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,, = u(z, y). Is the problem tlay-0 for 0 < x, y < 1,on the unit square Ω = [0, 1] × [0,1] , where the value of u is prescribed on the boundary 830 of the square, a well-posed problem? Discuss...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0)...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I]...
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0 a [0,2
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0...
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R....
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
6] Find the solution u(x, y) of the following boundary value problem. u(x,0) = i, u(x, 2) = 0, a(...
6] Find the solution u(x, y) of the following boundary value problem. u(x,0) = i, u(x, 2) = 0, a(0, y) = 0, u(3, y) = 3, 0 < x < 3 0 < y < 2.
6] Find the solution u(x, y) of the following boundary value problem. u(x,0) = i, u(x, 2) = 0, a(0, y) = 0, u(3, y) = 3, 0
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0,...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou...
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y)...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), usi...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
Problem 11. 12 marks] Consider the following two-point boundary value problem: y" + y' + ßy = 0, y(0) = 0, y(1) = 0, where ß is a real nurnber. we know the problern has a trivial solution...
Problem 11. 12 marks] Consider the following two-point boundary value problem: y" + y' + ßy = 0, y(0) = 0, y(1) = 0, where ß is a real nurnber. we know the problern has a trivial solution, i.e. y(x) = 0, Discuss how the value of B influences the nontrivial solutions of the boundary value problem, and get the nontrivial solutions (Find all the real eigenvalues β and the corresponding eigenfunctions.)
Problem 11. 12 marks] Consider the following two-point...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0 a [0,2
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0...
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
6] Find the solution u(x, y) of the following boundary value problem. u(x,0) = i, u(x, 2) = 0, a(0, y) = 0, u(3, y) = 3, 0 < x < 3 0 < y < 2.
6] Find the solution u(x, y) of the following boundary value problem. u(x,0) = i, u(x, 2) = 0, a(0, y) = 0, u(3, y) = 3, 0
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
Problem 11. 12 marks] Consider the following two-point boundary value problem: y" + y' + ßy = 0, y(0) = 0, y(1) = 0, where ß is a real nurnber. we know the problern has a trivial solution, i.e. y(x) = 0, Discuss how the value of B influences the nontrivial solutions of the boundary value problem, and get the nontrivial solutions (Find all the real eigenvalues β and the corresponding eigenfunctions.)
Problem 11. 12 marks] Consider the following two-point...
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