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2. Find the travelling wave solution of the regularized long wave equation U + uz +...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the...
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...
1. Let u be a solution of the wave equation u0. Let the points A, B, C, D be the vertices of the ...
1. Let u be a solution of the wave equation u0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines z--d-q,z-ct-c2, z + ct = di, z + ct d2. Show that u(A) u (C) u(B) +u(D) Use this to find u satisfying till- tLzz =0, u( s,s)-s, u(8,8)-82 for s > 0. For which (r, t) can you determine u (x, t) uniquely this way?
1. Let...
d1=8 d2=9 lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero in...
d1=8
d2=9
lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to th...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
PROBLEM 4.3. The one-dimensional wave equation is ə?u - 20u = 0, ət2 or where c>...
PROBLEM 4.3. The one-dimensional wave equation is ə?u - 20u = 0, ət2 or where c> 0 is constant. Show that any function of the form u(x, t) = f(x - ct)+9(2+ct), where f,g: RR are twice continuously differentiable, satisfies this equation. Explain why we call c the wave speed.
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velo...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) ,...
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral.
9. Use a suitable Fourier Transform...
d1= 3 & d2= 2 Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram bel...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
1. Let u be a solution of the wave equation ue20. Let the points A. B, C. D be the vertices of th...
1. Let u be a solution of the wave equation ue20. Let the points A. B, C. D be the vertices of the paralleogram formed by the two pairs of characteristic lines x-ct = cı, x-ct c2, x + ct = di, r + ct = d2. Show that u (A) + u (C) = u (B) + u (D). Use this to find u satisfying For which (r, t) can you determine u (x, t) uniquely this way?
1....
2. Consider the initial-boundary value problem 0 0, t >0 Uz uェェ, Uz(0, t) _ u(0, t) = 0. a(z, 0...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
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...
1. Let u be a solution of the wave equation u0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines z--d-q,z-ct-c2, z + ct = di, z + ct d2. Show that u(A) u (C) u(B) +u(D) Use this to find u satisfying till- tLzz =0, u( s,s)-s, u(8,8)-82 for s > 0. For which (r, t) can you determine u (x, t) uniquely this way?
1. Let...
d1=8
d2=9
lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
PROBLEM 4.3. The one-dimensional wave equation is ə?u - 20u = 0, ət2 or where c> 0 is constant. Show that any function of the form u(x, t) = f(x - ct)+9(2+ct), where f,g: RR are twice continuously differentiable, satisfies this equation. Explain why we call c the wave speed.
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral.
9. Use a suitable Fourier Transform...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
1. Let u be a solution of the wave equation ue20. Let the points A. B, C. D be the vertices of the paralleogram formed by the two pairs of characteristic lines x-ct = cı, x-ct c2, x + ct = di, r + ct = d2. Show that u (A) + u (C) = u (B) + u (D). Use this to find u satisfying For which (r, t) can you determine u (x, t) uniquely this way?
1....
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
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