Homework Answers
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have solved the first problem. the rest of the problems can be
solved exactly following the method by which I have solved the
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2. Use eigenfunction expansion to solve the following IBVP: u,(x, t) ="-(x,t) + (t-1)sin(m), 0 0 ...
2. Use eigenfunction expansion to solve the following IBVP: please answer v) (fifth one) 2. Use...
2. Use eigenfunction expansion to solve the following IBVP:
please answer v) (fifth one)
2. Use eigenfunction expansion to solve the following IBVP u,(x.t)-u(x.t)+(t-1)sin(a) 0<x<1 t>0 u(0,t)0, u(l,r) 0, t>0 u,(x,t)(x) cos(z), 0 <x<1 t>0 n(x,0) = 2-cos(32t) 0 < x < 1 u(0,0, u(l,t) 0, t>0 n(x,0) = 1 u,(x,0) = 0 0 < x < 1 IV Hm(x,y)+u" (x,y)--r', 0<x<1 0<y<2 u(x,0) = 0, u(x2) =-x 0 < x < 1 v) 7" 11(0,8) bounded , -π<θ<π
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u...
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t) + 2t sin (2na:) , 0 < x < 1, 0, u(1,t)=0, t > 0, sin(2π.r)-5 sin (4π.r) , 0 < x < 1. t > 0, = = =
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t)...
V) u(0,6) bounded , -π < θ < π
Use eigenfunction expansion to solve this IBVP.
v) u(0,6) bounded , -π < θ < π
v) u(0,6) bounded , -π
Solving IBVP using Eigenfunction Expansion
Solve the following IBVP by eigenfunction expansion.$$ u_{t t}=u_{x x}+1+t \cos (\pi x), \quad 0<x<1, quad="" t="">0 $$$$ u_{x}(0, t)=0 \quad \text { and } \quad u_{x}(1, t)=0, \quad t>0 $$$$ u(x, 0)=2 \quad \text { and } \quad u_{t}(x, 0)=-2 \cos (2 \pi x), \quad 0<x<1 $$
(3 points) Use eigenvalues and eigenfunction expansion expansion to solve the Dirichlet problem Δ...
(3 points) Use eigenvalues and eigenfunction expansion expansion to solve the Dirichlet problem Δυ(x,y)-0 on the rectangle {(x, y):0
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t)...
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solu...
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
3. Use separation of variables to compute the first five terms of the series solution of the IBVP...
3. Use separation of variables to compute the first five terms of the series solution of the IBVP: urr (r,0) + r-rur (r, θ u (1,0, t) 0, u (r, θ, t) , ur(r, θ, t) bounded as r-+0+,-π < θ < π, t > 0, u (r,0,0) = r sin θ, ut (r.0, 0) = 0, o < r < 1, -π < θ < π. Hint: Follow the example from Lecture 19 and use the fact that with...
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:
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b)...
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
2. Use eigenfunction expansion to solve the following IBVP:
please answer v) (fifth one)
2. Use eigenfunction expansion to solve the following IBVP u,(x.t)-u(x.t)+(t-1)sin(a) 0<x<1 t>0 u(0,t)0, u(l,r) 0, t>0 u,(x,t)(x) cos(z), 0 <x<1 t>0 n(x,0) = 2-cos(32t) 0 < x < 1 u(0,0, u(l,t) 0, t>0 n(x,0) = 1 u,(x,0) = 0 0 < x < 1 IV Hm(x,y)+u" (x,y)--r', 0<x<1 0<y<2 u(x,0) = 0, u(x2) =-x 0 < x < 1 v) 7" 11(0,8) bounded , -π<θ<π
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t) + 2t sin (2na:) , 0 < x < 1, 0, u(1,t)=0, t > 0, sin(2π.r)-5 sin (4π.r) , 0 < x < 1. t > 0, = = =
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t)...
Use eigenfunction expansion to solve this IBVP.
v) u(0,6) bounded , -π < θ < π
v) u(0,6) bounded , -π
(3 points) Use eigenvalues and eigenfunction expansion expansion to solve the Dirichlet problem Δυ(x,y)-0 on the rectangle {(x, y):0
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
3. Use separation of variables to compute the first five terms of the series solution of the IBVP: urr (r,0) + r-rur (r, θ u (1,0, t) 0, u (r, θ, t) , ur(r, θ, t) bounded as r-+0+,-π < θ < π, t > 0, u (r,0,0) = r sin θ, ut (r.0, 0) = 0, o < r < 1, -π < θ < π. Hint: Follow the example from Lecture 19 and use the fact that with...
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:
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
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