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Question: In the annulus the functions ,z , log l2l,1 (where k EN and any convergent series in these are harmonic....
4. (a) Indicate where the series is (i) absolutely convergent, n-1 where it is (ii) conditionally convergent, and where it is (iii) divergent. Justify your answers Find f,(z) if f(x) = arctan (e* ) +...
4. (a) Indicate where the series is (i) absolutely convergent, n-1 where it is (ii) conditionally convergent, and where it is (iii) divergent. Justify your answers Find f,(z) if f(x) = arctan (e* ) + arcsin V2x + 4. (b) (a) Set up (but do not evaluate) a definite integral that represents the area 5. of the region R inside the circle r = 4 sin θ and outside the circle r = 2. Carefully sketch the region R. (i)...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Sho...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
1 point) Suppose that the function f(x) is equal to the convergent powers series 51+1 -(x...
1 point) Suppose that the function f(x) is equal to the convergent powers series 51+1 -(x - 2)3n+1 n! n= Vhich of the following is equal to the value of f(13) (2)? A. 514 B. 4! C.O. u D. 13! 4! 514 E. 13! IM8 n?(5x - 1)" 2" n=0 Which of the following is the radius of convergence for the series? O A. R= IN OB. R= 2 C. R= . OD. R= 0. O 0 5 E. R=...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D {1 z€ |-п/2, т/2] 0 те (т...
QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D {1 z€ |-п/2, т/2] 0 те (т/2, т) on the mterval (-T,7T) (5.2) Use separation of varables to find a solution of the partial differential equation (7) ди ди =0, on z, y € (0, со), with boundarу value u(z, 1) - e(1-2)/z [12
QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D...
the below is the previous question solution: 1. Recall the following boundary-value problem on the interval [0, 1] from...
the below is the previous question solution:
1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Question 1 (1 mark) Attempt 1 Consider the boundary value problem Q Find the functions g, Φ 1 and Φ 2 so that u-3-a...
Question 1 (1 mark) Attempt 1 Consider the boundary value problem Q Find the functions g, Φ 1 and Φ 2 so that u-3-al Φ1-a2Φ2 is the approximate quadratic solution that satisfies the essential boundary condition Your answer should consist of three expressions, the first representing the term g, the second representing the term Ф1 and the third representing the term Ф2 All three expressions should be expressed in terms of the independent variable x. Your answers should be expressed...
4. (a) Indicate where the series is (i) absolutely convergent, n-1 where it is (ii) conditionally convergent, and where it is (iii) divergent. Justify your answers Find f,(z) if f(x) = arctan (e* ) + arcsin V2x + 4. (b) (a) Set up (but do not evaluate) a definite integral that represents the area 5. of the region R inside the circle r = 4 sin θ and outside the circle r = 2. Carefully sketch the region R. (i)...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
1 point) Suppose that the function f(x) is equal to the convergent powers series 51+1 -(x - 2)3n+1 n! n= Vhich of the following is equal to the value of f(13) (2)? A. 514 B. 4! C.O. u D. 13! 4! 514 E. 13! IM8 n?(5x - 1)" 2" n=0 Which of the following is the radius of convergence for the series? O A. R= IN OB. R= 2 C. R= . OD. R= 0. O 0 5 E. R=...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D {1 z€ |-п/2, т/2] 0 те (т/2, т) on the mterval (-T,7T) (5.2) Use separation of varables to find a solution of the partial differential equation (7) ди ди =0, on z, y € (0, со), with boundarу value u(z, 1) - e(1-2)/z [12
QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D...
the below is the previous question solution:
1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Question 1 (1 mark) Attempt 1 Consider the boundary value problem Q Find the functions g, Φ 1 and Φ 2 so that u-3-al Φ1-a2Φ2 is the approximate quadratic solution that satisfies the essential boundary condition Your answer should consist of three expressions, the first representing the term g, the second representing the term Ф1 and the third representing the term Ф2 All three expressions should be expressed in terms of the independent variable x. Your answers should be expressed...
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