6.8. Verify that u(x, y)= A sin(27Tx) sin(27ty) solves Poisson's equation V2u-Ron W (0, 1) x...
ar,a7, that V2u V.Vu 6.4. Verify directly from the gradient operator that V uK +uyy-see Definition 6.5 Definition 6.5 (Two-Dimensional Heat or Diffusion Equation). Consider the open do- main (x, y) W. Using the continuity equation (1.4) the flux rule (6.13) yields DV u+R (6.14) where V2u V.Vu u +uyy is the linear Laplacian operator The boundary conditions come in the three types: conditions on u, conditions on flux, and mixed as we are familiar with from Chapter 4. The...
ar,a7, that V2u V.Vu 6.4. Verify directly from the gradient operator that V uK +uyy-see Definition 6.5 Definition 6.5 (Two-Dimensional Heat or Diffusion Equation). Consider the open do- main (x, y) W. Using the continuity equation (1.4) the flux rule (6.13) yields DV u+R (6.14) where V2u V.Vu u +uyy is the linear Laplacian operator The boundary conditions come in the three types: conditions on u, conditions on flux, and mixed as we are familiar with from Chapter 4. The...
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:
= r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2
= r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2
MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING, Consider Vf(x, y, z) in terms of a new coordinate system, x= x(u, v, w), y=y(u, v, w), z=z(u, v, w). Let r(s) = x(s) i+y(s) + z(s) k be the position vector defining some continuous path as a function of the arc length. Similarly for the other partial derivatives in v and w. For spherical coordinates the following must also be true for any points, x = Rsin o cose,...
Please ONLY work parts a, d, e
4.4. Consider the standard equilibrium heat equation with a source u (D 1, cp on x E [0, L]. Given the following parameter values and boundary condi tions, determine (1) the equilibrium solution and draw a graph, and (2) compute the flux and indicate on the solution graph the direction and magnitude of the flux. Alternatively specify if the solution does not exist, or detail how it is not fully determined 0 uxxR(x)...
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) =...
please solve 17 for me thanks~~ :) !
temperature f(x) °C, where 5. f(x) = sin 0.1 x 6 f(x) = 4 - 08 |x - 5 7. fix) =x(10 - x) 8 Arbitrarytemperatures at ends. If the ends x = 0 and x= Lof the bar in the text are kept at constant 20. CAS PROJECT. Isotherms. Fim solutions (tempe rature s) in the squa with a 2 satisfying the followin tions. Graph isotherms. (a) u80 sin Tx on...
A string vibrates according to the equation 1. Y(x,t) = 10.0 cm * sin((7.00cm-')x)*cos(2.00 rad/s)t (THIS IS A STANDING WAVE. What is the angular wave number? What is the wavelength? What is the speed of the 2 waves making the standing wave? What is the amplitude of the 2 waves making the standing wave? What is the distance between the nodes? What is the angular speed of the 2 waves making the standing wave? What is the period of the...
Can you please help me answer Task 2.b?
Please show all work.
fs=44100; no_pts=8192;
t=([0:no_pts-1]')/fs;
y1=sin(2*pi*1000*t);
figure;
plot(t,y1);
xlabel('t (second)')
ylabel('y(t)')
axis([0,.004,-1.2,1.2]) % constrain axis so you can actually see
the wave
sound(y1,fs); % play sound using windows driver.
%%
% Check the frequency domain signal. fr is the frequency vector and
f1 is the magnitude of F{y1}.
fr=([0:no_pts-1]')/no_pts*fs; %in Hz
fr=fr(1:no_pts/2); % single-sided spectrum
f1=abs(fft(y1)); % compute fft
f1=f1(1:no_pts/2)/fs;
%%
% F is the continuous time Fourier. (See derivation...