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Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ,...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(...
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(r-2,0)-sin(20), 0 < θ < π. u(r Please also draw the sketch associated with this problem. You may assume that An -n2, Hn(s)sin(ns), n 1,2,3,. are the eigenpairs for the eigenvalue problem H(0) 0, H(T)0.
30] Find th e solution of the following boundary value problem. 1
#6 6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0 < θ < 2π BC 6. What is the solution to the following interior Dirichlet problem wi...
#6
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0 < θ < 2π BC
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R. 9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u...
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R.
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R.
The general solution y(t, p. 6) to the wave equation on a disc of radius R...
The general solution y(t, p. 6) to the wave equation on a disc of radius R with boundary condition v(t, R, 1) = 0 is given by vlt,0,0) = EE - ( ) [cos (ES) (Am.cos(nb) + Bu sin(no)) + (ME) (C..cos(no) + Dm (no) n=0 s=0 sin sin(ne)) 728 where Jn (2) is a Bessel function and x is the s'th root of In(x). (i) Derive the expressions for y and Oy/at at t = 0. (ii) Find all...
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) =...
Consider the differential equation y(t) + 69(r) + 5y( Q3. t)u(t), where y(0) (0)0 and iu(t) is a ...
Matlab code for the following problems.
Consider the differential equation y(t) + 69(r) + 5y( Q3. t)u(t), where y(0) (0)0 and iu(t) is a unit step. Deter- mine the solution y(t) analytically and verify by co-plotting the analytic solution and the step response obtained with the step function. Consider the mechanical system depicted in Figure 4. The input is given by f(t), and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot...
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant,...
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
where M=7 322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0...
where M=7
322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0 t<-M/2 M <t< - 5) Show that the Fourier transform of function f(t) sin 7 s (10 marks) au 6) Show that u = ln(x2 + xy + y2) satisfies the partial differential equation x x ди +y 2. (7 marks) au 7) Solve the partial differential equation = e-cos(x) where at du x = 0, at =tet ax at and t = 0,...
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0<r< R, t 0, u(R,t) 0, t>0, u(r,0) 0, 0sr <R. 10. Use Duhamel's principle to find a bounded soluti...
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0<r< R, t 0, u(R,t) 0, t>0, u(r,0) 0, 0sr <R.
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(r-2,0)-sin(20), 0 < θ < π. u(r Please also draw the sketch associated with this problem. You may assume that An -n2, Hn(s)sin(ns), n 1,2,3,. are the eigenpairs for the eigenvalue problem H(0) 0, H(T)0.
30] Find th e solution of the following boundary value problem. 1
#6
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0 < θ < 2π BC
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R.
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R.
The general solution y(t, p. 6) to the wave equation on a disc of radius R with boundary condition v(t, R, 1) = 0 is given by vlt,0,0) = EE - ( ) [cos (ES) (Am.cos(nb) + Bu sin(no)) + (ME) (C..cos(no) + Dm (no) n=0 s=0 sin sin(ne)) 728 where Jn (2) is a Bessel function and x is the s'th root of In(x). (i) Derive the expressions for y and Oy/at at t = 0. (ii) Find all...
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) =...
Matlab code for the following problems.
Consider the differential equation y(t) + 69(r) + 5y( Q3. t)u(t), where y(0) (0)0 and iu(t) is a unit step. Deter- mine the solution y(t) analytically and verify by co-plotting the analytic solution and the step response obtained with the step function. Consider the mechanical system depicted in Figure 4. The input is given by f(t), and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot...
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
where M=7
322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0 t<-M/2 M <t< - 5) Show that the Fourier transform of function f(t) sin 7 s (10 marks) au 6) Show that u = ln(x2 + xy + y2) satisfies the partial differential equation x x ди +y 2. (7 marks) au 7) Solve the partial differential equation = e-cos(x) where at du x = 0, at =tet ax at and t = 0,...
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0<r< R, t 0, u(R,t) 0, t>0, u(r,0) 0, 0sr <R.
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0
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