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Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
A metal rod of length a cm has initial temperature function f(x) = 2 sin 3x and its two ends are held at temperature zero for all time t>O The heat equation is given as: ди au 4 for 0 < x < it and t > 0 at @x? Boundary conditions: u(0,t) = u(1,t)=0, Initial conditions: u(x,0) = 2 sin 3x By using the method of separation of variables, calculate the general temperature u(x,t) for all cases, k =...
A metal rod of length a cm has initial temperature function f(x) = 2 sin 3x and its two ends are held at temperature zero for all time t>O The heat equation is given as: ди au 4 for 0 < x < it and t > 0 at @x? Boundary conditions: u(0,t) = u(1,t)=0, Initial conditions: u(x,0) = 2 sin 3x By using the method of separation of variables, calculate the general temperature u(x,t) for all cases, k =...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
Please provide very detailed steps Find the temperature u(x, t) in a bar of length T with thermoconductivity coefficient c2 1 (all the quantities are non-dimensional) under adiabatic boundary conditions (zero heat flux) at ends of the bar if the initial tem- perature u(z, 0) . 130 cos(3x) Find the temperature u(x, t) in a bar of length T with thermoconductivity coefficient c2 1 (all the quantities are non-dimensional) under adiabatic boundary conditions (zero heat flux) at ends of the...
Partial Differential Equation - Wave equation : Vibrating spring Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
d1=7 d2=8 Any help would be greatly appreciated. Question 3 Left end (r-0) of a copper rod of length 100mm is kept at a constant temperature of Temp-1 0 a 2 degrees and the right end and sides are insulated, so that the temperature in the ul ul where D = 111 mm2/s for copper. rod, u(x,t), obeys the heat partial DE, Ot Ox (a) Write the boundary conditions for il(x,t) of the problem above. Note that for the left...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = uti 0 < x < 6; t>0; B.C.: 4x(0,t) = 0; uz (6,t) = 0; t> 0; 1. C.: u(x,0) = 12 + 5cos (6x) – 4cos(27x); 0<x<6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is...
D1 = 7 D2 = 4 Any assistance would be greatly appreciated Question 3 Left end (x 0) of a copper rod of length 100mm is kept at a constant temperature of Temp - 10+d2 degrees and the right end and sides are insulated, so that the temperature in the rod, u(x,t). obeys the heat partial DE, CD11 mms copper. where D-1 mm's for copper (a) Write the boundary conditions for u(x, 1) of the problem above. Note that for...