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1) Does the problem = f(x), wr(0) = ur (1) 0 (1) da2 always have a...
Partial Differential Equations 1. (20 points) Consider the problem u" (x)+ u(x) (0.1) f(x) 1 (0.2)...
Partial Differential Equations
1. (20 points) Consider the problem u" (x)+ u(x) (0.1) f(x) 1 (0.2) u'(0)u(0) (u'(0) + u(l)) with f(x) (10 points) a) Is the solution unique? Justify your answer (10 points) b) Does a solution exist. or is there a condition that f (x) must satisfy for existence? Justify your answer given function a
1. (20 points) Consider the problem u" (x)+ u(x) (0.1) f(x) 1 (0.2) u'(0)u(0) (u'(0) + u(l)) with f(x) (10 points) a) Is...
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...
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...
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) =...
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G...
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G (0,y) = 6,(1, y) = 0 does not exist, but a modified Green's function Ĝ(x,y) satisfying a2G 22 = (x - y) -1, G.(0,y)=G.(1,y) = 0 does. How would you use G to solve problem (1) when f satisfies the condition that you found for a solution to exist? Hint: is f(x) = f(u) (8(x - y) - 1) dy?
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t,...
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t, x) € (0, +00), u(0,x) = 1, u(t,0) = e. For the unique solution u(x, t) find the following limit as a function of t: lim u(x, t).
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...
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1...
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1,t)-0, t > 0 and the initial condition u(r,0) f(a) We found the solution to be of the form where (2n-1)n 1,2,3,. TL 20 Now consider the heat conduction problem with the boundary conditions u(0, t) 1,u(T, t)0, t>0 and the initial condition ur,0) 0. Find u(r,t). Hint: First you must find the steady state.
3. In class we discussed the heat conduction problem...
5. Consider a diffusion problem on the interval 0 < x < 1 governed by the...
5. Consider a diffusion problem on the interval 0 < x < 1 governed by the pde ut = uxx. Assume that the solution satisfies the boundary conditions u(0) = 0.5 and u′(1) = −3(u(1) − 0.7). Find the steady state solution v(x) for this problem. Hint: The steady state solution must satisfy the pde and the boundary conditions.
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x =...
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
Partial Differential Equations
1. (20 points) Consider the problem u" (x)+ u(x) (0.1) f(x) 1 (0.2) u'(0)u(0) (u'(0) + u(l)) with f(x) (10 points) a) Is the solution unique? Justify your answer (10 points) b) Does a solution exist. or is there a condition that f (x) must satisfy for existence? Justify your answer given function a
1. (20 points) Consider the problem u" (x)+ u(x) (0.1) f(x) 1 (0.2) u'(0)u(0) (u'(0) + u(l)) with f(x) (10 points) a) Is...
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...
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...
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) =...
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G (0,y) = 6,(1, y) = 0 does not exist, but a modified Green's function Ĝ(x,y) satisfying a2G 22 = (x - y) -1, G.(0,y)=G.(1,y) = 0 does. How would you use G to solve problem (1) when f satisfies the condition that you found for a solution to exist? Hint: is f(x) = f(u) (8(x - y) - 1) dy?
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t, x) € (0, +00), u(0,x) = 1, u(t,0) = e. For the unique solution u(x, t) find the following limit as a function of t: lim u(x, t).
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...
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1,t)-0, t > 0 and the initial condition u(r,0) f(a) We found the solution to be of the form where (2n-1)n 1,2,3,. TL 20 Now consider the heat conduction problem with the boundary conditions u(0, t) 1,u(T, t)0, t>0 and the initial condition ur,0) 0. Find u(r,t). Hint: First you must find the steady state.
3. In class we discussed the heat conduction problem...
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
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