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The motion of a string with fixed ends in a viscous medium is described by: together with boundary conditions: u(0,t) 0 and u(2, t)0 and initially: u(z,0-sin(nz/2) and ut(z, 0)--sin(2πχ). (a) If u(x,...
The motion of a string with fixed ends in a viscous medium is described by: together...
The motion of a string with fixed ends in a viscous medium is described by: together with boundary conditions: u(0,t) 0 and u(2, t)0 and initially: u(z,0-sin(nz/2) and ut(z, 0)--sin(2πχ). (a) If u(x, t) - X(x)T(t) find the ordinary differential equations satisfied by X and T e ordinary different (c) Determine u(x, t).
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solu...
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
)Consider the wave equation for a vibrating string of semi-infnite length with a fixed end at z = 0, t > 0 a(0,t) = 0, and initial conditions 0 < x < oo u(z,0) = 1-cos(nz), ut(x,0) = 0,...
)Consider the wave equation for a vibrating string of semi-infnite length with a fixed end at z = 0, t > 0 a(0,t) = 0, and initial conditions 0 < x < oo u(z,0) = 1-cos(nz), ut(x,0) = 0, Complete the table below with the values of u(0.5, t) at the specified time instants 0.5 0.5 x 0.5 0.5 0.5 2 0.5 0.75 t 0.25 u(x,t)
)Consider the wave equation for a vibrating string of semi-infnite length with a fixed...
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 in...
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...
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the s...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
(1 point) Suppose we have ut = aʻuzz, 0 < x <1,t > 0, boundary conditions...
(1 point) Suppose we have ut = aʻuzz, 0 < x <1,t > 0, boundary conditions are u(0,t) = u(1,t) = 0, and the initial condition is u(x,0) = sin(T2). What will be the behavior of u(x, t) as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. A. The solution behaves unpredictably. B. The solution increases to infinity. OC. For a fixed t, the solution will...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t)...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
Problem 2. A string of a guitar is fixed at the two ends, x = 0...
Problem 2. A string of a guitar is fixed at the two ends, x = 0 and r = a. The string is set in motion with initial position f(x) = (h/a)., 0 <r <a, where h > 0, and then it is released with no initial velocity. The displacement u of the string is described by the PDE au 1 au ar2 2 212 0<x<a, t> 0. (i) State the boundary value initial value problem that u satisfies. (ii)...
The motion of a string with fixed ends in a viscous medium is described by: together with boundary conditions: u(0,t) 0 and u(2, t)0 and initially: u(z,0-sin(nz/2) and ut(z, 0)--sin(2πχ). (a) If u(x, t) - X(x)T(t) find the ordinary differential equations satisfied by X and T e ordinary different (c) Determine u(x, t).
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
)Consider the wave equation for a vibrating string of semi-infnite length with a fixed end at z = 0, t > 0 a(0,t) = 0, and initial conditions 0 < x < oo u(z,0) = 1-cos(nz), ut(x,0) = 0, Complete the table below with the values of u(0.5, t) at the specified time instants 0.5 0.5 x 0.5 0.5 0.5 2 0.5 0.75 t 0.25 u(x,t)
)Consider the wave equation for a vibrating string of semi-infnite length with a fixed...
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...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
(1 point) Suppose we have ut = aʻuzz, 0 < x <1,t > 0, boundary conditions are u(0,t) = u(1,t) = 0, and the initial condition is u(x,0) = sin(T2). What will be the behavior of u(x, t) as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. A. The solution behaves unpredictably. B. The solution increases to infinity. OC. For a fixed t, the solution will...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
Problem 2. A string of a guitar is fixed at the two ends, x = 0 and r = a. The string is set in motion with initial position f(x) = (h/a)., 0 <r <a, where h > 0, and then it is released with no initial velocity. The displacement u of the string is described by the PDE au 1 au ar2 2 212 0<x<a, t> 0. (i) State the boundary value initial value problem that u satisfies. (ii)...
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