(a) The temperature distribution u(x, t) of the one-dimensional silver rod is governed by the heat equation as follows. ди 0.5 at Given the boundary conditions u(0, t) = t2, u(0.8, t) = 6t, for 0 < t< 0.04 s and the initial condition u(x, 0) = x(0.8 – x) for 0 s
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- Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.Solve for the equilibrium temperature distribution using the 2D Laplace equation on an L x H sized rectangular domain with the following boundary conditions: 1. Left: u(0,y) = f(y) (fixed temperature) 2. Bottom: u₂(x,0) = 0 (insulating) 3. Top: u₂(x, H) = 0 (insulating) 4. Right: u(L, y) = 0 (zero temperature) Solve for a general boundary temperature f(y). Also solve for the particular temperature distribution f(y) = sin(4Ty/H). u(0,y)-f(y) U₂, (x,H) = 0 7² - 0 u(L, y) - 0 u₂(x,0) = 0 Without too much extra work, tell me how this solution would change if we also made the right boundary condition insulating?3 If the pollution index of a point (x, y) is given by p(x, y) = x². Calculate the total pollution of the region bounded by y = x² and y = 2x. Find the total pollution and the centre of pollution in this region.
- (g) fa(2, 3) = (h) = fy(2,3)Two transverse sinusoidal waves combining in a medium are described by the wave functions Y1 = 5.00 sin T(x + 0.300t) Y2 = 5.00 sin T(x - 0.300t) where x, y1, and y, are in centimeters and t is in seconds. Determine the maximum transverse position of an element of the medium at the following positions. (a) x = 0.130 cm Iymaxl = cm (b) x = 0.360 cm Iymaxl = cm (c) x = 1.70 cm Iymaxl = cm (d) Find the three smallest values of x corresponding to antinodes. (Enter your answers from smallest to largest.) cm cm cmSolve the one-dimensional Laplace equation for the equilibrium temperature distribution in a finite rod of length L for the following boundary conditions: 1. u(0) = -2 and u(L) = 2 2. u(0) = A and u(L) + ux (L) = 0 Please also draw a sketch for each boundary condition.
- 1. The heat equation is an equation involving the partial derivatives of a function u(r,t): du Pu Show that u(r, t) = cos(nr)e-nt satisfies the heat equation for any constant n. 2. For which points (r, g) is the tangent plane to z = sin(r) sin(y) horizontal?A two-dimensional rectangular plate is subjected to prescribed boundary conditions, T1 = 50°C, T2 = 140°C. The temperature distribution equation, derived by applying separation of variable methods to a two-dimensional conduction problem for a thin rectangular plate or long rectangular rod, is as follows. (-1)*+1) + 1 -sin L sinh (nty/L) sinh (naW/L) nAX 0(x, y) = = Σ n n=1 Using this expression, calculate the temperature at the point (x,y) = (0.75, 0.5) by considering the first five nonzero terms of the infinite series that must be evaluated. Assume that L = 1.5 m. у (m) T2 1 T = 50°C- T = 50°C →x (m) L L -T = 50°C