and initial conditions of the problem. iii. By applying the boundary condition u(0, t) = 0, show that u(x, t) = Be-4x²t, sin X.x. iv. By applying the boundary condition u(10, t) = 0, show that u(x,t) = Σ Bne sin ¹ (107) ² X. n=1 v. Compute B₁ by applying the initial condition f(x) = 4-x. (15 marks)