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3. Use u(), r) and a mathematical equation to describe each of the above signal atn...
1. Use combinations of STEP FUNCTIONS to describe each continuous-time signal shown below. f(t) 0 2 4 6 0 1 2 3 0 1 2 3 4 2. Sketch the following signals: (a) x (t)=1 [u(t+2)-u(t-1)] (c) X(t)=\fety (b) X(t)=t.e (d) x (t) = u(t) u(t-1).ult-2).u(t-3) 3. Determine whether the systems below are linear and time invariant. Justify your answer! (a) y(t) = x(31) (b) y(t)= 2x(1-t) y(t)=cos(x(t)] 4. Simplify the expressions: (a) y(t)=1.8(t+2)+(t +1) 8(1-1)+(t+3). 8(t) (b) y(t) =...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
7 Draw the continuous time signal. x(t)={r(t)-r(t-2)-r(t-4)+r(t-6)}+{u(t+4)-2u(t+2)+2u(t)-u(t-6)} where [u(t) is unit step signal and r(t) is unit ramp signal]. And sketch the following i. yl(t)=x[-1-2) ii. y2(t)=x[3-t] 15 Marks
Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject to the nonhomogenoous boundary conditions u(0, t) 1, u(r, t) 0, t>o and the initial condition u(, 0)in() Find the solution u (z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution t) (b) Den ote u(x, t)-u(x, t)-ue(x). Derive the IBVP for the function u(x,t). (c) Find v(x, t) (d) Find u(x,t) Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject...
Problem 6. Describe the surface r(u, u)-R cos u x + R sin u ý + uz where 0 < u < 2π and 0 < u-H. and R and H are positive constants. What is the surface element and what is the total surface area? Show that Or/au, or/àv are continuous across the "cut at 2T coS W T
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by completing each of the following steps (a) Find the equilibrium temperature distribution u ( (b) Denote v, t)t) - u(). Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t>...
(3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)- 30sin (5r) (r, t) E (0, ) x (0, 0o) t >0 t > 0 1 (3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)-...
do question 3 with the info provided f 0 Question 3 Given the graph above represents a string being plucked at point (g). The wave equation generated when the string is released after being plucked, is given by the wave equation in question 1, and that additionally: 1. u(0, t) 0 u(4, t) 2. u(x, 0) f(x) as in question 2 au 3. atlt-0 Solve wave equation subject to the restrictions above. [10] Question 2 a) In the General Fourier...
The equation for a causal full-wave rectified signal is given by c(t) = 12 sin(1 t)u(t) a) The even component of r(t) is shown by xe(t). 1) Find the equation for e(t). ve(t) = 2) From the figures 1 to 4 shown below, select the graph that matches xet). Figure: ? 3) Is ze() causal? ? 4) Is zelt) periodic? ? b) The odd component of 3(t) is shown by xo(t). 1) Find the equation for 2 (t). 2.(t) =...
(1) Find the solution for each of the following BVP for the heat equation at)u(r, t)0 (r, t) E (0, 20) x (0, co) (0, t) 0 u(20, t)0 u(r, 0) f(r) E [0, 10 E (10, 20 t > 0 1 where f(r) a. t > 0 (1) Find the solution for each of the following BVP for the heat equation at)u(r, t)0 (r, t) E (0, 20) x (0, co) (0, t) 0 u(20, t)0 u(r, 0) f(r)...