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() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1...
x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t)...
x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t) Assume that the impulse response function is given by g(t,e-)-2e2-) Derive the least dimensional quadruplet {A(t), B(t),C(t),D(t)}
x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t) Assume that the impulse response function is given by g(t,e-)-2e2-) Derive the least dimensional quadruplet {A(t), B(t),C(t),D(t)}
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z). x(t)-A()x(t)...
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solutio...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d)...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, S...
partial differential equations
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that dt and conclude that Use this estimate to bound the difference between two solutions in terms of the difference between the initial functions. Does this problem have a unique solution for each initial function f?
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show th...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
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...
1. Write the state-space equations for the system shown below ri (t) +2 (t) u (t) Figure 1: Syste...
1. Write the state-space equations for the system shown below ri (t) +2 (t) u (t) Figure 1: System of Problem#1 2. Evaluate the state transition matrix eA for the matrix below and find the homogenous solution given x (0) 1 1 ] A=10-21 3. Find the power lution in powers of x. Show the details of your work. s (b) y" +4y=0 4. Determine if either the Frobenus or regular power series could be the method of your choice...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous
on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite)
exists. Show that f is Riemann integrable.
1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t) Assume that the impulse response function is given by g(t,e-)-2e2-) Derive the least dimensional quadruplet {A(t), B(t),C(t),D(t)}
x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t) Assume that the impulse response function is given by g(t,e-)-2e2-) Derive the least dimensional quadruplet {A(t), B(t),C(t),D(t)}
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
partial differential equations
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that dt and conclude that Use this estimate to bound the difference between two solutions in terms of the difference between the initial functions. Does this problem have a unique solution for each initial function f?
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
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...
1. Write the state-space equations for the system shown below ri (t) +2 (t) u (t) Figure 1: System of Problem#1 2. Evaluate the state transition matrix eA for the matrix below and find the homogenous solution given x (0) 1 1 ] A=10-21 3. Find the power lution in powers of x. Show the details of your work. s (b) y" +4y=0 4. Determine if either the Frobenus or regular power series could be the method of your choice...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous
on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite)
exists. Show that f is Riemann integrable.
1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
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