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2. Show that the Dirichlet problem for the disc t(z,y): +y S R2), where f(0) is the boundary func...
Solve the Dirichlet problem in an infinite strip uxx + uyy=0 for x ϵ R and 0 <y <b , u(x,0)=f(x) , u(x,b)=g(x). (...
Solve the Dirichlet problem in an infinite strip
uxx + uyy=0
for x ϵ R and 0 <y <b ,
u(x,0)=f(x) ,
u(x,b)=g(x). (Hint: first
do the case f=0. The case g=0 reduces to this one
by the substitution y→ b-y , and the case general is
obtained by superposition)
4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
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...
Solve the following Dirichlet problem in the upper semi-plane 2 X 2 (c) Au=f(x1,x); ulon=s(x,) where...
Solve the following Dirichlet problem in the upper
semi-plane
2 X 2 (c) Au=f(x1,x); ulon=s(x,) where f(x1,xx)=sin x, e?*; s(x1)=cos X1 ·
2. The solution to the boundary value problem y' + way=0, y(0) =0, y(1) - y'(1)...
2. The solution to the boundary value problem y' + way=0, y(0) =0, y(1) - y'(1) = 0 is y(x) = an sin(Zral) T=1 where the an are Fourier coefficients and the Zn are zeros of tan(w) To compute the zeros we can solve the fixed point problem w= tan(w). (i) Draw a graph of y=w and y=tan(w) on the interval (-37, 37). (ii) How many zeros of f(w) =tan(w) - w do we expect for all w. (iii) As...
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...
2. Consider the initial-boundary value problem 0 0, t >0 Uz uェェ, Uz(0, t) _ u(0, t) = 0. a(z, 0...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find al...
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a)...
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a) Jls (cosz-s (b) fdl where y is the line segment from (2,-1,3) to (0, 1, 4) and f (x,y,z)-y+2 sin y) dA 3 marks 3 marks (c) Jc F dr where C is the unit circle centred at the origin, traversed once anticlockwise and F R2R2 is given by F(r,y)- (x2.x + y) 3 marks JJR eVEdA where R is the region enclosed by...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
3. If z = f(x,y), where x = r cos, y=r sin 0 show that 222...
3. If z = f(x,y), where x = r cos, y=r sin 0 show that 222 222 1 222 1.az + + +) ar2 ду? ar2 a02 rar
Solve the Dirichlet problem in an infinite strip
uxx + uyy=0
for x ϵ R and 0 <y <b ,
u(x,0)=f(x) ,
u(x,b)=g(x). (Hint: first
do the case f=0. The case g=0 reduces to this one
by the substitution y→ b-y , and the case general is
obtained by superposition)
4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
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...
Solve the following Dirichlet problem in the upper
semi-plane
2 X 2 (c) Au=f(x1,x); ulon=s(x,) where f(x1,xx)=sin x, e?*; s(x1)=cos X1 ·
2. The solution to the boundary value problem y' + way=0, y(0) =0, y(1) - y'(1) = 0 is y(x) = an sin(Zral) T=1 where the an are Fourier coefficients and the Zn are zeros of tan(w) To compute the zeros we can solve the fixed point problem w= tan(w). (i) Draw a graph of y=w and y=tan(w) on the interval (-37, 37). (ii) How many zeros of f(w) =tan(w) - w do we expect for all w. (iii) As...
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...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a) Jls (cosz-s (b) fdl where y is the line segment from (2,-1,3) to (0, 1, 4) and f (x,y,z)-y+2 sin y) dA 3 marks 3 marks (c) Jc F dr where C is the unit circle centred at the origin, traversed once anticlockwise and F R2R2 is given by F(r,y)- (x2.x + y) 3 marks JJR eVEdA where R is the region enclosed by...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
3. If z = f(x,y), where x = r cos, y=r sin 0 show that 222 222 1 222 1.az + + +) ar2 ду? ar2 a02 rar
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