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7. (a) Find the harmonic function in the semi-infinite strip {0 < x < π, 0sy oo) that satisfies t...
7. Find a conformal map of the infinite strip 0 < y < π onto the semi-infinite strip 7. Find a conformal map of the infinite strip 0
7. Find a conformal map of the infinite strip 0 < y < π onto the semi-infinite strip
7. Find a conformal map of the infinite strip 0
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0 0, and the addit...
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies...
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
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.
25 points) Find the solution of the Laplace equation ur the domain 0-x-π and 0-y-T. The...
25 points) Find the solution of the Laplace equation ur the domain 0-x-π and 0-y-T. The boundary condition at the left boundary is given by u(0, y)-sin(y/2). The boundary conditions at al other boundaries are zero. Express the solution as an infinite series = 0, over
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...
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...
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)...
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann cond...
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that with solves the equation ou- u=0, 0<z<a, t > to > 0.
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that...
Question 57.5 from Fourier series and Boundary value problems Brown and Churchill
Question 57.5 from Fourier series and Boundary value problems
Brown and Churchill
S Find the bounded harmonic function ux, y) in the semi-infinite strip0< 1,y that satisfies the conditions 2 Answer: u(x, y)E sinh ax cos α f(s) cos as ds da.
S Find the bounded harmonic function ux, y) in the semi-infinite strip0
7. Find a conformal map of the infinite strip 0 < y < π onto the semi-infinite strip
7. Find a conformal map of the infinite strip 0
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
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.
25 points) Find the solution of the Laplace equation ur the domain 0-x-π and 0-y-T. The boundary condition at the left boundary is given by u(0, y)-sin(y/2). The boundary conditions at al other boundaries are zero. Express the solution as an infinite series = 0, over
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...
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...
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that with solves the equation ou- u=0, 0<z<a, t > to > 0.
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that...
Question 57.5 from Fourier series and Boundary value problems
Brown and Churchill
S Find the bounded harmonic function ux, y) in the semi-infinite strip0< 1,y that satisfies the conditions 2 Answer: u(x, y)E sinh ax cos α f(s) cos as ds da.
S Find the bounded harmonic function ux, y) in the semi-infinite strip0
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