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2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
*1. (This problem is to be solved manually, but you can use MATLAB or any other software as a calculator only) Consider the problem of finding the minimum of the following function for x>0 0.65an...
*1. (This problem is to be solved manually, but you can use MATLAB or any other software as a calculator only) Consider the problem of finding the minimum of the following function for x>0 0.65an 0.75 fx) 0.65- 1+x2 a) First find a bracket for the minimum. b) Using the bracket found in Part (a) above, perform two iterations of:. Golden section search method . Quadratic interpolation method
*1. (This problem is to be solved manually, but you can use...
Problem 1. x(t) = 2 cos(210.8t) + 3cos(270.2t) 1) Sketch x(t) for 0<t<2 2) Find the...
Problem 1. x(t) = 2 cos(210.8t) + 3cos(270.2t) 1) Sketch x(t) for 0<t<2 2) Find the Fourier Series coefficients for x(t)
Exercise 7.3. Consider the nonlinearly constrained problem minimize xER2 to (7.1) a x2 1 = 0. subject 1)T is a feasible...
Exercise 7.3. Consider the nonlinearly constrained problem minimize xER2 to (7.1) a x2 1 = 0. subject 1)T is a feasible path for the nonlinear constraint (a) Show that x(a) x x - 1 = 0 of problem (7.1). Compute the tangent to the feasible path at E = (0, 0)7 (sin a, cos a - + X (b) Find another feasible path for the constraint x? + (x2 + 1)2 - 1 = 0. Compute the tangent to the...
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for...
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for minimization will terminate in exactly one iteration for any initial points Xo, X1, provided that x1 + xo: 6. Consider the sequence {x(k)} given by i. Write down the value of the limit of {x(k)}. ii. Find the order of convergence of {x(k)}. 7. Consider the function f(x) = x4 – 14x3 + 60x2 – 70x in the interval (0, 2). Use the bisection...
Problem 2: Consider the following differential equation: 0 and with u = e-31. Solve for x(t)...
Problem 2: Consider the following differential equation: 0 and with u = e-31. Solve for x(t) using with initial conditions x(0)-x(0) Laplace transforms.
[10pt] 5. Consider the IVP :' = t +x?, *(0) = 1. Complete the following table...
[10pt] 5. Consider the IVP :' = t +x?, *(0) = 1. Complete the following table for the numerical solutions of given IVP with step-size h = 0.05. t by Euler's Method by Improved Euler's Method 0 0.05 0.1 6. Which of the followings is the solution of the IVP
III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve...
III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve for x(t) ("by hand"). Now suppose we measure both states as a linear combination, namely y(t) = [ 1 1 ]x(t). Find y(t)
III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve for x(t) ("by hand"). Now suppose we measure both states as a linear combination, namely y(t) = [ 1 1 ]x(t)....
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t) + sin(t)2(t) = 0 for all t E (0,T). (a) S...
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t) + sin(t)2(t) = 0 for all t E (0,T). (a) Show that this defines a FODE for at least one T>0. 1 mark 2 mark (c) Find the potential and conclude (briefly) what is the solution space for the FODE for (b) Transform (possibly inverting) the DE into an exact DE. T. 2 mark
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t)...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
*1. (This problem is to be solved manually, but you can use MATLAB or any other software as a calculator only) Consider the problem of finding the minimum of the following function for x>0 0.65an 0.75 fx) 0.65- 1+x2 a) First find a bracket for the minimum. b) Using the bracket found in Part (a) above, perform two iterations of:. Golden section search method . Quadratic interpolation method
*1. (This problem is to be solved manually, but you can use...
Problem 1. x(t) = 2 cos(210.8t) + 3cos(270.2t) 1) Sketch x(t) for 0<t<2 2) Find the Fourier Series coefficients for x(t)
Exercise 7.3. Consider the nonlinearly constrained problem minimize xER2 to (7.1) a x2 1 = 0. subject 1)T is a feasible path for the nonlinear constraint (a) Show that x(a) x x - 1 = 0 of problem (7.1). Compute the tangent to the feasible path at E = (0, 0)7 (sin a, cos a - + X (b) Find another feasible path for the constraint x? + (x2 + 1)2 - 1 = 0. Compute the tangent to the...
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for minimization will terminate in exactly one iteration for any initial points Xo, X1, provided that x1 + xo: 6. Consider the sequence {x(k)} given by i. Write down the value of the limit of {x(k)}. ii. Find the order of convergence of {x(k)}. 7. Consider the function f(x) = x4 – 14x3 + 60x2 – 70x in the interval (0, 2). Use the bisection...
Problem 2: Consider the following differential equation: 0 and with u = e-31. Solve for x(t) using with initial conditions x(0)-x(0) Laplace transforms.
[10pt] 5. Consider the IVP :' = t +x?, *(0) = 1. Complete the following table for the numerical solutions of given IVP with step-size h = 0.05. t by Euler's Method by Improved Euler's Method 0 0.05 0.1 6. Which of the followings is the solution of the IVP
III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve for x(t) ("by hand"). Now suppose we measure both states as a linear combination, namely y(t) = [ 1 1 ]x(t). Find y(t)
III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve for x(t) ("by hand"). Now suppose we measure both states as a linear combination, namely y(t) = [ 1 1 ]x(t)....
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t) + sin(t)2(t) = 0 for all t E (0,T). (a) Show that this defines a FODE for at least one T>0. 1 mark 2 mark (c) Find the potential and conclude (briefly) what is the solution space for the FODE for (b) Transform (possibly inverting) the DE into an exact DE. T. 2 mark
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t)...
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